# Option Volatility Investment Strategy: The Combination of Neural Network and Classical Volatility Prediction Model

Authors: Yuanyang Teng; Yicun Li; Xiaobo Wu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This study focuses on the volatility prediction and option volatility investment. By investigating the traditional Volatility Prediction Model and machine learning algorithms, this study tries to merge these two aspects together. This work setup a bridge of previous financial studies and machine learning studies by proposing an algorithm integrating neural network and three traditional volatility models, called “Quantile based neural network and model integration combination algorithm.” The algorithm effectively lowers the volatility prediction error (measured by root of mean square error, shorted for RMSE: 0.319724) and beat the Wavenet (RMSE: 0.44) which is the benchmark and surpasses integrated model (RMSE: 0.348346) in test set. In terms of option investment strategy, this paper constructs a CSI 300 index option portfolio which hedges the underlying asset price risk and exposes the volatility risk. Then propose the “Option strategy of volatility prediction with dynamic thresholds.” With the new algorithm above, the strategy improves the return-risk ratio in test set (measured by Sharpe ratio: 1.99–2.07).


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## Body

## 1. Introduction

The investment of linear financial instruments such as stocks and futures mainly focuses on the price prediction of financial assets. With the development of financial derivatives and the birth of more and more complex nonlinear financial instruments such as VIX Index, option and volatility swap, current capital market provides many tools for volatility investment. This paper tries to investigate volatility investment.There are a lot of previous foreign works about the volatility investment problem. One approach is to search for new indicators. Audrino and Knaus [1] focused on the impact of sentiment indicators and investor concern variables on the realized volatility of the stock market. After controlling manymacroeconomic and financial prediction variables, researchers use a very innovative emotion classification system to mine whether emotion indicators contain additional information of realized volatility. Then, using the penalty constrained regression model, they determined the most relevant variables for volatility prediction. Their results show that although investors pay attention to concern variables and sentiment indicators to improve volatility prediction, they are not statistically significant. Another approach is to merge different features by machine learning. Vrontos et al. [2] comprehensively investigated 32 different characteristics and designed different investment strategies of VIX Index funds with a variety of machine learning algorithms. A variety of machine learning algorithms make binary prediction, which is used to predict the rise and fall direction of volatility. When predicting the decline of volatility, short the VIX Index, and when predicting the rise of volatility, long the VIX Index, and record the evaluation indexes such as annual return, max drawdown and Sortino ratio. Then, compared with the traditional financial economics prediction model, it is found that machine learning algorithm has more advantages in volatility prediction. Moreover, some researchers are interested in the strategy designing. Hirsa et al. [3] used a variety of machine learning algorithms to copy VIX Index and VIX futures with S&P index options, totake advantage of the potential arbitrage opportunities between VIX Index and its underlying assets.With the development of China’s derivatives market, a number of options have been listed in recent years, accompanied by domestic researchers’ research in the fields of options and volatility. Because of the immature market, Yang and Lijuan [4] conducted an empirical study on China's iVIX index using the HAR model showing that China's iVIX index has a very obvious “Monday effect,” according to which we can build an investment strategy with good performance. Xi [5] focused on the stock market with stationary process and developed an option arbitrage strategy based on the assumption of stationary process. The strategy believes that when the annualized yield of the underlying asset is higher than the risk-free interest rate, the premium of European put options and call options should be lowered and higher than the pricing under the BS model framework respectively, which forms an arbitrage opportunity. Lei [6] established a long-term volatility prediction model based on investor sentiment. The results show that volatility prediction is significantly affected by investor sentiment. If investor sentiment is included in volatility prediction, the prediction performance of implied volatility is higher than that of realized volatility. Yang Bei [7] studied the investment performance of different option strategies on 50ETF options when the trading volume of 50ETF options shrank under the background of low volatility, and put forward some suggestions on risk management.Besides the growing interests of the investment of options and other financial derivatives, time series prediction problem based on machine learning is also very popular. Singh et al. [8] study a parallel SVM method based on the Cascade SVM model to forecast a forest fire. This method provides an ensemble of SVM, splits large data sets into smaller data sets and improve the performance of prediction. Ribeiro et al. [9] does research in epidemiological time series forecasting using a series combination of machine learning methods. He uses the ensemble empirical mode decomposition (EEMD) to decompose data into a series of components. Then, each component is used as the input of five different forecasting models, and five different forecasting models give different forecasting result which need to be weighted integrated, Non-Dominated Sorting Genetic Algorithm (version II) and the Technique for Order of Preference by similarity to Ideal Solution (TOPSIS) is applied to choose the adequate set of weights to get the final optimized prediction. Vos et al. [10] put machine learning into practice in the problem of fault detection. He combines LSTM and SVM by two different architectures to detect faults of gear faults and bearing failures. Ribeiro et al. [11] and Ribeiro et al. [12] do research about load forecasting which is related with financial return and electrical systems planning. He conducts a series of methods such as bootstrapping, cross-validation like, inputs decimation, constructive selection, stacked generalization algorithms for ensemble aggregation of the Wavenet learners. Then, he compares this method with existing models from GEFCom competition using hourly load values. Zouhri et al. [13] considers a key problem about feature uncertainty in machine learning. As known, the performance of machine learning is essentially affected by data sets, that is, features. The research goes into data sensitivity analysis is very valuable. Zouhri et al. [13] propose SVM based on Sobol’ sensitivity analysis which calculates a Sobol total-effect indices to evaluate how SVM accuracy is sensitivity to feature uncertainty.To sum up, there are two important applied research directions for the investment of options and other financial derivatives and one methodology research direction about machine learning for time series prediction: firstly, collect, test and explore new effective characteristics. For example, Audrino et al. [14] focused on the impact of sentiment indicators on volatility prediction. Vrontos et al. [2] comprehensively investigated 32 different characteristics, and then combined with machine learning algorithm to invest in VIX Index Fund. Secondly, design option strategy. Yanjun and Lijuan [4] used the HAR model and studied China’s iVIX index to establish an investment strategy using the “Monday effect.” Yang Bei [7] studied the investment strategy and risk management strategy of 50ETF option under the background of low volatility. Thirdly, improve machine learning algorithms. Singh et al. [8] proposes a parallel SVM method, Zouhri et al. [13] focuses on feature uncertainty measurement, Ribeiro et al. [9] combines a series of algorithms to optimize predictions, Ribeiro et al. [11] and [12] use an ensemble of the Wavenet learners to forecast load values.The limitations of the previous studies are following points: (1) some researchers are still following traditional financial research and only focus on one point in their studies. Audrino et al. [14] focus on indicators, Yanjun and Lijuan [4] find one effect called “Monday effect.” This kind of research doing one or two points cannot construct a systematical story. (2) Some other researchers try to construct an investment strategy by financial properties or theoretical characteristics. This kind of researches doing the financial problems only in the financial field. For example, Xi [5] and Yang Bei [7]. It’s acceptablebut not novel. (3) Some researchers find the advantages of machine learning, Vrontos et al. [2] uses machine learning algorithms investigates 32 different characteristics and designed different investment strategies of VIX Index funds. This is something novel, but not deep enough. He does not improve the algorithms, and I do not think the contributions of classical financial models can be replaced by machine learning algorithms. (4) There are some studies go deep into the machine learning algorithms and improve the performance of algorithms (Singh et al. [8]; Ribeiro et al. [9]; Vos et al. [10]). However, they just treat the machine learning algorithms as the data mining tools, that is to say, they do not combine the professional knowledge of fields of problems with their algorithms.The paper try to fix the limitations above: (1) this paper sets up a generalized machine learning procedure for data mining. (2) This paper sets up a generalized framework to improve an investment strategy which is under developing. (3) Proposes a hybrid algorithm combining classical volatility models and machine learning algorithms, improve the machine learning algorithms by the financial professional knowledge. (4) Proposes a volatility investment strategy. Specifically, the contents are as follows:(1) Three feature selection algorithms are used to comprehensively investigate 30 financial and economic features. (2) A variety of machine learning algorithms are used to predict the realized volatility of CSI 300 index. (3) Propose a new hybrid algorithm called “quantile based neural network and model integration algorithm” to improve volatility prediction and take Wavenet (Ribeiro et al. [11] and [12]) as a benchmark, make comparison between “Quantile based neural network and model integration algorithm” and Wavenet learners. (4) Based on the prediction results, multiple groups of call put option portfolios and different investment strategies are constructed, Hedge the price risk of the underlying asset, expose the volatility risk and conduct volatility speculation. Finally, after many strategic improvements, the “option strategy of volatility prediction with dynamic thresholds” is constructed, and good returns are obtained.This paper consists of 5 sections. This section is introduction, which reviews the previous works, sums up the research approaches and proposes the novelty of this study. The second section is theoretical methodology, which introduces all the models and algorithms used in the next section. The third section is empirical research. This section is main part of this study and proposes the new algorithms and the new investment strategy. The fourth part is discussion, stating the disadvantages of this study. The final part is conclusion, sums up the work of this paper.


## 2. Methodology

### 2.1. Theory of Machine Learning

#### 2.1.1. Support Vector Machine (SVM)

SVM is a statistical pattern recognition algorithm, whose ultimate goal is to get the equation coefficients of classification hyperplane. The algorithm searches the suitable support vector, and then calculates the final classification hyperplane based on the support vector. This algorithm has a complete mathematical proof given by Li [15].Farther the sample point is from the classification surface, the more reliable the classification prediction results are. Therefore, to find such a classification surface, the distance between the nearest sample and the hyperplane in the training sample is maximized, that is, the distance between the two samples to the hyperplane arethe same. When such a classification hyperplane cannot be found in the original dimension space, we need to project the sample set into the high-dimensional space, and try to find a hyperplane with the above effect in the high-dimensional space.The goal of the algorithm is to maximize the minimum distance between all training sample points and the classification surface, that is,(1)maxw,bmini=1,…mγi,where m is the number of training samples. This paper considers the correct classification surface of all sample points, and maximizes the distance from the nearest point to the plane. γi is the distance between support vector and classification surface, w,b is parameter to determine classification surface.After substitutingγi=yiw/wTxi+b/w into the problem, the maximization of the objective function can be rewritten as the following optimization problem:(2)maxw,bγs.t.∀i∈1,2,…m,yiwwTxi+bw≥γ.Lagrange multiplier method is used, and construct Lagrange operator:(3)Lw,b,αi=12w2−∑i=1mαiyiwTxi+b−1.Among them, Lagrange multiplierαi≥0.At this point, the target function to optimize becomes:(4)minw,bmaxαiLw,b,αi.


#### 2.1.2. Random Forests

The Random Forests is a classical machine learning algorithm, which has been reintroduced by Li [15]. It uses a bagging algorithm to integrate decision trees, which can be used for both classification and regression. The algorithm trains multiple decision trees in an independent way. When the training results are obtained, the voting principle is used for classification. In all the decision tree classification results, which type appears most, for the problem of regression, the final regression value of the Random Forests is equal to the average of the regression values of each decision tree. In other words, Random Forests is equivalent to paralleling each decision tree, and each decision trees is independent of each other.The method to decidea decision tree is Gini Index. The algorithm of Gini Index is as follows:(5)Ginip=∑k=1Kpk1−pk=1−∑k=1Kpk2,where pk is the probability that the selected sample belongs to k category, then the probability that the sample is wrongly divided is 1−pk.For a given sample setD, the Gini Index is:(6)GiniD=1−∑k=1KCkD2,Ck is the sample of class k in D, and K is the number of classes.If the sample setD is divided into D1 and D2 according to whether the feature A takes a possible value or not, that is:(7)D1=x,y∈D|Ax=a,D2=D−D1.Then under the condition of feature A, the Gini Index of setD is defined as:(8)GiniD,A=D1DGiniD1+D2DGiniD2.Gini IndexGiniD represents the uncertainty of set D, and Gini Index GiniD,A represents the uncertainty of set D divided by A = a. The larger the Gini index, the greater the uncertainty of the sample set. This is the method to generate a tree.After getting trees, how to assemble trees to a forest? Algorithm flow of Bagging method:(1)
Forj = 1: Nt, Nt is the total number of decision trees(1.1
) Generate a bootstrap sampleSj.(1.2
) Training a decision treeCj on Sj(2.1)
For each regression model, set a threshold, the samples whose regression error is greater than the threshold are recorded as - 1, and those whose regression error is less than the threshold are recorded as 1, so as to obtain the recognition vector of each model, and the length of the vector is the sample size.(9)λ=quantileyi−Cjxiyii=1N,0.75,Cj is the j-th model. The 0.75 quantile of regression error under each model is the threshold, which can be adjusted. For the recognition vector of each model, if the sample error is greater than the threshold, it is marked as - 1, otherwise it is 1.(2.2)
calculate the overall recognition vector(10)Cens=1J∑j=1JCj,Cens is the overall recognition vector, Cj is the recognition vector of the j-th model, J is the number of classifiers.(2.3)
Calculate the reference vector(11)Cref=o−dotCens,o÷normCens2×Cens,Here,o is the unit vector with all elements equal, that is, the vector with all elements equal to 1 divided by its module, and the Cref can be obtained from the above formula.(3)
There are three models for aggregatingNt. Calculate cosine value of Cj and Cref. Reserve Cj which the cosine value is greater than 0, because it means the angle is less than 90 degrees, and other unsuitable decision trees are deleted. The specific threshold of cosine can be adjusted through experiments.


#### 2.1.3. Forward Neural Network

Neural network is a famous machine learning algorithm which has a general application in many fields. Forward neural network is a kind of Neural Network which consists of multilayer perceptron introduced by Li [15]. They are a kind of multilayer feedforward networks trained by error back propagation. The basic idea is the gradient descent method. The gradient search technique is used to minimize the mean square error between the actual output value and the expected output value.The hidden layer node of feedforward neural network uses the inner product of input pattern and weight vector as the independent variable of activation function, and the activation function usually uses sigmoid function. Each node has the same influence on the output of the feedforward neural network, so the feedforward neural network is the global approximation of nonlinear mapping.The forward neural network consists of the following seven steps:(1)
Network initialization Feedforward neural network is a kind of hierarchical network. In order to avoid the influence of extreme initial value on network training, it usually randomizes the weights of each layer.(2)
Input layer Accept the sample data input and calculate the output through the input layer.(3)
Hidden layer Accept the output of the input layer, and calculate the output of the hidden layer through the hidden layer.(4)
Output layer Accept the output of the hidden layer, and calculate the output of the output layer through the output layer, that is, the final prediction label.(5)
Calculation of error The error between the predicted result and the true label is calculated.(6)
Update of weight The new network weights are calculated according to the error correction formula.(7)
Update of offset According to the error correction formula, the new network weight bias term is calculated.(8)
Judge whether the algorithm iteration ends or not According to the number of iterations or error convergence, whether to end the network training is judged.


### 2.2. Classical Volatility Prediction Model

#### 2.2.1. Fractional Brownian Motion Model (fBm Model)

fBm model assumes that the logarithmic increment of volatility follows fractional Brownian motion [16], which can be expressed by (12)logσt+1−logσt=νWt+□H−WtH,where WH is a fractional Brownian motion with Hurst index H, where Hurst index H can be used to measure roughness of σt, ν Is a positive number.However, measured by equation (12) σt is non-stationary, which supposes logσt follows the fractional O-U process, and then modeling.Stationary fractional O-U processXt can be defined by (13)dXt=νdWtH−αXt−mdt,where m is a real number, ν and α Is a positive number.According to equation (13), Xt can be obtained by (14)Xt=ν∫−∞te−αt−sdWtH+m.The integral here is a Riemann–Stieltjes Integral [17]. In this way, the volatility integral in the time interval [0, t] is obtained:(15)σt=eXt,t∈0,T,Xt in equation (15) obeys equation (14), and m is a real number, ν and α are positive numbers, H < 0.5.


#### 2.2.2. ARMA (p q) Model

Autoregressive moving average model is abbreviated as ARMA model. The model consists of two parts: autoregressive model (AR model) and moving average model (MA model). The AR model was proposed by Yule [18], and then Slutsky [19] supplemented the AR model and proposed the MA model. Subsequently, in 1938, Wold [20] formally proposed the ARMA model by combining the AR model and MA model.Autoregressive model is a model that assumes that the variable to be predicted is related to its own lag term and not related to other variables, then the p-order AR model can be expressed as follows:(16)RVt=β0+β1RVt−1+β2RVt−2+…+βpRVt−p+εt,whereεtis a sequence ofindependent identically distributedrandom variables,andEεt=0.The moving average model assumes that the time series to be predicted is a linear combination of pastq-order white noise, which can be expressed as(17)RVt=α0+α1εt−1+α2εt−2+…+αqεt−q,where εt is the white noise sequence.Combining the abovep-order AR model and q-order MA model can form ARMA (p, q) model, which can be expressed as(18)RVt=β0+β1RVt−1+β2RVt−2+…+βpRVt−p+α0+α1εt−1+α2εt−2+…+αqεt−q.


#### 2.2.3. Brownian Semistationary with Gamma Kernel

Brownian semistationary with Gamma kernel is a volatility prediction model based on kernel function [21]. It defines Xt following the random process:(19)Xt=∫−∞tt−sαe−λt−svsdWs,t>0.In equation (20), Xt>0, α∈−0.5,0.5, λ∈0,∞, W is a standard Brownian motion, and v is an adapted, covariance-stationary volatility process.Define the autocorrelation function (ACF):(20)ρh,α,λ=CorrXt+h,Xt=2−α+0.5Гα+0.5λhα+0,5Kα+0.5λh,h∈R,where α It can be estimated by(21)α^p≔log2COFp,np−0.5,where COFp,n=∑k=5nXk−2Xk−2+Xk−4p/∑k=3nXk−2Xk−1+Xk−2p and p > 0 is a power parameter to be chosen by the researcher.Next, estimate the optimalλ^ value, equation (22):(22)λ^=argminλ∑i=1Nρ^h−ρh,α,λ2,where the empirical estimate of ρ^h can be replaced by linear correlation coefficient. N is the number of linear correlation coefficients calculated on different lag terms.Then, calculateГ22 and Г12 by equations (23) and (24), respectively, and the matrix form of (25):(23)Г22≔1CorrXm,Xm−1CorrXm−1,Xm1⋯CorrXm,X0⋮CorrXm−1,Xm⋮⋮CorrX0,XmCorrX0,Xm−1⋱⋮⋯1,(24)Г12≔CorrXm+1,XmCorrXm+1,Xm−1⋯CorrXm+1,X0,(25)a=Xm,Xm−1,⋯,X0,So, combinea, Г22, and Г12. The expected value of the one-step predicted value can be calculated by the expression of μ, as follows:(26)μ=Г12Г22−1a.At the same time, the estimated value of variance can also be obtained:(27)ξ2=VarXt1−Г12Г22−1Г21,where Г21=Г12T.


## 2.1. Theory of Machine Learning

### 2.1.1. Support Vector Machine (SVM)

SVM is a statistical pattern recognition algorithm, whose ultimate goal is to get the equation coefficients of classification hyperplane. The algorithm searches the suitable support vector, and then calculates the final classification hyperplane based on the support vector. This algorithm has a complete mathematical proof given by Li [15].Farther the sample point is from the classification surface, the more reliable the classification prediction results are. Therefore, to find such a classification surface, the distance between the nearest sample and the hyperplane in the training sample is maximized, that is, the distance between the two samples to the hyperplane arethe same. When such a classification hyperplane cannot be found in the original dimension space, we need to project the sample set into the high-dimensional space, and try to find a hyperplane with the above effect in the high-dimensional space.The goal of the algorithm is to maximize the minimum distance between all training sample points and the classification surface, that is,(1)maxw,bmini=1,…mγi,where m is the number of training samples. This paper considers the correct classification surface of all sample points, and maximizes the distance from the nearest point to the plane. γi is the distance between support vector and classification surface, w,b is parameter to determine classification surface.After substitutingγi=yiw/wTxi+b/w into the problem, the maximization of the objective function can be rewritten as the following optimization problem:(2)maxw,bγs.t.∀i∈1,2,…m,yiwwTxi+bw≥γ.Lagrange multiplier method is used, and construct Lagrange operator:(3)Lw,b,αi=12w2−∑i=1mαiyiwTxi+b−1.Among them, Lagrange multiplierαi≥0.At this point, the target function to optimize becomes:(4)minw,bmaxαiLw,b,αi.


### 2.1.2. Random Forests

The Random Forests is a classical machine learning algorithm, which has been reintroduced by Li [15]. It uses a bagging algorithm to integrate decision trees, which can be used for both classification and regression. The algorithm trains multiple decision trees in an independent way. When the training results are obtained, the voting principle is used for classification. In all the decision tree classification results, which type appears most, for the problem of regression, the final regression value of the Random Forests is equal to the average of the regression values of each decision tree. In other words, Random Forests is equivalent to paralleling each decision tree, and each decision trees is independent of each other.The method to decidea decision tree is Gini Index. The algorithm of Gini Index is as follows:(5)Ginip=∑k=1Kpk1−pk=1−∑k=1Kpk2,where pk is the probability that the selected sample belongs to k category, then the probability that the sample is wrongly divided is 1−pk.For a given sample setD, the Gini Index is:(6)GiniD=1−∑k=1KCkD2,Ck is the sample of class k in D, and K is the number of classes.If the sample setD is divided into D1 and D2 according to whether the feature A takes a possible value or not, that is:(7)D1=x,y∈D|Ax=a,D2=D−D1.Then under the condition of feature A, the Gini Index of setD is defined as:(8)GiniD,A=D1DGiniD1+D2DGiniD2.Gini IndexGiniD represents the uncertainty of set D, and Gini Index GiniD,A represents the uncertainty of set D divided by A = a. The larger the Gini index, the greater the uncertainty of the sample set. This is the method to generate a tree.After getting trees, how to assemble trees to a forest? Algorithm flow of Bagging method:(1)
Forj = 1: Nt, Nt is the total number of decision trees(1.1
) Generate a bootstrap sampleSj.(1.2
) Training a decision treeCj on Sj(2.1)
For each regression model, set a threshold, the samples whose regression error is greater than the threshold are recorded as - 1, and those whose regression error is less than the threshold are recorded as 1, so as to obtain the recognition vector of each model, and the length of the vector is the sample size.(9)λ=quantileyi−Cjxiyii=1N,0.75,Cj is the j-th model. The 0.75 quantile of regression error under each model is the threshold, which can be adjusted. For the recognition vector of each model, if the sample error is greater than the threshold, it is marked as - 1, otherwise it is 1.(2.2)
calculate the overall recognition vector(10)Cens=1J∑j=1JCj,Cens is the overall recognition vector, Cj is the recognition vector of the j-th model, J is the number of classifiers.(2.3)
Calculate the reference vector(11)Cref=o−dotCens,o÷normCens2×Cens,Here,o is the unit vector with all elements equal, that is, the vector with all elements equal to 1 divided by its module, and the Cref can be obtained from the above formula.(3)
There are three models for aggregatingNt. Calculate cosine value of Cj and Cref. Reserve Cj which the cosine value is greater than 0, because it means the angle is less than 90 degrees, and other unsuitable decision trees are deleted. The specific threshold of cosine can be adjusted through experiments.


### 2.1.3. Forward Neural Network

Neural network is a famous machine learning algorithm which has a general application in many fields. Forward neural network is a kind of Neural Network which consists of multilayer perceptron introduced by Li [15]. They are a kind of multilayer feedforward networks trained by error back propagation. The basic idea is the gradient descent method. The gradient search technique is used to minimize the mean square error between the actual output value and the expected output value.The hidden layer node of feedforward neural network uses the inner product of input pattern and weight vector as the independent variable of activation function, and the activation function usually uses sigmoid function. Each node has the same influence on the output of the feedforward neural network, so the feedforward neural network is the global approximation of nonlinear mapping.The forward neural network consists of the following seven steps:(1)
Network initialization Feedforward neural network is a kind of hierarchical network. In order to avoid the influence of extreme initial value on network training, it usually randomizes the weights of each layer.(2)
Input layer Accept the sample data input and calculate the output through the input layer.(3)
Hidden layer Accept the output of the input layer, and calculate the output of the hidden layer through the hidden layer.(4)
Output layer Accept the output of the hidden layer, and calculate the output of the output layer through the output layer, that is, the final prediction label.(5)
Calculation of error The error between the predicted result and the true label is calculated.(6)
Update of weight The new network weights are calculated according to the error correction formula.(7)
Update of offset According to the error correction formula, the new network weight bias term is calculated.(8)
Judge whether the algorithm iteration ends or not According to the number of iterations or error convergence, whether to end the network training is judged.


## 2.1.1. Support Vector Machine (SVM)

SVM is a statistical pattern recognition algorithm, whose ultimate goal is to get the equation coefficients of classification hyperplane. The algorithm searches the suitable support vector, and then calculates the final classification hyperplane based on the support vector. This algorithm has a complete mathematical proof given by Li [15].Farther the sample point is from the classification surface, the more reliable the classification prediction results are. Therefore, to find such a classification surface, the distance between the nearest sample and the hyperplane in the training sample is maximized, that is, the distance between the two samples to the hyperplane arethe same. When such a classification hyperplane cannot be found in the original dimension space, we need to project the sample set into the high-dimensional space, and try to find a hyperplane with the above effect in the high-dimensional space.The goal of the algorithm is to maximize the minimum distance between all training sample points and the classification surface, that is,(1)maxw,bmini=1,…mγi,where m is the number of training samples. This paper considers the correct classification surface of all sample points, and maximizes the distance from the nearest point to the plane. γi is the distance between support vector and classification surface, w,b is parameter to determine classification surface.After substitutingγi=yiw/wTxi+b/w into the problem, the maximization of the objective function can be rewritten as the following optimization problem:(2)maxw,bγs.t.∀i∈1,2,…m,yiwwTxi+bw≥γ.Lagrange multiplier method is used, and construct Lagrange operator:(3)Lw,b,αi=12w2−∑i=1mαiyiwTxi+b−1.Among them, Lagrange multiplierαi≥0.At this point, the target function to optimize becomes:(4)minw,bmaxαiLw,b,αi.


## 2.1.2. Random Forests

The Random Forests is a classical machine learning algorithm, which has been reintroduced by Li [15]. It uses a bagging algorithm to integrate decision trees, which can be used for both classification and regression. The algorithm trains multiple decision trees in an independent way. When the training results are obtained, the voting principle is used for classification. In all the decision tree classification results, which type appears most, for the problem of regression, the final regression value of the Random Forests is equal to the average of the regression values of each decision tree. In other words, Random Forests is equivalent to paralleling each decision tree, and each decision trees is independent of each other.The method to decidea decision tree is Gini Index. The algorithm of Gini Index is as follows:(5)Ginip=∑k=1Kpk1−pk=1−∑k=1Kpk2,where pk is the probability that the selected sample belongs to k category, then the probability that the sample is wrongly divided is 1−pk.For a given sample setD, the Gini Index is:(6)GiniD=1−∑k=1KCkD2,Ck is the sample of class k in D, and K is the number of classes.If the sample setD is divided into D1 and D2 according to whether the feature A takes a possible value or not, that is:(7)D1=x,y∈D|Ax=a,D2=D−D1.Then under the condition of feature A, the Gini Index of setD is defined as:(8)GiniD,A=D1DGiniD1+D2DGiniD2.Gini IndexGiniD represents the uncertainty of set D, and Gini Index GiniD,A represents the uncertainty of set D divided by A = a. The larger the Gini index, the greater the uncertainty of the sample set. This is the method to generate a tree.After getting trees, how to assemble trees to a forest? Algorithm flow of Bagging method:(1)
Forj = 1: Nt, Nt is the total number of decision trees(1.1
) Generate a bootstrap sampleSj.(1.2
) Training a decision treeCj on Sj(2.1)
For each regression model, set a threshold, the samples whose regression error is greater than the threshold are recorded as - 1, and those whose regression error is less than the threshold are recorded as 1, so as to obtain the recognition vector of each model, and the length of the vector is the sample size.(9)λ=quantileyi−Cjxiyii=1N,0.75,Cj is the j-th model. The 0.75 quantile of regression error under each model is the threshold, which can be adjusted. For the recognition vector of each model, if the sample error is greater than the threshold, it is marked as - 1, otherwise it is 1.(2.2)
calculate the overall recognition vector(10)Cens=1J∑j=1JCj,Cens is the overall recognition vector, Cj is the recognition vector of the j-th model, J is the number of classifiers.(2.3)
Calculate the reference vector(11)Cref=o−dotCens,o÷normCens2×Cens,Here,o is the unit vector with all elements equal, that is, the vector with all elements equal to 1 divided by its module, and the Cref can be obtained from the above formula.(3)
There are three models for aggregatingNt. Calculate cosine value of Cj and Cref. Reserve Cj which the cosine value is greater than 0, because it means the angle is less than 90 degrees, and other unsuitable decision trees are deleted. The specific threshold of cosine can be adjusted through experiments.


## 2.1.3. Forward Neural Network

Neural network is a famous machine learning algorithm which has a general application in many fields. Forward neural network is a kind of Neural Network which consists of multilayer perceptron introduced by Li [15]. They are a kind of multilayer feedforward networks trained by error back propagation. The basic idea is the gradient descent method. The gradient search technique is used to minimize the mean square error between the actual output value and the expected output value.The hidden layer node of feedforward neural network uses the inner product of input pattern and weight vector as the independent variable of activation function, and the activation function usually uses sigmoid function. Each node has the same influence on the output of the feedforward neural network, so the feedforward neural network is the global approximation of nonlinear mapping.The forward neural network consists of the following seven steps:(1)
Network initialization Feedforward neural network is a kind of hierarchical network. In order to avoid the influence of extreme initial value on network training, it usually randomizes the weights of each layer.(2)
Input layer Accept the sample data input and calculate the output through the input layer.(3)
Hidden layer Accept the output of the input layer, and calculate the output of the hidden layer through the hidden layer.(4)
Output layer Accept the output of the hidden layer, and calculate the output of the output layer through the output layer, that is, the final prediction label.(5)
Calculation of error The error between the predicted result and the true label is calculated.(6)
Update of weight The new network weights are calculated according to the error correction formula.(7)
Update of offset According to the error correction formula, the new network weight bias term is calculated.(8)
Judge whether the algorithm iteration ends or not According to the number of iterations or error convergence, whether to end the network training is judged.


## 2.2. Classical Volatility Prediction Model

### 2.2.1. Fractional Brownian Motion Model (fBm Model)

fBm model assumes that the logarithmic increment of volatility follows fractional Brownian motion [16], which can be expressed by (12)logσt+1−logσt=νWt+□H−WtH,where WH is a fractional Brownian motion with Hurst index H, where Hurst index H can be used to measure roughness of σt, ν Is a positive number.However, measured by equation (12) σt is non-stationary, which supposes logσt follows the fractional O-U process, and then modeling.Stationary fractional O-U processXt can be defined by (13)dXt=νdWtH−αXt−mdt,where m is a real number, ν and α Is a positive number.According to equation (13), Xt can be obtained by (14)Xt=ν∫−∞te−αt−sdWtH+m.The integral here is a Riemann–Stieltjes Integral [17]. In this way, the volatility integral in the time interval [0, t] is obtained:(15)σt=eXt,t∈0,T,Xt in equation (15) obeys equation (14), and m is a real number, ν and α are positive numbers, H < 0.5.


### 2.2.2. ARMA (p q) Model

Autoregressive moving average model is abbreviated as ARMA model. The model consists of two parts: autoregressive model (AR model) and moving average model (MA model). The AR model was proposed by Yule [18], and then Slutsky [19] supplemented the AR model and proposed the MA model. Subsequently, in 1938, Wold [20] formally proposed the ARMA model by combining the AR model and MA model.Autoregressive model is a model that assumes that the variable to be predicted is related to its own lag term and not related to other variables, then the p-order AR model can be expressed as follows:(16)RVt=β0+β1RVt−1+β2RVt−2+…+βpRVt−p+εt,whereεtis a sequence ofindependent identically distributedrandom variables,andEεt=0.The moving average model assumes that the time series to be predicted is a linear combination of pastq-order white noise, which can be expressed as(17)RVt=α0+α1εt−1+α2εt−2+…+αqεt−q,where εt is the white noise sequence.Combining the abovep-order AR model and q-order MA model can form ARMA (p, q) model, which can be expressed as(18)RVt=β0+β1RVt−1+β2RVt−2+…+βpRVt−p+α0+α1εt−1+α2εt−2+…+αqεt−q.


### 2.2.3. Brownian Semistationary with Gamma Kernel

Brownian semistationary with Gamma kernel is a volatility prediction model based on kernel function [21]. It defines Xt following the random process:(19)Xt=∫−∞tt−sαe−λt−svsdWs,t>0.In equation (20), Xt>0, α∈−0.5,0.5, λ∈0,∞, W is a standard Brownian motion, and v is an adapted, covariance-stationary volatility process.Define the autocorrelation function (ACF):(20)ρh,α,λ=CorrXt+h,Xt=2−α+0.5Гα+0.5λhα+0,5Kα+0.5λh,h∈R,where α It can be estimated by(21)α^p≔log2COFp,np−0.5,where COFp,n=∑k=5nXk−2Xk−2+Xk−4p/∑k=3nXk−2Xk−1+Xk−2p and p > 0 is a power parameter to be chosen by the researcher.Next, estimate the optimalλ^ value, equation (22):(22)λ^=argminλ∑i=1Nρ^h−ρh,α,λ2,where the empirical estimate of ρ^h can be replaced by linear correlation coefficient. N is the number of linear correlation coefficients calculated on different lag terms.Then, calculateГ22 and Г12 by equations (23) and (24), respectively, and the matrix form of (25):(23)Г22≔1CorrXm,Xm−1CorrXm−1,Xm1⋯CorrXm,X0⋮CorrXm−1,Xm⋮⋮CorrX0,XmCorrX0,Xm−1⋱⋮⋯1,(24)Г12≔CorrXm+1,XmCorrXm+1,Xm−1⋯CorrXm+1,X0,(25)a=Xm,Xm−1,⋯,X0,So, combinea, Г22, and Г12. The expected value of the one-step predicted value can be calculated by the expression of μ, as follows:(26)μ=Г12Г22−1a.At the same time, the estimated value of variance can also be obtained:(27)ξ2=VarXt1−Г12Г22−1Г21,where Г21=Г12T.


## 2.2.1. Fractional Brownian Motion Model (fBm Model)

fBm model assumes that the logarithmic increment of volatility follows fractional Brownian motion [16], which can be expressed by (12)logσt+1−logσt=νWt+□H−WtH,where WH is a fractional Brownian motion with Hurst index H, where Hurst index H can be used to measure roughness of σt, ν Is a positive number.However, measured by equation (12) σt is non-stationary, which supposes logσt follows the fractional O-U process, and then modeling.Stationary fractional O-U processXt can be defined by (13)dXt=νdWtH−αXt−mdt,where m is a real number, ν and α Is a positive number.According to equation (13), Xt can be obtained by (14)Xt=ν∫−∞te−αt−sdWtH+m.The integral here is a Riemann–Stieltjes Integral [17]. In this way, the volatility integral in the time interval [0, t] is obtained:(15)σt=eXt,t∈0,T,Xt in equation (15) obeys equation (14), and m is a real number, ν and α are positive numbers, H < 0.5.


## 2.2.2. ARMA (p q) Model

Autoregressive moving average model is abbreviated as ARMA model. The model consists of two parts: autoregressive model (AR model) and moving average model (MA model). The AR model was proposed by Yule [18], and then Slutsky [19] supplemented the AR model and proposed the MA model. Subsequently, in 1938, Wold [20] formally proposed the ARMA model by combining the AR model and MA model.Autoregressive model is a model that assumes that the variable to be predicted is related to its own lag term and not related to other variables, then the p-order AR model can be expressed as follows:(16)RVt=β0+β1RVt−1+β2RVt−2+…+βpRVt−p+εt,whereεtis a sequence ofindependent identically distributedrandom variables,andEεt=0.The moving average model assumes that the time series to be predicted is a linear combination of pastq-order white noise, which can be expressed as(17)RVt=α0+α1εt−1+α2εt−2+…+αqεt−q,where εt is the white noise sequence.Combining the abovep-order AR model and q-order MA model can form ARMA (p, q) model, which can be expressed as(18)RVt=β0+β1RVt−1+β2RVt−2+…+βpRVt−p+α0+α1εt−1+α2εt−2+…+αqεt−q.


## 2.2.3. Brownian Semistationary with Gamma Kernel

Brownian semistationary with Gamma kernel is a volatility prediction model based on kernel function [21]. It defines Xt following the random process:(19)Xt=∫−∞tt−sαe−λt−svsdWs,t>0.In equation (20), Xt>0, α∈−0.5,0.5, λ∈0,∞, W is a standard Brownian motion, and v is an adapted, covariance-stationary volatility process.Define the autocorrelation function (ACF):(20)ρh,α,λ=CorrXt+h,Xt=2−α+0.5Гα+0.5λhα+0,5Kα+0.5λh,h∈R,where α It can be estimated by(21)α^p≔log2COFp,np−0.5,where COFp,n=∑k=5nXk−2Xk−2+Xk−4p/∑k=3nXk−2Xk−1+Xk−2p and p > 0 is a power parameter to be chosen by the researcher.Next, estimate the optimalλ^ value, equation (22):(22)λ^=argminλ∑i=1Nρ^h−ρh,α,λ2,where the empirical estimate of ρ^h can be replaced by linear correlation coefficient. N is the number of linear correlation coefficients calculated on different lag terms.Then, calculateГ22 and Г12 by equations (23) and (24), respectively, and the matrix form of (25):(23)Г22≔1CorrXm,Xm−1CorrXm−1,Xm1⋯CorrXm,X0⋮CorrXm−1,Xm⋮⋮CorrX0,XmCorrX0,Xm−1⋱⋮⋯1,(24)Г12≔CorrXm+1,XmCorrXm+1,Xm−1⋯CorrXm+1,X0,(25)a=Xm,Xm−1,⋯,X0,So, combinea, Г22, and Г12. The expected value of the one-step predicted value can be calculated by the expression of μ, as follows:(26)μ=Г12Г22−1a.At the same time, the estimated value of variance can also be obtained:(27)ξ2=VarXt1−Г12Г22−1Г21,where Г21=Г12T.


## 3. Empirical Research

### 3.1. Feature Introduction and Selection

The prediction ability of machine learning algorithm largely depends on the effectiveness of training features. This section comprehensively refers to Vrontos et al. [2] and Christensen et al. [22] and collects 30 features, including the multi-factor model from the classical Fama French, the factors based on the stock index, the factors based on the national debt and the interest rate, the indexes based on financial economy, and the lag terms of realized volatility. Table 1 summarizes the introduction of 30 features.Table 1
Name, description, and data set of 30 features.
FeatureExplanationFeature setrvdDaily lag of HS300 RVM1,M2,M3,M4rvwWeekly lag of HS300 RVM1,M2,M3,M4momentum_5HS300 momentum factor with parameters of 5 daysM1,M2,M3,M4hmlFama-French HML factor daily portfolio returns–high minus low value factorM4mktHS300 market factor portfolio returnsM4mkt_premiumMarket factor – 1 year treasury yieldM3,M4bicBusiness index of ChinaM2,M3,M4momentum_60HS300 momentum factor with parameters of 60 daysM1,M2,M3,M4treasury_bond_momentum_51 year treasury bond momentum factor with parameters of 5 daysM3,M4rvmMonthly lag of HS300 RVM2,M3,M4epuChinese economic policy uncertainty index,M4treasury_bond_momentum_601 year treasury bond momentum factor with parameters of 60 daysM1,M2,M3,M4Skewness _120Skewness of return of HS300 with parameters of 120 daysM1,M2,M3,M4hisSquared hang seng index log-returnM3,M4treasury_bond_momentum_1201 year treasury bond momentum factor with parameters of 120 daysM1,M2,M3,M4mkt_premium_lag1Market factor: 1 year treasury yield lagged by 1 dayM2,M3,M4ShiborShibor overnight interest rateM1,M2,M3,M41mmHS300 1 week momentumM2,M3,M4treasury_bond_spread10 year treasury yield−1 year treasury yieldM3,M4treasury_bond_momentum_201 year treasury bond momentum factor with parameters of 20 daysM3,M4skewness_20Skewness of return of HS300 with parameters of 20 daysM3,M4SC futureShanghai oil future indexM3,M4AmtdifFirst differences of daily trading amount of HS300 stock indexM4momentum_120HS300 momentum factor with parameters of 120 daysM3,M4cmaFama-French CMA portfolio returns–conserve minus aggressive investment factorM4smbFama-French SMB portfolio returns–small minus big size factorM4momentum_20HS300 momentum factor with parameters of 20 daysM3,M4rmwFama-French RMW portfolio returns–robust minus weak quality factorM4skewness_60Skewness of return of HS300 with parameters of 60 daysM4The prediction ability to the realized volatility of the 30 features are uneven, which needs to be further screened and compared. In this study, 30 features are scored and sorted by three methods: Select-K-Best method, Random Forest Regressor feature importance, Permutation Importance.Select-K-Best method measures the effectiveness of features through a certain index, such as Chi-square value, information entropy and so on. The algorithm calculates the evaluation index of each feature from an ex-ante perspective, and rank features by evaluation index.Random Forest Regressor feature importance is based on the contribution of features in each decision tree. The contribution of a feature is measured by calculating the Gini index or out of bag data (OOB) error rate of the feature at each split point of each decision tree, and then the contributions of all decision trees are added as the importance evaluation of the feature.Permutation Importance scores the importance of features from an ex-post perspective. First, all the features are used to train the model, then the features are disrupted one by one, and the disrupted features are used to train the model again. If the effectiveness of the disrupted model is reduced compared with the model before disruption, it means that the feature before disruption is an effective feature. The more the effectiveness of model decreases, the more important the feature is, which is used as the basis for feature scoring.The ideas of the above three feature evaluation are different, so the ranking of feature importance will also be different. Table2 gives the feature scores and weights of the three algorithms. Select-K-Best method and Random Forest Regressor feature importance directly give the score or weight of the feature, while Permutation Importance gives the mean and standard deviation of the feature weight. According to these weights and scoring data, the features can be sorted. By taking the intersection of the top 17 features of the three algorithms, 8 features can be obtained to form the feature set M1. Taking the intersection of the top 20 features of the three algorithms, 12 features can be obtained to form the feature set M2. Taking the intersection of the top 27 features of the three algorithms, 21 features can be obtained to form the feature set M3. Finally, all 30 features form the feature set M4. The attribution of each feature in the four feature sets is shown in the feature sets column of Table 2. Top features in different algorithms are highlighted in each column.Table 2
Scoring and ranking of feature importance by 3 different feature selection algorithms.
Feature nameSelect-K-BestRandom Forest Regressor feature importancePermutation ImportanceFeature setrvd163.09130.2905970.2228 ± 0.1104M1,M2,M3,M4rvw86.812950.0808110.0158 ± 0.0145M1,M2,M3,M4momentum_515.336580.0449440.0119 ± 0.0199M1,M2,M3,M4hml12.795450.030411−0.0044 ± 0.0078M4mkt10.983620.023517−0.0057 ± 0.0058M4mkt_premium10.976160.027876−0.0017 ± 0.0207M3,M4bic10.569030.0208150.0297 ± 0.0154M2,M3,M4momentum_608.6242850.0390440.0437 ± 0.0221M1,M2,M3,M4treasury_bond_momentum_56.9908730.013640.0000 ± 0.0020M3,M4rvm6.6544990.0336350.0015 ± 0.0061M2,M3,M4epu6.543260.0085320.0015 ± 0.0047M4treasury_bond_momentum_606.083490.0221050.0061 ± 0.0117M1,M2,M3,M4skewness_1205.7443280.0237140.0096 ± 0.0097M1,M2,M3,M4his5.6088240.0148280.0005 ± 0.0050M3,M4treasury_bond_momentum_1204.2273070.0232980.0246 ± 0.0178M1,M2,M3,M4mkt_premium_lag13.5037030.0186770.0025 ± 0.0087M2,M3,M4shibor3.4894080.0273680.0111 ± 0.0095M1,M2,M3,M41mm1.5741030.0192350.0070 ± 0.0114M2,M3,M4treasury_bond_spread1.5326280.0161240.0088 ± 0.0050M3,M4treasury_bond_momentum_201.4622630.015703−0.0006 ± 0.0065M3,M4skewness_201.3022150.0250740.0236 ± 0.0078M3,M4SC future1.2137010.0334170.0098 ± 0.0073M3,M4amtdif0.5894990.016307−0.0047 ± 0.0053M4momentum_1200.3581620.0160340.0064 ± 0.0091M3,M4cma0.2520260.0075630.0001 ± 0.0016M4smb0.2520260.009307−0.0001 ± 0.0018M4momentum_200.0278060.0272420.0084 ± 0.0214M3,M4rmw0.0239640.016459−0.0029 ± 0.0159M4skewness_600.0121750.0236040.0087 ± 0.0101M4mkt_premium_lag20.0055490.0301170.0051 ± 0.0171M4


### 3.2. Option Strategy for Directly Predicting the Rise and Fall of Volatility

In this section, the four feature sets M1, M2, M3, and M4 constructed in Table1 are used to train six machine learning models. The six machine learning models are random forest, support vector machine and four forward neural networks with different network structures, labeled random forest, SVM, NN (4), NN (8, 4), NN (16, 8, 4) and NN (32, 16, 8, 4). The structures of four neural networks are: NN (4) has a hidden layer with 4 neurons, NN (8, 4) has a two-layer hidden layer with 8 and 4 neurons, NN (16, 8, 4) has a three-layer hidden layer with 16, 8 and 4 neurons, NN (32, 16, 8, 4) has a four-layer hidden layer with 32, 16, 8 and 4 neurons.In the BS model, the option pricing is mainly determined by five Greek letters, which are respective sensitivity to the price change of the underlying assetΔ, implied volatility of options σ, differential of Δ which is г, sensitivity to changes in interest rates ρ, sensitivity to expiration θ。The maturity time is naturally expiring, and the interest rate usually changes little in the short term, the second derivative of the price change of the underlying asset г is the acceleration of the underlying asset price change. Therefore, the most significant impact on the option price change is the price change sensitivity Δ and implied volatility of options σ. The strategy of this paper mainly focuses on the implied volatility of options σ.The above 6 machine learning models and 4 sets of feature sets are used to train and predict the realized volatility respectively, and then construct the “option strategy for directly predicting the rise and fall of volatility”: if the machine learning model predicts the rise of realized volatility the next day, the portfolio of call and put option at the money is bought at the closing time of the day; If the machine learning model predicts the realized volatility decline of the next day, the portfolio of call and put option at the money is sold at the closing time of the day, close the positions of the previous day at the closing time of the next day, and conduct the trading according to this rule repeatedly day by day. This strategy hedgesΔ risk by buying or selling both call and put options at the same time, because the transaction is traded at the same price it also can hedge г Risk to a certain extent, so the implied volatility risk σ is exposed. Then, if we can accurately predict the rise and fall direction of realized volatility, we can make profits through this strategy. The investment evaluation results of “option strategy for directly predicting the rise and fall of volatility” are summarized in Tables 3–6.Table 3
Option strategy for directly predicting the rise and fall of volatility (use 8 features M1).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.110.520.120.130.470.351.060.5059.00Next month0.130.420.200.210.410.301.040.5159.00Month after next0.020.42−0.08−0.090.390.291.030.5054.50Current quarter−1.130.55−2.15−2.080.570.490.660.3828.00Next quarter−1.770.59−3.10−3.020.600.590.480.4228.50Quarter after next−0.230.53−0.54−0.510.550.320.620.5724.50Support vector machineCurrent month0.230.470.380.410.430.291.140.5059.00Next month0.230.400.460.480.380.261.180.4959.00Month after next−0.030.41−0.20−0.220.370.240.960.5054.50Current quarter−0.750.47−1.69−1.650.490.390.730.4128.00Next quarter−1.460.54−2.83−2.750.550.530.480.4628.50Quarter after next−0.280.54−0.62−0.590.570.330.590.5724.50NN (4)Current month0.150.510.200.220.480.320.980.5359.00Next month0.230.410.440.440.410.271.000.5359.50Month after next−0.010.43−0.15−0.160.400.300.970.5054.50Current quarter−0.740.47−1.67−1.610.490.370.690.4328.00Next quarter−1.490.54−2.85−2.790.550.530.500.4428.50Quarter after next−0.330.55−0.70−0.660.580.330.620.5525.00NN (8, 4)Current month−0.100.57−0.27−0.300.510.421.090.4759.00Next month−0.260.48−0.64−0.680.450.381.120.4459.00Month after next−0.170.45−0.50−0.550.400.311.040.4754.50Current quarter−0.990.53−1.98−1.910.550.450.680.3928.00Next quarter−2.070.60−3.53−3.410.620.650.430.4028.50Quarter after next−0.340.51−0.77−0.730.540.330.610.5524.50NN (16, 8, 4)Current month0.000.55−0.09−0.100.510.351.030.4959.00Next month0.030.45−0.03−0.040.420.271.130.4759.00Month after next−0.140.46−0.41−0.460.410.271.100.4654.50Current quarter−0.940.51−1.94−1.890.520.430.740.3828.00Next quarter−1.710.58−3.06−2.950.600.580.430.4628.50Quarter after next−0.350.55−0.72−0.690.580.340.610.5524.50NN (32, 16, 8, 4)Current month−0.580.71−0.89−0.940.670.481.040.4359.00Next month−0.810.59−1.45−1.510.570.481.030.4159.00Month after next−0.520.49−1.17−1.210.470.330.920.4654.50Current quarter−1.120.52−2.26−2.200.530.480.680.3628.00Next quarter−2.240.65−3.57−3.510.650.680.550.3328.50Quarter after next−0.700.64−1.18−1.130.670.430.610.4924.50Table 4
Option strategy for directly predicting the rise and fall of volatility (use 12 features M2).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.000.53−0.09−0.100.470.381.040.4959.00Next month−0.050.43−0.24−0.250.410.341.040.4859.00Month after next−0.020.42−0.16−0.170.380.301.050.4955.00Current quarter−1.040.53−2.06−1.990.550.460.640.3928.00Next quarter−1.850.60−3.16−3.090.620.610.510.4028.50Quarter after next−0.150.50−0.39−0.370.520.310.720.5524.50Support vector machineCurrent month−0.240.51−0.58−0.550.540.450.940.4859.00Next month−0.140.43−0.44−0.450.420.361.060.4759.00Month after next−0.230.41−0.68−0.690.400.350.770.5354.50Current quarter−0.670.47−1.54−1.480.480.380.680.4528.00Next quarter−1.250.51−2.54−2.510.520.490.530.4628.50Quarter after next−0.470.54−0.97−0.920.570.360.600.5324.50NN (4)Current month0.090.530.070.080.480.361.120.4859.00Next month0.090.430.100.100.420.321.020.5159.00Month after next−0.050.44−0.22−0.240.410.320.990.5054.50Current quarter−0.910.51−1.90−1.820.530.440.610.4328.00Next quarter−1.770.60−3.07−2.960.620.600.450.4428.50Quarter after next−0.440.57−0.87−0.820.600.380.570.5524.50NN (8, 4)Current month−0.350.66−0.60−0.660.610.451.020.4659.00Next month−0.260.50−0.62−0.630.490.360.880.5059.00Month after next−0.410.51−0.90−0.960.470.340.880.4954.50Current quarter−0.900.51−1.88−1.820.520.420.720.3928.00Next quarter−1.480.59−2.59−2.560.600.520.550.4428.50Quarter after next−0.440.58−0.85−0.800.610.340.530.5724.50NN (16, 8, 4)Current month0.220.470.370.420.410.291.170.4959.00Next month0.030.42−0.06−0.060.410.290.950.5259.50Month after next−0.010.40−0.16−0.170.370.270.910.5254.50Current quarter−0.740.47−1.68−1.610.490.370.640.4528.00Next quarter−1.340.52−2.67−2.630.530.490.520.4628.50Quarter after next−0.310.54−0.66−0.630.570.320.630.5525.00NN (32, 16, 8, 4)Current month−0.310.67−0.55−0.600.610.451.040.4659.00Next month−0.500.55−1.01−1.060.520.421.010.4459.00Month after next−0.260.47−0.65−0.700.440.320.970.4854.50Current quarter−1.150.57−2.12−2.050.590.500.620.3928.00Next quarter−2.460.74−3.40−3.380.740.710.580.3328.50Quarter after next−0.990.68−1.54−1.480.700.490.660.4324.50Table 5
Option strategy for directly predicting the rise and fall of volatility (use 21 features M3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.020.53−0.13−0.140.480.401.020.4959.00Next month−0.120.44−0.38−0.400.430.371.000.4859.00Month after next−0.080.43−0.31−0.340.390.301.010.4954.50Current quarter−1.130.55−2.15−2.080.570.490.660.3828.00Next quarter−1.730.58−3.08−2.990.590.580.490.4228.50Quarter after next−0.210.52−0.51−0.480.550.320.630.5724.50Support vector machineCurrent month0.050.490.000.000.530.390.990.5159.00Next month0.040.42−0.02−0.020.410.321.130.4759.00Month after next0.170.400.300.310.380.250.870.5654.50Current quarter−0.350.46−0.86−0.840.470.320.790.4828.00Next quarter−0.760.49−1.65−1.670.480.350.630.4928.50Quarter after next−0.500.54−1.02−0.980.560.380.630.5124.50NN (4)Current month−0.200.59−0.43−0.470.530.401.080.4659.00Next month−0.280.48−0.69−0.730.450.351.100.4459.00Month after next−0.050.43−0.24−0.250.410.280.850.5355.00Current quarter−0.880.50−1.87−1.820.510.410.720.3928.00Next quarter−1.760.59−3.09−3.030.600.590.520.4028.50Quarter after next−0.240.53−0.55−0.530.550.320.720.5324.50NN (8, 4)Current month0.180.480.280.310.430.391.190.4859.00Next month0.160.390.300.310.370.331.140.4959.00Month after next0.290.360.660.710.330.241.050.5354.50Current quarter−0.650.45−1.54−1.500.460.390.780.4128.00Next quarter−1.010.46−2.31−2.300.460.470.580.4628.50Quarter after next0.090.450.080.080.470.280.690.6124.50NN (16, 8, 4)Current month0.510.431.081.220.380.221.260.5259.00Next month0.460.361.141.220.340.221.200.5359.00Month after next0.340.380.780.880.330.211.070.5354.50Current quarter−0.170.38−0.57−0.550.390.270.740.5428.00Next quarter−0.590.42−1.54−1.540.420.360.550.5428.50Quarter after next0.100.450.100.090.480.250.530.6724.50NN (32, 16, 8, 4)Current month0.270.470.460.520.420.261.330.4759.00Next month0.420.380.981.030.360.201.290.5059.00Month after next0.340.360.800.860.340.171.010.5554.50Current quarter−0.430.42−1.14−1.110.430.320.720.4828.00Next quarter−1.260.51−2.58−2.540.520.470.540.4628.50Quarter after next−0.290.54−0.63−0.600.570.320.590.5724.50Table 6
Option strategy for directly predicting the rise and fall of volatility (use 30 features M4).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.050.52−0.01−0.010.460.381.100.4859.00Next month−0.050.43−0.24−0.250.410.341.080.4759.00Month after next0.000.42−0.12−0.130.380.291.020.5054.50Current quarter−0.980.52−1.99−1.910.540.440.620.4128.00Next quarter−1.730.58−3.07−2.990.590.580.490.4228.50Quarter after next−0.080.49−0.27−0.260.510.310.700.5725.00Support vector machineCurrent month−0.010.52−0.12−0.110.550.411.030.4959.00Next month0.020.44−0.06−0.060.430.331.120.4759.00Month after next0.220.400.430.440.390.250.830.5855.00Current quarter−0.420.48−0.98−0.960.490.330.800.4628.00Next quarter−0.740.49−1.63−1.640.480.340.590.5128.50Quarter after next−0.620.57−1.18−1.140.590.390.630.4924.50NN (4)Current month0.060.540.010.010.510.330.930.5359.00Next month−0.050.44−0.22−0.220.440.320.880.5359.50Month after next−0.350.49−0.81−0.850.470.330.930.4854.50Current quarter−0.880.50−1.85−1.790.520.440.720.3928.00Next quarter−1.900.57−3.43−3.280.590.630.390.4428.50Quarter after next−0.440.53−0.92−0.860.570.370.480.5925.00NN (8, 4)Current month0.150.520.200.200.500.280.910.5459.00Next month0.120.440.160.160.430.271.000.5259.00Month after next−0.050.44−0.23−0.240.420.270.860.5354.50Current quarter−0.620.45−1.48−1.430.470.340.710.4528.00Next quarter−1.330.51−2.68−2.590.530.500.440.4928.50Quarter after next−0.380.56−0.77−0.730.590.350.550.5724.50NN (16, 8, 4)Current month0.100.550.090.090.530.270.880.5459.00Next month0.170.440.270.280.420.231.090.5059.50Month after next−0.090.46−0.31−0.320.440.250.810.5454.50Current quarter−0.580.45−1.41−1.360.460.330.680.4628.00Next quarter−0.980.50−2.07−2.030.510.390.520.5128.50Quarter after next−0.330.55−0.70−0.660.580.340.570.5724.50NN (32, 16, 8, 4)Current month0.330.470.600.670.420.211.320.4759.00Next month0.390.400.850.950.360.171.500.4559.50Month after next0.310.380.670.750.340.151.130.5154.50Current quarter−0.940.51−1.93−1.890.520.430.770.3828.00Next quarter−0.940.47−2.13−2.140.460.410.560.4928.50Quarter after next−0.030.48−0.16−0.150.500.270.670.5924.50From Tables3–6, the evaluation results of 6 machine learning models under 6 different expiration month option contracts by 4 different feature sets are given respectively. A total of 9 evaluation indicators including annual return, annual risk, Sharpe ratio, Sortino ratio, downside risk, max drawdown, profit loss ratio, win ratio and trading times are recorded. It can be seen that in the annual return column, except for a few current month and next month option contracts of Random Forest and Neural Network, most machine learning algorithms are doing badly on the 4 data sets; The annualized standard deviation fluctuates between 0.36 and 0.69, which is far greater than the annualized rate of return; Sharpe ratio and Sortino ratio are basically negative numbers, which are highlighted in a bold font; Downside risk, max drawdown, profit-loss ratio and win ratio are also not optimistic; The lowest number of trading times is 24.5 and the highest is 59.The above 9 evaluation indicators can almost unanimously judge that the “option strategy of directly predicting the rise and fall of volatility” is a failed investment strategy, which requires rethinking the strategy design and further improving and redesigning the strategy.According to the investment experience in the industrial practice, most investment strategies with stable profitability have a certain tradeoff among profit-loss ratio, win ratio, trading times and data sampling frequency (this strategy is daily frequency). The trading frequency of the strategy, that is, the number of transactions, is often lower than the data sampling frequency. Taking the speculative strategy of daily frequency as an example, the position holding period is usually even longer at the weekly frequency. In contrast, the “option strategy of directly predicting the rise and fall of volatility” has a maximum of 59 transactions in half a year, which is obviously much higher than the practical experience of the industry. If the trading frequency of an investment strategy is high, it will put forward higher requirements for the trading win rate. Otherwise, under a short position holding period, the price change of the investment target is not large enough. Multiplying by the same win rate and profit-loss ratio, the low trading frequency strategy means that the expected return is higher than that of the high trading frequency strategy. In addition, higher trading frequency will bring higher trading costs, then the comprehensive result is that the strategy with high trading frequency will be inferior to the strategy with low trading frequency. Observe Tables3–6, according to the profit -loss ratio and win ratio, the “option strategy of directly predicting the rise and fall of volatility” does not dominate these two indicators, so it is obvious that the trading frequency is too high for the current strategy.This provides an idea for the improvement of investment strategy: reduce the trading times by adding some restrictions on trading conditions, so as to improve the profit loss ratio and win ratio of investment strategy.


### 3.3. Option Strategy of Predicting the Rise and Fall of Volatility by Threshold

It can be seen from the analysis in the previous section that it is necessary to reduce the trading times by adding some restrictions on trading conditions, so the direct solution is to set a threshold parameter: it will buy or sell option portfolio only when the rise and fall of the realized volatility predicted by the machine learning model relative to the real realized volatility of yesterday exceeds a certain threshold parameter, otherwise the strategy does not implement. Other strategy conditions are the same as the “option strategy of directly predicting the rise and fall of volatility.” In this way, the trading times can be significantly reduced by increasing the threshold parameter.Drawing on the design of confidence strategy 2 and confidence strategy 3 of Vrontos et al. [2] on VIX Index Investment. He set a probability interval based on the classification confidence probability of machine learning to trade selectively. The “option strategy of predicting the rise and fall of volatility by thresholds” in this section is essentially similar to that of confidence strategy 2 and confidence strategy 3 designed by Vrontos et al. [2]. We use the predicted volatility obtained by machine learning algorithm and set a threshold interval as the trading condition, only when predicted volatility exceeds threshold parameter, the strategy triggers.Tables7 to 18 summarizes the strategy evaluation results of “option strategy of predicting the rise and fall of volatility by thresholds” under the setting of different threshold parameters.Table 7
Option strategy of predicting the rise and fall of volatility by threshold (use 8 features M1, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.340.340.851.780.160.171.670.4435.00Next month0.300.280.881.590.150.121.630.4435.00Month after next0.250.290.701.130.180.141.270.4932.50Current quarter−0.180.18−1.32−1.620.140.141.100.3915.50Next quarter−0.950.31−3.24−3.360.300.380.480.4116.00Quarter after next0.120.180.400.550.130.111.090.5412.00Support vector machineCurrent month0.330.211.332.340.120.081.260.5625.00Next month0.390.211.612.910.120.091.440.5425.00Month after next0.190.190.731.170.120.121.110.5522.00Current quarter0.100.130.400.640.080.081.570.4612.00Next quarter−0.210.21−1.23−1.430.180.140.750.4812.50Quarter after next0.310.161.702.630.100.101.100.6710.50NN (4)Current month−0.300.59−0.59−0.700.490.431.130.4232.00Next month−0.180.45−0.52−0.580.400.331.130.4432.00Month after next−0.280.43−0.76−0.870.380.260.940.4729.00Current quarter−0.950.50−2.00−1.990.500.430.560.3515.50Next quarter−1.280.50−2.64−2.620.510.470.360.4716.00Quarter after next−0.470.56−0.93−0.910.570.330.410.5913.50NN (8, 4)Current month0.400.301.152.560.140.111.860.4530.00Next month0.370.251.302.310.140.081.530.4930.50Month after next0.330.251.161.860.150.081.150.5528.00Current quarter−0.070.17−0.69−0.880.130.101.070.4514.50Next quarter−0.510.26−2.17−2.370.240.260.580.4715.00Quarter after next0.280.151.522.480.090.061.150.6312.00NN (16, 8, 4)Current month0.540.311.613.690.130.121.730.4932.50Next month0.440.261.552.890.140.091.800.4632.50Month after next0.360.251.272.300.140.081.500.4929.50Current quarter−0.120.16−1.03−1.450.110.111.300.3814.50Next quarter−0.490.25−2.20−2.430.220.220.650.4315.00Quarter after next0.180.180.721.020.120.101.120.5612.50NN (32, 16, 8, 4)Current month−0.250.45−0.68−0.690.430.320.900.4830.50Next month−0.210.41−0.65−0.690.380.291.050.4430.50Month after next−0.210.37−0.69−0.730.350.240.800.5127.50Current quarter−0.740.46−1.72−1.710.460.360.550.3713.50Next quarter−1.720.56−3.16−3.140.560.580.410.3214.00Quarter after next−0.760.65−1.26−1.260.640.450.550.4511.00Table 8
Option strategy of predicting the rise and fall of volatility by threshold (use 12 features M2, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.320.340.791.640.160.181.560.4634.00Next month0.270.280.791.420.150.141.520.4634.50Month after next0.270.280.771.240.180.121.220.5131.50Current quarter−0.160.18−1.19−1.470.140.131.070.4116.00Next quarter−0.870.30−3.04−3.180.290.360.490.4216.50Quarter after next0.140.170.510.690.130.101.040.5612.50Support vector machineCurrent month0.190.170.851.270.110.061.010.5920.50Next month0.260.171.231.910.110.071.110.5920.50Month after next0.040.16−0.06−0.090.110.090.970.5318.00Current quarter0.020.12−0.28−0.410.080.071.300.4510.00Next quarter−0.250.20−1.48−1.660.180.150.550.5210.50Quarter after next0.250.141.402.040.100.100.950.689.50NN (4)Current month0.010.51−0.09−0.100.440.321.090.4836.50Next month−0.010.40−0.15−0.160.370.271.050.4936.00Month after next−0.140.40−0.48−0.520.360.250.870.5133.50Current quarter−0.650.45−1.56−1.540.450.350.540.4516.50Next quarter−1.010.47−2.25−2.250.470.400.500.4417.00Quarter after next−0.230.50−0.56−0.550.500.280.470.6214.50NN (8, 4)Current month−0.230.58−0.49−0.570.490.361.140.4337.00Next month−0.220.45−0.59−0.640.410.291.020.4637.50Month after next−0.280.43−0.77−0.830.390.280.850.4934.50Current quarter−1.000.52−2.03−2.020.520.450.660.3419.00Next quarter−1.740.59−3.03−3.000.600.580.450.3819.50Quarter after next−0.570.60−1.04−1.020.610.390.520.5316.00NN (16, 8, 4)Current month0.000.51−0.09−0.110.430.331.060.4937.00Next month−0.100.42−0.35−0.380.390.310.910.5136.50Month after next−0.240.42−0.69−0.740.390.290.730.5433.50Current quarter−0.760.47−1.73−1.710.470.370.620.4118.50Next quarter−1.280.48−2.78−2.740.490.480.380.4719.00Quarter after next−0.290.51−0.68−0.650.520.300.470.6116.50NN (32, 16, 8, 4)Current month0.430.331.172.370.160.131.630.4735.00Next month0.400.271.322.270.160.101.420.5134.50Month after next0.370.271.171.910.160.091.310.5232.00Current quarter−0.220.19−1.45−1.770.150.160.980.4116.00Next quarter−0.810.30−2.89−3.050.280.330.520.4216.50Quarter after next0.190.180.801.050.130.090.720.6713.50Table 9
Option strategy of predicting the rise and fall of volatility by threshold (use 21 features M3, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.290.340.711.490.160.181.600.4532.50Next month0.260.280.761.370.150.131.580.4532.50Month after next0.220.280.620.990.170.121.220.5030.00Current quarter−0.190.17−1.39−1.680.140.141.000.4015.00Next quarter−0.790.30−2.85−2.990.280.330.470.4515.50Quarter after next0.140.170.510.690.130.101.040.5612.50Support vector machineCurrent month0.240.161.141.720.110.050.980.6219.50Next month0.330.171.672.720.100.051.260.5919.50Month after next0.130.150.520.740.110.071.020.5617.00Current quarter0.060.120.080.120.080.071.220.5010.00Next quarter−0.120.19−0.90−1.040.170.110.580.5710.50Quarter after next0.250.141.382.000.100.100.950.689.50NN (4)Current month−0.070.54−0.22−0.250.460.331.220.4437.50Next month−0.110.44−0.38−0.410.400.251.150.4537.00Month after next0.010.38−0.11−0.120.360.210.690.5934.50Current quarter−0.660.45−1.56−1.540.460.340.600.4519.00Next quarter−1.310.49−2.78−2.720.500.490.410.4419.50Quarter after next−0.310.51−0.71−0.690.530.310.470.6116.50NN (8, 4)Current month0.100.490.100.120.420.341.110.4934.50Next month0.030.40−0.05−0.050.360.291.220.4634.00Month after next0.150.370.260.300.330.251.030.5232.50Current quarter−0.490.42−1.29−1.290.420.330.720.4318.50Next quarter−1.060.45−2.49−2.490.450.470.430.4719.00Quarter after next−0.240.49−0.59−0.580.490.300.550.5815.50NN (16, 8, 4)Current month0.590.291.864.320.120.091.920.4933.50Next month0.570.232.274.400.120.081.770.5233.50Month after next0.600.242.344.190.130.081.340.5931.50Current quarter0.200.141.071.480.100.071.200.5914.50Next quarter−0.270.22−1.45−1.550.210.220.430.6015.00Quarter after next0.380.122.635.400.060.051.100.7411.50NN (32, 16, 8, 4)Current month−0.170.44−0.51−0.530.420.310.840.5133.50Next month−0.030.39−0.20−0.210.370.251.010.4933.50Month after next−0.030.37−0.21−0.220.340.240.720.5829.50Current quarter−0.620.44−1.52−1.510.440.330.530.4514.50Next quarter−1.290.48−2.79−2.740.490.480.310.4715.00Quarter after next−0.450.54−0.92−0.900.550.330.370.6213.00Table 10
Option strategy of predicting the rise and fall of volatility by threshold (use 30 features M4, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.350.330.911.890.160.151.590.4633.50Next month0.310.270.941.690.150.111.550.4633.50Month after next0.260.270.791.260.170.121.190.5231.00Current quarter−0.160.18−1.19−1.470.140.131.070.4116.00Next quarter−0.870.30−3.04−3.180.290.360.490.4216.50Quarter after next0.140.170.510.690.130.101.040.5612.50Support vector machineCurrent month−0.230.39−0.72−0.700.390.250.480.6119.00Next month−0.090.34−0.40−0.400.340.220.650.5819.00Month after next−0.260.33−0.96−0.950.330.220.540.5617.00Current quarter−0.430.39−1.23−1.220.390.250.440.5010.00Next quarter−0.550.37−1.62−1.620.370.300.310.5710.50Quarter after next−0.320.46−0.80−0.790.470.270.280.689.50NN (4)Current month0.360.320.991.970.160.121.320.5232.00Next month0.360.261.222.100.150.101.320.5231.50Month after next0.230.280.671.060.170.101.280.4930.00Current quarter−0.300.19−1.87−2.280.150.191.200.3215.50Next quarter−1.040.32−3.42−3.540.310.410.490.3816.00Quarter after next−0.070.17−0.74−0.930.130.100.790.5212.50NN (8, 4)Current month0.180.260.500.810.160.181.190.5036.00Next month0.230.250.711.210.150.141.510.4535.50Month after next−0.030.27−0.30−0.440.180.191.070.4832.50Current quarter−0.140.18−1.07−1.340.140.121.110.4117.00Next quarter−0.870.31−3.00−3.180.290.350.580.4017.50Quarter after next0.010.19−0.22−0.280.150.090.760.5714.00NN (16, 8, 4)Current month0.290.340.691.350.170.121.460.4736.50Next month0.270.290.781.360.160.131.620.4436.50Month after next0.170.290.430.660.190.171.120.5134.50Current quarter−0.280.19−1.74−2.100.160.180.870.4218.00Next quarter−0.830.30−2.95−3.070.290.340.510.4318.50Quarter after next0.080.180.160.210.140.081.010.5316.00NN (32, 16, 8, 4)Current month0.520.311.543.580.130.131.730.4934.00Next month0.610.262.144.280.130.111.730.5134.50Month after next0.620.242.384.660.120.131.700.5331.00Current quarter−0.190.15−1.56−2.120.110.161.330.3315.00Next quarter−0.680.27−2.73−2.990.240.290.750.3515.50Quarter after next0.080.160.170.240.110.121.090.5212.50Table 11
Option strategy of predicting the rise and fall of volatility by threshold (use 8 features M1, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.220.200.841.490.110.101.750.4716.00Next month0.420.182.004.390.080.052.420.5016.00Month after next0.440.192.033.440.110.050.870.7214.50Current quarter−0.010.13−0.45−0.580.100.090.960.507.00Next quarter−0.230.20−1.37−1.500.190.160.400.607.50Quarter after next0.240.101.985.070.040.031.590.735.50Support vector machineCurrent month0.050.09−0.03−0.050.070.060.880.598.50Next month0.080.090.290.400.070.061.300.538.50Month after next−0.060.08−1.31−1.570.070.050.630.537.50Current quarter0.120.071.002.110.030.032.280.564.50Next quarter−0.170.14−1.62−1.760.130.120.440.505.00Quarter after next0.160.081.423.410.030.021.220.754.00NN (4)Current month−0.400.48−0.94−1.030.440.381.100.3818.50Next month−0.290.41−0.82−0.900.370.311.280.3618.00Month after next−0.360.39−1.05−1.130.360.290.730.4917.50Current quarter−0.720.47−1.63−1.640.470.360.490.388.00Next quarter−1.190.48−2.57−2.550.490.450.270.418.50Quarter after next−0.750.59−1.35−1.360.590.390.400.477.50NN (8, 4)Current month−0.130.24−0.78−1.200.150.201.490.3621.00Next month0.010.22−0.16−0.260.140.141.660.3821.00Month after next0.160.220.480.720.150.100.960.5619.50Current quarter−0.260.16−1.96−2.290.130.170.790.379.50Next quarter−0.660.25−2.80−2.930.240.280.470.4010.00Quarter after next0.040.15−0.08−0.120.100.091.430.448.00NN (16, 8, 4)Current month0.640.262.337.330.080.062.150.5521.00Next month0.510.222.154.470.100.072.090.5021.00Month after next0.430.201.964.310.090.071.470.5818.00Current quarter0.100.120.450.760.070.041.270.539.50Next quarter0.020.16−0.17−0.210.140.090.700.6010.00Quarter after next0.400.122.989.110.040.022.330.718.50NN (32, 16, 8, 4)Current month−0.460.46−1.10−1.140.440.330.740.4519.00Next month−0.400.40−1.13−1.170.380.260.770.4519.00Month after next−0.310.36−0.99−1.040.340.220.630.5216.50Current quarter−0.660.44−1.63−1.630.440.330.540.339.00Next quarter−1.340.48−2.89−2.890.480.490.370.329.50Quarter after next−0.530.57−1.03−1.040.560.370.400.546.50Table 12
Option strategy of predicting the rise and fall of volatility by threshold (use 12 features M2, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.080.220.150.250.140.131.500.4318.50Next month0.300.201.262.440.100.071.960.4618.50Month after next0.400.211.732.750.130.070.970.6716.50Current quarter−0.150.16−1.30−1.590.130.121.010.399.00Next quarter−0.490.25−2.16−2.320.230.220.480.479.50Quarter after next0.040.14−0.07−0.090.100.110.990.537.50Support vector machineCurrent month0.020.09−0.34−0.480.060.060.980.538.50Next month0.060.090.130.180.070.061.200.538.50Month after next−0.040.09−1.00−1.270.070.060.820.508.00Current quarter0.090.070.621.330.030.022.380.505.00Next quarter−0.050.14−0.75−0.890.120.090.670.555.50Quarter after next0.230.091.985.600.030.021.390.784.50NN (4)Current month−0.150.52−0.39−0.470.430.311.130.4419.50Next month−0.010.39−0.15−0.170.340.221.160.4619.50Month after next−0.030.37−0.22−0.250.330.210.690.5818.00Current quarter−0.680.44−1.68−1.680.440.350.300.477.50Next quarter−0.710.38−2.00−1.980.390.340.270.508.00Quarter after next−0.390.47−0.95−0.940.470.280.220.676.00NN (8, 4)Current month−0.210.20−1.29−1.690.160.160.980.4320.00Next month−0.110.21−0.78−1.090.150.111.220.4119.50Month after next−0.220.22−1.25−1.600.170.170.950.4318.50Current quarter−0.350.16−2.53−2.770.140.170.760.3310.50Next quarter−0.790.29−2.87−3.060.270.330.480.4111.00Quarter after next−0.020.17−0.44−0.560.130.130.850.539.50NN (16, 8, 4)Current month−0.090.49−0.29−0.350.400.290.960.4921.50Next month−0.150.38−0.53−0.580.340.280.810.5121.50Month after next−0.080.35−0.38−0.410.320.240.660.5819.00Current quarter−0.470.39−1.34−1.340.390.290.470.4810.50Next quarter−0.690.36−2.04−2.030.360.370.300.5511.00Quarter after next−0.270.45−0.72−0.700.460.270.260.718.50NN (32, 16, 8, 4)Current month0.160.330.340.690.160.151.700.4120.50Next month0.150.230.420.690.140.101.360.4820.50Month after next0.130.240.310.470.160.081.110.5118.50Current quarter−0.170.16−1.37−1.630.140.140.870.429.50Next quarter−0.690.27−2.78−2.890.260.290.460.4010.00Quarter after next0.080.170.190.260.120.090.820.607.50Table 13
Option strategy of predicting the rise and fall of volatility by threshold (use 21 features M3, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.110.230.270.460.130.151.570.4318.50Next month0.260.201.062.050.100.082.040.4318.50Month after next0.350.201.482.420.120.081.020.6416.50Current quarter−0.130.15−1.22−1.520.120.121.050.399.00Next quarter−0.490.22−2.42−2.560.210.230.500.429.50Quarter after next0.160.120.981.910.060.051.650.537.50Support vector machineCurrent month−0.010.09−0.61−0.850.060.061.110.477.50Next month0.040.09−0.16−0.220.070.061.370.477.50Month after next−0.080.08−1.53−1.820.070.050.640.507.00Current quarter0.090.070.651.400.030.022.500.504.00Next quarter−0.060.14−0.82−0.970.110.090.600.564.50Quarter after next0.220.091.795.050.030.021.520.754.00NN (4)Current month0.030.49−0.04−0.050.400.301.250.4525.50Next month0.060.370.030.030.330.221.150.4825.50Month after next0.150.330.310.340.300.190.660.6423.50Current quarter−0.460.40−1.29−1.280.400.310.420.5413.00Next quarter−0.660.37−1.93−1.900.380.370.330.5613.50Quarter after next−0.170.46−0.48−0.470.470.290.340.7011.50NN (8, 4)Current month0.080.270.100.200.130.111.400.4418.00Next month0.030.20−0.11−0.170.130.091.240.4618.00Month after next−0.020.21−0.34−0.470.150.080.910.5216.50Current quarter−0.170.13−1.68−1.880.120.100.770.418.50Next quarter−0.410.23−2.03−2.140.210.220.350.569.00Quarter after next0.010.11−0.34−0.440.090.050.700.607.50NN (16, 8, 4)Current month0.460.251.674.320.090.051.930.5419.50Next month0.420.182.134.780.080.032.000.5519.00Month after next0.300.201.272.100.120.081.170.5818.00Current quarter0.110.110.520.680.080.061.300.568.00Next quarter−0.220.18−1.50−1.630.170.190.490.538.50Quarter after next0.280.092.566.710.030.021.110.836.00NN (32, 16, 8, 4)Current month0.030.38−0.04−0.040.350.221.100.4920.50Next month0.090.350.120.130.320.201.070.5120.50Month after next0.220.310.530.570.290.170.610.6819.00Current quarter−0.390.40−1.10−1.100.390.240.520.478.50Next quarter−0.700.39−1.96−1.930.390.340.240.569.00Quarter after next−0.380.48−0.90−0.890.480.280.340.607.50Table 14
Option strategy of predicting the rise and fall of volatility by threshold (use 30 features M4, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.120.23−0.73−1.090.150.191.240.4017.50Next month0.110.200.310.520.120.101.610.4317.50Month after next0.240.210.891.320.140.090.810.6515.50Current quarter−0.180.16−1.48−1.770.130.140.830.418.50Next quarter−0.640.25−2.77−2.890.240.280.450.399.00Quarter after next0.070.150.140.200.100.101.260.507.00Support vector machineCurrent month−0.010.09−0.61−0.850.060.061.110.477.50Next month0.040.09−0.16−0.220.070.061.370.477.50Month after next−0.080.08−1.53−1.820.070.050.640.507.00Current quarter0.090.070.651.400.030.022.500.504.00Next quarter−0.060.14−0.82−0.970.110.090.600.564.50Quarter after next0.220.091.795.050.030.021.520.754.00NN (4)Current month0.230.310.591.280.140.111.630.4520.00Next month0.150.250.410.740.140.111.650.4320.00Month after next0.100.270.170.270.170.121.250.4719.00Current quarter−0.190.16−1.49−1.790.130.130.980.3810.50Next quarter−0.700.28−2.74−2.830.270.300.390.4511.00Quarter after next−0.050.16−0.61−0.770.120.080.780.538.50NN (8, 4)Current month0.000.22−0.24−0.380.140.141.310.4322.00Next month0.060.210.060.100.130.101.580.4122.00Month after next0.040.22−0.06−0.090.150.111.010.5119.50Current quarter−0.100.15−0.96−1.190.120.101.230.3810.50Next quarter−0.510.24−2.31−2.460.230.230.580.4111.00Quarter after next0.090.160.230.320.110.091.000.569.00NN (16, 8, 4)Current month0.230.280.631.300.130.101.560.4721.50Next month0.310.231.142.030.130.081.880.4521.00Month after next0.250.230.881.320.150.080.980.5919.50Current quarter−0.310.15−2.33−2.570.140.140.730.3611.00Next quarter−0.680.27−2.70−2.790.260.290.370.4811.50Quarter after next−0.120.14−1.18−1.320.130.100.630.539.50NN (32, 16, 8, 4)Current month−0.100.23−0.68−1.210.130.151.710.3318.00Next month0.040.21−0.04−0.060.130.111.490.4218.00Month after next0.230.190.991.670.110.071.090.5816.50Current quarter−0.120.14−1.22−1.670.100.121.430.339.00Next quarter−0.250.19−1.58−1.850.160.130.730.449.00Quarter after next0.080.140.190.260.100.090.990.568.00Table 15
Option strategy of predicting the rise and fall of volatility by threshold (use 8 features M1, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.020.06−1.31−1.820.040.041.210.405.00Next month0.070.080.190.260.060.041.750.505.00Month after next0.180.071.723.950.030.020.630.894.50Current quarter0.000.06−0.86−1.140.050.041.460.402.50Next quarter−0.120.10−1.70−1.800.100.070.230.602.50Quarter after next0.030.05−0.39−0.550.030.020.550.752.00Support vector machineCurrent month−0.010.07−0.88−1.200.050.031.360.402.50Next month−0.010.06−0.96−1.100.050.040.590.602.50Month after next−0.010.03−1.93−2.390.030.020.770.502.00Current quarter0.000.000.000.000.000.000.000.001.00Next quarter0.000.000.000.000.000.000.000.001.50Quarter after next0.000.000.000.000.000.000.000.001.00NN (4)Current month−0.380.42−1.02−1.080.400.270.540.477.50Next month−0.310.37−0.98−1.030.350.240.630.477.50Month after next−0.250.34−0.89−0.940.320.200.370.626.50Current quarter−0.620.43−1.56−1.580.420.310.490.334.50Next quarter−0.750.40−2.02−2.010.400.330.160.444.50Quarter after next−0.820.57−1.52−1.530.570.380.310.384.00NN (8, 4)Current month−0.120.19−0.93−1.470.120.161.950.269.50Next month0.010.15−0.27−0.480.080.091.770.379.50Month after next−0.070.18−0.70−0.940.130.110.720.538.50Current quarter−0.050.12−0.84−1.060.100.071.270.384.00Next quarter−0.360.19−2.18−2.290.180.170.370.444.50Quarter after next0.120.090.771.810.040.031.740.573.50NN (16, 8, 4)Current month0.350.241.284.260.070.052.080.5211.50Next month0.310.181.462.970.090.072.260.4811.50Month after next0.360.161.934.450.070.061.160.7010.00Current quarter0.100.090.520.880.050.031.170.605.00Next quarter0.140.120.791.190.080.051.360.605.00Quarter after next0.120.080.952.150.030.022.430.573.50NN (32, 16, 8, 4)Current month0.090.160.240.550.070.092.480.367.00Next month0.050.140.010.010.080.071.210.507.00Month after next0.080.130.210.420.060.051.410.506.00Current quarter0.110.070.904.190.010.014.900.503.00Next quarter−0.330.16−2.43−2.490.150.160.480.293.50Quarter after next0.170.091.437.950.020.014.850.602.50Table 16
Option strategy of predicting the rise and fall of volatility by threshold (use 12 features M2, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.010.05−1.16−1.470.040.040.940.505.00Next month0.070.080.230.290.060.051.160.605.00Month after next0.180.081.743.880.030.020.610.894.50Current quarter0.000.06−0.80−1.050.050.041.000.503.00Next quarter−0.100.11−1.39−1.500.100.060.250.673.00Quarter after next0.030.05−0.37−0.530.040.020.550.752.00Support vector machineCurrent month−0.010.07−0.88−1.200.050.031.360.402.50Next month−0.010.06−0.96−1.100.050.040.590.602.50Month after next−0.010.03−1.93−2.390.030.020.770.502.00Current quarter0.000.000.000.000.000.000.000.001.00Next quarter0.000.000.000.000.000.000.000.001.50Quarter after next0.000.000.000.000.000.000.000.001.00NN (4)Current month−0.100.11−1.35−1.470.100.080.700.477.50Next month0.030.10−0.23−0.300.070.051.280.477.50Month after next0.010.12−0.35−0.410.100.060.570.647.00Current quarter−0.100.10−1.45−1.590.090.070.510.504.00Next quarter−0.030.11−0.76−0.860.100.060.490.634.00Quarter after next0.090.050.802.080.020.011.150.802.50NN (8, 4)Current month−0.260.14−2.25−2.430.130.130.510.449.00Next month−0.100.13−1.21−1.470.100.071.120.399.00Month after next−0.130.14−1.28−1.410.130.090.480.568.00Current quarter−0.230.12−2.31−2.450.120.110.640.305.00Next quarter−0.220.16−1.64−1.790.150.140.450.505.00Quarter after next−0.030.11−0.70−0.840.090.090.520.634.00NN (16, 8, 4)Current month0.040.07−0.09−0.130.050.041.300.5010.00Next month0.120.080.931.500.050.031.600.5510.00Month after next0.200.081.793.840.040.030.930.768.50Current quarter0.090.060.811.870.020.021.930.634.00Next quarter−0.070.13−0.91−0.990.120.110.350.674.50Quarter after next0.140.051.775.860.020.012.430.754.00NN (32, 16, 8, 4)Current month0.130.120.631.020.070.051.400.5212.50Next month0.170.130.921.420.090.051.090.6012.50Month after next0.100.140.350.500.090.061.090.5511.00Current quarter−0.050.09−1.10−1.330.080.071.030.426.00Next quarter−0.400.20−2.26−2.340.190.180.380.466.50Quarter after next0.000.13−0.40−0.520.100.090.830.555.50Table 17
Option strategy of predicting the rise and fall of volatility by threshold (use 21 features M3, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.040.05−1.96−2.360.040.040.750.455.50Next month0.020.08−0.41−0.500.060.051.390.455.50Month after next0.020.08−0.35−0.460.060.040.600.674.50Current quarter0.000.06−0.78−1.040.050.041.030.502.00Next quarter−0.200.14−1.78−1.820.140.100.170.602.50Quarter after next−0.030.04−1.97−2.190.040.040.530.502.00Support vector machineCurrent month0.000.07−0.79−1.080.050.030.980.503.00Next month−0.020.06−1.06−1.210.060.040.820.503.00Month after next−0.050.04−2.37−2.650.040.020.650.402.50Current quarter0.000.000.000.000.000.000.000.001.50Next quarter−0.150.12−1.65−1.710.110.090.260.502.00Quarter after next0.000.000.000.000.000.000.000.001.50NN (4)Current month0.080.180.150.270.100.081.740.4113.50Next month0.120.180.400.630.110.081.400.4813.50Month after next0.280.161.492.500.090.061.020.6712.00Current quarter0.020.11−0.30−0.440.070.051.090.507.00Next quarter−0.200.18−1.38−1.500.170.150.510.537.50Quarter after next0.130.120.691.050.080.051.070.626.50NN (8, 4)Current month0.150.091.152.360.040.031.260.677.50Next month0.240.092.057.420.030.024.850.537.50Month after next0.220.082.035.500.030.021.260.797.00Current quarter0.100.060.882.250.020.012.290.564.50Next quarter0.110.130.440.540.100.070.420.805.00Quarter after next0.170.071.845.340.020.011.390.784.50NN (16, 8, 4)Current month0.390.211.617.670.040.032.230.6510.00Next month0.310.112.3813.350.020.024.520.639.50Month after next0.230.141.352.890.060.051.310.639.50Current quarter0.110.051.104.380.010.013.020.605.00Next quarter−0.100.14−1.10−1.200.130.130.370.645.50Quarter after next0.250.082.5813.390.010.011.670.894.50NN (32, 16, 8, 4)Current month0.030.12−0.14−0.300.050.052.450.3211.00Next month0.090.140.260.450.080.051.890.4111.00Month after next0.220.131.282.310.070.070.870.7010.00Current quarter−0.020.06−1.11−1.450.050.051.590.333.00Next quarter−0.320.16−2.29−2.330.160.160.230.433.50Quarter after next0.000.05−0.92−1.250.040.031.020.503.00Table 18
Option strategy of predicting the rise and fall of volatility by threshold (use 30 features M4, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.010.04−1.46−2.010.030.031.130.444.50Next month0.030.07−0.23−0.290.060.041.700.444.50Month after next0.080.060.570.990.030.020.810.754.00Current quarter0.000.06−0.78−1.040.050.041.030.502.00Next quarter−0.200.14−1.78−1.820.140.100.170.602.50Quarter after next−0.030.04−1.97−2.190.040.040.530.502.00Support vector machineCurrent month0.000.07−0.79−1.080.050.030.980.503.00Next month−0.020.06−1.06−1.210.060.040.820.503.00Month after next−0.050.04−2.37−2.650.040.020.650.402.50Current quarter0.000.000.000.000.000.000.000.001.50Next quarter−0.150.12−1.65−1.710.110.090.260.502.00Quarter after next0.000.000.000.000.000.000.000.001.50NN (4)Current month−0.050.09−1.14−1.430.070.040.700.538.50Next month0.000.12−0.44−0.570.090.050.880.538.50Month after next−0.040.13−0.72−0.950.100.060.870.508.00Current quarter−0.040.09−1.01−1.170.070.060.300.713.50Next quarter−0.030.14−0.53−0.610.120.100.360.713.50Quarter after next−0.080.10−1.32−1.470.090.080.340.602.50NN (8, 4)Current month−0.210.14−1.86−2.150.120.130.810.3812.00Next month−0.070.14−0.85−1.090.110.091.140.4212.00Month after next−0.080.15−0.88−1.050.120.090.530.6010.00Current quarter−0.150.11−1.79−1.960.100.071.030.275.50Next quarter−0.310.18−1.95−2.020.180.160.350.506.00Quarter after next−0.060.08−1.27−1.530.070.060.710.505.00NN (16, 8, 4)Current month0.350.211.397.520.040.033.510.5010.00Next month0.350.132.3012.250.020.036.200.479.50Month after next0.320.132.086.780.040.031.710.718.50Current quarter−0.010.05−1.23−2.010.030.031.710.334.50Next quarter−0.100.13−1.15−1.250.120.070.400.605.00Quarter after next0.060.060.150.260.030.021.410.564.50NN (32, 16, 8, 4)Current month0.170.180.671.510.080.082.760.379.50Next month0.240.171.102.140.090.091.710.539.50Month after next0.250.151.302.520.080.050.870.718.50Current quarter0.050.10−0.02−0.040.060.062.240.365.50Next quarter−0.120.14−1.23−1.440.120.110.540.555.50Quarter after next0.180.091.352.850.040.020.870.784.50Tables7 to 9 shows the trading results with the threshold parameter of 0.3 under the four feature sets of M1, M2, M3 and M4. As you can see, compared with Tables 19–22, the minimum number of trading times decreased from 24.5 to 10, and the maximum number of trading times decreased from 59 to 37.5. With the decrease of the number of trading times, the annual return, annual standard deviation, Sharpe ratio, Sortino ratio and other indicators have also improved.Table 19
Investment evaluation on train data set with parameters [0.05, −0.2].
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.410.251.463.150.110.111.790.4516.50Next month0.590.252.174.870.110.072.780.3916.50Month after next0.540.251.993.870.130.091.480.5315.00Current quarter−0.190.13−1.85−2.330.100.091.280.337.50Next quarter−0.150.17−1.17−1.440.140.100.870.477.50Quarter after next0.340.142.163.780.080.041.260.647.00Support vector machineCurrent month0.850.243.3010.670.070.042.580.5413.00Next month0.650.252.425.590.110.062.460.4613.00Month after next0.500.251.813.900.120.071.740.5012.00Current quarter−0.060.11−1.01−1.360.080.041.890.305.00Next quarter−0.150.15−1.35−1.560.130.050.660.505.00Quarter after next0.050.100.020.030.080.040.800.605.00NN (4)Current month0.420.251.462.980.120.101.860.4417.00Next month0.500.261.743.780.120.072.610.3817.00Month after next0.690.242.715.240.120.051.340.5916.00Current quarter−0.290.13−2.57−3.200.100.091.270.297.00Next quarter−0.240.17−1.70−2.030.140.100.840.437.00Quarter after next0.160.110.981.560.070.031.360.546.50NN (8, 4)Current month0.780.243.026.340.120.091.720.5417.50Next month0.950.243.829.020.100.072.770.4917.50Month after next0.960.233.998.120.110.041.500.6416.50Current quarter0.070.160.130.190.100.081.690.4010.00Next quarter0.210.220.761.180.140.081.080.5510.00Quarter after next0.500.172.585.220.090.041.320.639.50NN (16, 8, 4)Current month0.930.263.469.930.090.062.120.5516.50Next month0.960.253.658.700.100.052.800.4816.50Month after next0.970.253.668.160.110.061.710.6015.00Current quarter0.050.12−0.01−0.010.080.041.840.386.50Next quarter0.080.170.200.260.130.050.750.626.50Quarter after next0.320.132.083.450.080.041.000.696.50NN (32, 16, 8, 4)Current month0.550.242.065.320.090.062.400.4315.00Next month0.600.262.144.930.110.052.830.4015.00Month after next0.790.253.016.350.120.071.710.5714.00Current quarter−0.070.12−0.99−1.320.090.051.640.336.00Next quarter−0.190.17−1.39−1.630.150.060.680.506.00Quarter after next0.070.120.190.270.080.050.870.586.00Table 20
Investment evaluation on test data set with parameters [0.05, −0.2].
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.320.400.671.620.160.102.750.3310.50Next month0.200.250.611.190.130.081.590.4510.00Month after next0.080.290.120.200.170.091.510.429.50Current quarter−0.150.16−1.23−1.510.130.101.500.305.00Next quarter−0.450.21−2.40−2.620.190.140.610.405.00Quarter after next0.480.123.5529.130.010.013.010.863.50Support vector machineCurrent month0.170.220.520.970.120.101.580.477.50Next month0.150.180.531.040.090.081.560.477.50Month after next−0.060.17−0.61−1.160.090.071.410.386.50Current quarter0.020.12−0.25−0.440.060.061.800.384.00Next quarter−0.620.21−3.25−3.310.200.160.420.334.50Quarter after next0.320.132.147.730.040.023.690.504.00NN (4)Current month−0.250.47−0.64−1.410.210.212.620.2414.50Next month−0.480.33−1.60−2.580.200.221.490.3114.50Month after next−0.350.32−1.25−1.870.210.161.180.3813.00Current quarter−0.510.19−2.99−3.450.160.161.070.217.00Next quarter−0.860.26−3.55−3.730.240.220.510.367.00Quarter after next0.270.171.301.990.110.081.050.645.50NN (8, 4)Current month0.410.380.962.550.140.072.730.3611.00Next month0.250.250.841.750.120.071.540.4811.00Month after next0.180.280.460.760.170.071.220.5010.00Current quarter−0.180.16−1.40−1.720.130.102.350.205.00Next quarter−0.410.21−2.16−2.350.190.140.470.505.00Quarter after next0.450.133.2513.050.030.011.450.863.50NN (16, 8, 4)Current month0.390.380.892.380.140.073.250.3212.50Next month0.310.241.122.470.110.052.620.3812.50Month after next0.310.270.951.530.170.071.100.5711.50Current quarter−0.020.15−0.49−0.610.120.071.100.455.50Next quarter−0.250.20−1.50−1.640.180.120.510.555.50Quarter after next0.580.124.49NaN0.000.001.004.00NN (32, 16, 8, 4)Current month0.210.400.411.040.150.103.740.2510.00Next month0.280.240.941.990.110.051.800.4510.00Month after next0.230.240.741.230.140.071.110.568.00Current quarter−0.150.16−1.25−1.540.130.091.690.275.50Next quarter−0.370.21−2.04−2.230.190.130.590.455.50Quarter after next0.470.123.4520.960.020.014.270.754.00Table 21
Realized volatility prediction evaluation of the Wavenet in test set.
RMSEMAEpr_square0.4443180.3463831.20672−0.20431Table 22
Error evaluation of three volatility prediction models and their integrated models.
ModelRMSEMAEpr_squarefBm0.3522210.2704880.6777280.323525185arma0.350530.2692830.6712350.33000595BSSwithGammakernel0.3509990.2674310.6730350.328208943Integrated model0.3483460.2665680.6628980.338326951Specifically, Table23 which presents “Option strategy of predicting the rise and fall of volatility by thresholds” with 21 features performance better than other data sets. Neural Networks with different structures are better than Random Forest and SVM. Go through Tables 7 to 10, the more features, the Neural Networks with more complex structure performance better. For example, NN (8, 4) and NN (16, 8, 4) are better than NN (32, 16, 8, 4) in Table 23 while NN (32, 16, 8, 4) is better in Table 24.Table 23
The values of the optimal lower bound and upper bound for grid optimization and the values in the corresponding RV distribution.
LBUBQuantile0.010.68RV1.9522.8982Table 24
Option strategy of volatility prediction with dynamic thresholds based on quantile neural network and model integration algorithm.
Annual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesQuantile neural network and ensemble modelCurrent month1.050.511.996.790.150.072.680.4718.00Next month1.120.522.076.820.160.082.890.4718.00Month after next0.710.381.764.200.160.091.460.5618.00Current quarter−0.030.51−0.15−0.180.430.190.800.5510.00Next quarter0.690.341.914.990.130.061.340.6510.00Quarter after next0.510.281.633.840.120.041.090.6510.00NN (8, 4)Current month0.410.380.962.550.140.072.730.3611.00Next month0.250.250.841.750.120.071.540.4811.00Month after next0.180.280.460.760.170.071.220.5010.00Current quarter−0.180.16−1.40−1.720.130.102.350.205.00Next quarter−0.410.21−2.16−2.350.190.140.470.505.00Quarter after next0.450.133.2513.050.030.011.450.863.50Next, we adjust the threshold parameter, increase the threshold from 0.3 to 0.5, and observe the performance of the strategy. From Tables11 to 14, the relevant results show.Tables11 to 14 shows the investment evaluation with the threshold parameter set to 0.5 under four different feature sets. Compared with the results from Tables 7 to 10 it can be seen that the investment evaluation with the threshold parameter set at 0.5 is better than the investment evaluation with the threshold parameter set at 0.3. For example, the number of trading times has been reduced to 4–25.5, which is more in line with the empirically profitable daily frequency strategy; The Sortino ratio fluctuates from −3.05 to 9.11 under the threshold parameter of 0.5 which is higher than that under the threshold parameter of 0.3 floating from −3.36 to 5.19; In addition, most other indicators are also better under the threshold parameter of 0.5.Specifically, Tables11–13 show Random Forest is better than other models except for NN (16, 8, 4). This result is different from the result with parameter set at 0.3 where Neural Networks are better than Random Forest.Next, increase the threshold from 0.5 to 0.7 again and observe strategy performance from Tables15 to 18.Tables15 to 18 shows the evaluation of investment results with the threshold parameter set to 0.7. Because the threshold parameters are set too strictly and the number of trading times is too low, many favorable trading opportunities are missed. Therefore, most evaluation indicators are relatively lower than those from Tables 11 to 14.Although result with parameter set to 0.7 deteriorates, we still find that NN (8, 4) and NN (16, 8, 4) in Table17 which shows the result with 21 features (feature set M3) performance good. This indicates that M3 maybe the best feature set, and the Neural Network with proper structures is more robust than other models.Comprehensive comparison Tables7 to 18, the setting of threshold parameters can indeed greatly affect the performance of investment strategy. From the trading conditions never set the threshold parameter, then the threshold parameters are 0.3, 0.5, and finally 0.7. The trading conditions are gradually strict, and the investment result evaluation shows a trend of first rising and then falling. This shows that too loose and too strict trading restrictions are unfavorable to the performance of the strategy. Only by selecting an appropriate threshold parameter can we get the optimal investment result. This provides ideas for further improvement.


### 3.4. Option Strategy of Volatility Prediction with Dynamic Thresholds

This section further obtains clues to strategy improvement by observing the characteristics and the performance of the predicted variables. In most cases, predictive research is an inference process from known independent variables to unknown dependent variables, which can be compared with the mapping of functionsX⟶fY. In this cognitive framework, if you want to improve the final prediction accuracy of Y, there are only three breakthroughs to improve, either X, Y, or prediction method f. In this study, the research topic is the application of machine learning method in option investment, so the prediction method f has been given, then only X or Y can be improved.First, further observe the variable Y to be predicted, that is, the realized volatility. Figure1 is the histogram of realized volatility.Figure 1
Histogram of logarithmic realized volatility of CSI 300 index.From Figure1, it can be observed that the logarithmic realized volatility distribution of the real CSI 300 index is not symmetrical, and its right tail is longer than the left tail, which provides the first idea for the improvement of investment strategy: the setting of threshold parameter should be consistent with the logarithmic realized volatility distribution of the real CSI 300 index, and be set as positive threshold parameter and negative threshold parameter, the threshold parameter on both sides should also be set asymmetrically.Secondly, starting from the independent variableX, although the three feature screening algorithms have different evaluation on the feature, they all give high scores on the evaluation of RVD. Figure 2 is the histogram of feature important score of random forest.Figure 2
Feature important score of random forest.From Figure2 it can be seen that the feature important score of RVD is much higher than that of other features, and the feature important score of RVD alone is close to 0.3. The RVW ranked second is also the weekly lag term of logarithmic realized volatility. This makes it worthwhile for us to further observe the relationship between the logarithmic realized volatility of the prediction target CSI 300 index and its own lag term.Figure3 shows the partial autocorrelation coefficient diagram of logarithmic realized volatility of CSI 300 index. The partial autocorrelation coefficient is the correlation coefficient of the dependent variable with respect to its own k-order lag term. It can be seen from the figure that the correlation between the logarithmic realized volatility of CSI 300 index and the first-order lag term is still strongest. The correlation of the second-order lag term is more than half that of the first-order lag term, and the correlation shows a power-law decline with the increase of the lag order.Figure 3
Partial autocorrelation coefficient of logarithmic realized volatility of CSI 300 index.From the above analysis, it can be seen that considering the significant importance of RVD, it is a reliable idea to establish an investment strategy improvement starting from the first-order lag term RVD. Specifically, we set the threshold parameter of the strategy to some form associated with the first-order lag term of the logarithmic realized volatility of the CSI 300 index, so that the relationship between the value of the logarithmic realized volatility on dayT and the value of the logarithmic realized volatility on day T − 1 can reproduce the relationship between the logarithmic realized volatility in the partial autocorrelation coefficient and its first-order lag term. Therefore, the threshold parameter is set as a certain percentage of the logarithmic realized volatility of T − 1 day to make the threshold dynamic.After the above two steps of improvement, the two parameters of positive percentile threshold parameter and negative percentile threshold parameter are obtained. Next, the reasonable threshold parameters need to be determined through parameter optimization.The feature set is divided into training feature set, test feature set, training label and test label according to the ratio of 50%: 50%. Figures4 and 5 is the visualization of the results of grid optimization algorithm obtained by using training feature set and training label. The positive percentile threshold parameter is Positive Threshold axis, the negative percentile threshold parameter is negative threshold axis, and Obv. Is the optimization goal of grid optimization algorithm. In this study, it is set as the sum of the Sortino ratio of the investment results on the all 6 option contracts.Figure 4
Visualization of results of grid optimization algorithm 1.Figure 5
Visualization of results of grid optimization algorithm 2.Figures4 and 5 is the visual result of grid optimization of training data. The result surface of parameter optimization is observed from two different angles. The values of positive threshold and negative threshold corresponding to the high point of the surface in the figure are the optimized values of positive opening threshold and negative opening threshold. It can be observed from the figure that the positive opening threshold is about 0.05 and the negative opening threshold is about −0.2. Two threshold parameters are asymmetric, which confirms the conclusion of the histogram of logarithmic realized volatility of CSI 300 index. Let’s look at Tables 19 and 20. These two tables record the investment strategy evaluation on the training set and the test set.Table19 is the result evaluation of the investment strategy with the threshold parameter [0.05, −0.2] under the training data set. It can be seen that after grid optimization, in the evaluation of investment strategy results, the Sortino ratio ranges from 2.98 to 10.97, which is higher than the previous static threshold.Table20 is the result evaluation of the investment strategy with the threshold parameter [0.05, −0.2] under the test data set. In the test set, the Sortino ratio of NN (8, 4) and NN (16, 8, 4) are generally better, and NN (32, 16, 8, 4) on quarter after next contract reaches 20.96. The results of the test set are not inferior to those of the training set, which means that the strategy model constructed in this study has no over fitting problem in parameter optimization.Figure6 shows the net investment value curve of each algorithm on the option contract of each month. The parameters used are still [0.05, −0.2]. The curve of the solid line part is the net investment value curve on the training set, and the dotted line is the net investment value curve on the test set. As can be seen from the figure, although some curves are loss, the strategy is profitable in most cases. Especially NN (8, 4) and NN (16, 8, 4), better investment results are obtained.Figure 6
Net value curve of 6 machine learning algorithms on 6 option contracts under [0.05, −0.2] parameters.From the frameworkX⟶fY, we have improved the investment strategy from the perspective of X and Y. In the following section, we consider improving the prediction method f.


### 3.5. Benchmark: Wavenet

Since Ribeiro et al. [11] and Ribeiro et al. [12] proves the performance of the Wavenet learners in load forecasting. In this section, we test the Wavenet learners in realized volatility prediction. Take daily realized volatility of CSI 300 index from 2005 to 04-11 to 2021-11-16 to have a try. We split the data set to get train set and leave last 500 observations as test set. Train Wavenet for 100 epochs until the loss function converges, show as Figure 7.Figure 7
Loss function converges with time of epochs.Then, we use the test set to test the performance of the Wavenet in the realized volatility prediction, Figure8 show the result of prediction performance, we can see that prediction series fit the target test set to a certain extent, but it is not as good as we presume.Figure 8
Performance of the Wavenet in the test set.Table21 gives the evaluation matrix, RMSE, MAE, P ratio, R square all show that the Wavenet does not do good job in realized volatility prediction.


### 3.6. Quantile Based Neural Network and Model Integration Algorithm

Machine learning has some advantages in volatility prediction that the traditional financial economics model does not have, but the traditional financial economics model also has its irreplaceable value. This section attempts to combine two models with different theoretical backgrounds to build an integrated combination of quantile based neural network and model for volatility prediction, and then apply it to the “option strategy of volatility prediction with dynamic thresholds” in Section3.4.We choose three classical volatility prediction models, namely fractional Brownian motion (fBm), autoregressive moving average model (ARMA) and Brownian semi-stationary with gamma kernel (BSSwithGammakernel). Then, integrate the three models, that is, calculate the average prediction value of the three models as the final prediction. The prediction errors of the four models are shown in Table22.It can be seen from Table22 that the utility brought by the integration model is positive. The RMSE and other indicators of the integrated model are improved compared with the three basic models.Next, continue to observe the performance of model prediction error under different quantiles of realized volatility. Figure9 describes the prediction error of three volatility prediction models and the integrated model in 10 quantiles from low to high.Figure 9
Quantile error of realized volatility of three volatility prediction models and their integrated model.The three volatility prediction models and the integrated model have similar performance. The RMSE is low in the middle and high on both sides, which means that the prediction efficiency of the distribution tail of realized volatility by the traditional financial and economic model is not high. It is understandable that most of the traditional financial and economic models are linear structures, so the nonlinear mapping ability is poor. However, at the tail of the volatility distribution, the change of volatility is nonlinear, which can be improved by combining neural network.The calculation process of the quantile based neural network and model integration algorithm is as follows:Training process: first, set the grid range of the quantile of the left tail and the quantile of the right tail, and then train the model under the parameters of different grids. When the two grid parameters are selected, the neural network is trained separately with the data of the tail from the two quantiles, while the traditional financial and economic model is trained with full sample data in the way of expanding, and the one-step prediction sequence of realized volatility is obtained. Then, minimize the RMSE, the two optimal grid parameters are selected.Prediction process: judge whether the volatility value is within or outside the quantile parameters on both sides according to the realized volatility of the previous day. If it is outside the two quantile parameters, the neural network is called for training and prediction. If the realized volatility of the previous day is within the two quantiles, the prediction sequence of the integrated model of the traditional financial and economic model is called for prediction.The pseudo code of the above algorithm is as follows:Training process:Setting the starting data window paddingFor: the date series of realized volatility from paddingTrain fBm, ARMA and BSSwithGammakernel models day by day in the way of expanding in the set window, and make one-step prediction, and then take the mean value of the predicted values of the three models as the predicted value of the integrated modelEndSet the value range of quantile parameters on both sides to build a two-dimensional parameter gridFor: each grid pointSelect two quantile parameters of a grid pointFor: the date series of realized volatility from paddingJudge whether the realized volatility of the previous day falls between two quantile parameters. If yes, skip. If no, train the neural network with data other than the two quantiles in the realized volatility from the first day to the previous day. Then the trained neural network is used to predict the realized volatility of next one dayEndEndFor: each grid pointSelect the one-step prediction sequence of neural network under each grid point parameter and fill the skipped realized volatility prediction value with the prediction of the integrated model on that day, so as to obtain a complete Realized Volatility Prediction sequenceEndCompare the RMSE of all realized volatility prediction sequences, and finally select the grid parameter with the smallest RMSEForecast process:The grid parameter with the smallest RMSE is selected as the quantile parameters on both sides of the distributionFor: the date series of realized volatility from paddingJudge whether the realized volatility of the previous day falls between two quantiles. If yes, the prediction of the integrated model is used for next day prediction. If not, the neural network is trained with the data other than the two quantiles in the distribution of realized volatility from the first day to the previous day. Then the trained neural network is used to predict the realized volatility for next dayEndFigure10 shows the results of grid optimization. The color bar on the right side of the figure shows the colors corresponding to different RMSEs. It can be seen that the darker areas corresponding to Lower Bound tend to have smaller values, while the darker areas corresponding to Upper Bound tend to have larger trends. From Tables 2 and 19, two optimal parameters with the smallest RMSE are selected.Figure 10
Grid optimization results of quantile parameters on both sides.Table23 shows that the value of the optimal Lower Bound quantile is 0.01, the value of the optimal Upper Bound quantile is 0.68, and the value of the realized volatility distribution is 1.952 and 2.8982, respectively.Then, according to Table23, the realized volatility is predicted by using the quantile based neural network and model integration algorithm, the prediction sequence of volatility is obtained, and the prediction error is measured, as shown in Table 25.Table 25
Prediction error of quantile based neural network and model integration algorithm.
RMSEMAEpr_square0.3197240.2606460.4913090.515611042From Table25, it can be seen that the “Quantile based neural network and model integration algorithm” can be compared with the RMSE and other indicators of the Integrated model in Table 22. All measurements in Table 25 are improved compared with those of integrated model in Table 22, which means that it is of positive significance to combine the neural network with the integrated model by using the quantile parameters to segment the distribution of volatility. Besides, the performance of “Quantile based neural network and model integration algorithm” is much better than that of the Wavenet showed in Table 21.Although Table25 shows that the prediction error of quantile based neural network and model integration algorithm has a positive improvement compared with the integrated model, whether the improvement is significant still needs to be verified by a statistical test. Therefore, Table 26 is given to show the results of analysis of ANOVA.Table 26
ANOVA analysis of quantile based neural network and model integration algorithm and integrated model.
SourceSSdfMSFp value (F)Group0.13032910.1303292.8425878759.40E-02Error6.4188241400.045849[][]Sum up6.549153141[][][]From the test results of ANOVA, thep value is significant at the 10% confidence level. The statistical significance of the improvement is not too high, which can also be explained. This is because the integrated model itself is an improvement on the single model. Therefore, although the superimposed neural network does improve the final result, the significance of the improvement will decrease.Table24 shows the investment results of the option strategy that brings the quantile neural network and model integration algorithm proposed in this section into the option strategy of volatility prediction with dynamic thresholds in the previous section, and the strategy parameters still use [0.05, −0.2]. Compared with using neural NN (8, 4) alone as the volatility prediction for the option strategy, we can see that Sortino ratio, annual return, max drawdown and other indicators have improved.Finally, Figure11, the net value curve of the investment results is visualized. It can be seen from Figure 11 that the investment results are good in most option contracts, especially in the contracts of the current month and the next month. This is because the option contract month is closer, and the expiration time has less impact on the option price, so the exposed volatility risk has a greater impact on the investment strategy. In this way, the quantile neural network and model integration algorithm for the purpose of predicting volatility has greater advantages.Figure 11
The investment results of option strategy of volatility prediction with dynamic thresholds in different option contracts.


## 3.1. Feature Introduction and Selection

The prediction ability of machine learning algorithm largely depends on the effectiveness of training features. This section comprehensively refers to Vrontos et al. [2] and Christensen et al. [22] and collects 30 features, including the multi-factor model from the classical Fama French, the factors based on the stock index, the factors based on the national debt and the interest rate, the indexes based on financial economy, and the lag terms of realized volatility. Table 1 summarizes the introduction of 30 features.Table 1
Name, description, and data set of 30 features.
FeatureExplanationFeature setrvdDaily lag of HS300 RVM1,M2,M3,M4rvwWeekly lag of HS300 RVM1,M2,M3,M4momentum_5HS300 momentum factor with parameters of 5 daysM1,M2,M3,M4hmlFama-French HML factor daily portfolio returns–high minus low value factorM4mktHS300 market factor portfolio returnsM4mkt_premiumMarket factor – 1 year treasury yieldM3,M4bicBusiness index of ChinaM2,M3,M4momentum_60HS300 momentum factor with parameters of 60 daysM1,M2,M3,M4treasury_bond_momentum_51 year treasury bond momentum factor with parameters of 5 daysM3,M4rvmMonthly lag of HS300 RVM2,M3,M4epuChinese economic policy uncertainty index,M4treasury_bond_momentum_601 year treasury bond momentum factor with parameters of 60 daysM1,M2,M3,M4Skewness _120Skewness of return of HS300 with parameters of 120 daysM1,M2,M3,M4hisSquared hang seng index log-returnM3,M4treasury_bond_momentum_1201 year treasury bond momentum factor with parameters of 120 daysM1,M2,M3,M4mkt_premium_lag1Market factor: 1 year treasury yield lagged by 1 dayM2,M3,M4ShiborShibor overnight interest rateM1,M2,M3,M41mmHS300 1 week momentumM2,M3,M4treasury_bond_spread10 year treasury yield−1 year treasury yieldM3,M4treasury_bond_momentum_201 year treasury bond momentum factor with parameters of 20 daysM3,M4skewness_20Skewness of return of HS300 with parameters of 20 daysM3,M4SC futureShanghai oil future indexM3,M4AmtdifFirst differences of daily trading amount of HS300 stock indexM4momentum_120HS300 momentum factor with parameters of 120 daysM3,M4cmaFama-French CMA portfolio returns–conserve minus aggressive investment factorM4smbFama-French SMB portfolio returns–small minus big size factorM4momentum_20HS300 momentum factor with parameters of 20 daysM3,M4rmwFama-French RMW portfolio returns–robust minus weak quality factorM4skewness_60Skewness of return of HS300 with parameters of 60 daysM4The prediction ability to the realized volatility of the 30 features are uneven, which needs to be further screened and compared. In this study, 30 features are scored and sorted by three methods: Select-K-Best method, Random Forest Regressor feature importance, Permutation Importance.Select-K-Best method measures the effectiveness of features through a certain index, such as Chi-square value, information entropy and so on. The algorithm calculates the evaluation index of each feature from an ex-ante perspective, and rank features by evaluation index.Random Forest Regressor feature importance is based on the contribution of features in each decision tree. The contribution of a feature is measured by calculating the Gini index or out of bag data (OOB) error rate of the feature at each split point of each decision tree, and then the contributions of all decision trees are added as the importance evaluation of the feature.Permutation Importance scores the importance of features from an ex-post perspective. First, all the features are used to train the model, then the features are disrupted one by one, and the disrupted features are used to train the model again. If the effectiveness of the disrupted model is reduced compared with the model before disruption, it means that the feature before disruption is an effective feature. The more the effectiveness of model decreases, the more important the feature is, which is used as the basis for feature scoring.The ideas of the above three feature evaluation are different, so the ranking of feature importance will also be different. Table2 gives the feature scores and weights of the three algorithms. Select-K-Best method and Random Forest Regressor feature importance directly give the score or weight of the feature, while Permutation Importance gives the mean and standard deviation of the feature weight. According to these weights and scoring data, the features can be sorted. By taking the intersection of the top 17 features of the three algorithms, 8 features can be obtained to form the feature set M1. Taking the intersection of the top 20 features of the three algorithms, 12 features can be obtained to form the feature set M2. Taking the intersection of the top 27 features of the three algorithms, 21 features can be obtained to form the feature set M3. Finally, all 30 features form the feature set M4. The attribution of each feature in the four feature sets is shown in the feature sets column of Table 2. Top features in different algorithms are highlighted in each column.Table 2
Scoring and ranking of feature importance by 3 different feature selection algorithms.
Feature nameSelect-K-BestRandom Forest Regressor feature importancePermutation ImportanceFeature setrvd163.09130.2905970.2228 ± 0.1104M1,M2,M3,M4rvw86.812950.0808110.0158 ± 0.0145M1,M2,M3,M4momentum_515.336580.0449440.0119 ± 0.0199M1,M2,M3,M4hml12.795450.030411−0.0044 ± 0.0078M4mkt10.983620.023517−0.0057 ± 0.0058M4mkt_premium10.976160.027876−0.0017 ± 0.0207M3,M4bic10.569030.0208150.0297 ± 0.0154M2,M3,M4momentum_608.6242850.0390440.0437 ± 0.0221M1,M2,M3,M4treasury_bond_momentum_56.9908730.013640.0000 ± 0.0020M3,M4rvm6.6544990.0336350.0015 ± 0.0061M2,M3,M4epu6.543260.0085320.0015 ± 0.0047M4treasury_bond_momentum_606.083490.0221050.0061 ± 0.0117M1,M2,M3,M4skewness_1205.7443280.0237140.0096 ± 0.0097M1,M2,M3,M4his5.6088240.0148280.0005 ± 0.0050M3,M4treasury_bond_momentum_1204.2273070.0232980.0246 ± 0.0178M1,M2,M3,M4mkt_premium_lag13.5037030.0186770.0025 ± 0.0087M2,M3,M4shibor3.4894080.0273680.0111 ± 0.0095M1,M2,M3,M41mm1.5741030.0192350.0070 ± 0.0114M2,M3,M4treasury_bond_spread1.5326280.0161240.0088 ± 0.0050M3,M4treasury_bond_momentum_201.4622630.015703−0.0006 ± 0.0065M3,M4skewness_201.3022150.0250740.0236 ± 0.0078M3,M4SC future1.2137010.0334170.0098 ± 0.0073M3,M4amtdif0.5894990.016307−0.0047 ± 0.0053M4momentum_1200.3581620.0160340.0064 ± 0.0091M3,M4cma0.2520260.0075630.0001 ± 0.0016M4smb0.2520260.009307−0.0001 ± 0.0018M4momentum_200.0278060.0272420.0084 ± 0.0214M3,M4rmw0.0239640.016459−0.0029 ± 0.0159M4skewness_600.0121750.0236040.0087 ± 0.0101M4mkt_premium_lag20.0055490.0301170.0051 ± 0.0171M4


## 3.2. Option Strategy for Directly Predicting the Rise and Fall of Volatility

In this section, the four feature sets M1, M2, M3, and M4 constructed in Table1 are used to train six machine learning models. The six machine learning models are random forest, support vector machine and four forward neural networks with different network structures, labeled random forest, SVM, NN (4), NN (8, 4), NN (16, 8, 4) and NN (32, 16, 8, 4). The structures of four neural networks are: NN (4) has a hidden layer with 4 neurons, NN (8, 4) has a two-layer hidden layer with 8 and 4 neurons, NN (16, 8, 4) has a three-layer hidden layer with 16, 8 and 4 neurons, NN (32, 16, 8, 4) has a four-layer hidden layer with 32, 16, 8 and 4 neurons.In the BS model, the option pricing is mainly determined by five Greek letters, which are respective sensitivity to the price change of the underlying assetΔ, implied volatility of options σ, differential of Δ which is г, sensitivity to changes in interest rates ρ, sensitivity to expiration θ。The maturity time is naturally expiring, and the interest rate usually changes little in the short term, the second derivative of the price change of the underlying asset г is the acceleration of the underlying asset price change. Therefore, the most significant impact on the option price change is the price change sensitivity Δ and implied volatility of options σ. The strategy of this paper mainly focuses on the implied volatility of options σ.The above 6 machine learning models and 4 sets of feature sets are used to train and predict the realized volatility respectively, and then construct the “option strategy for directly predicting the rise and fall of volatility”: if the machine learning model predicts the rise of realized volatility the next day, the portfolio of call and put option at the money is bought at the closing time of the day; If the machine learning model predicts the realized volatility decline of the next day, the portfolio of call and put option at the money is sold at the closing time of the day, close the positions of the previous day at the closing time of the next day, and conduct the trading according to this rule repeatedly day by day. This strategy hedgesΔ risk by buying or selling both call and put options at the same time, because the transaction is traded at the same price it also can hedge г Risk to a certain extent, so the implied volatility risk σ is exposed. Then, if we can accurately predict the rise and fall direction of realized volatility, we can make profits through this strategy. The investment evaluation results of “option strategy for directly predicting the rise and fall of volatility” are summarized in Tables 3–6.Table 3
Option strategy for directly predicting the rise and fall of volatility (use 8 features M1).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.110.520.120.130.470.351.060.5059.00Next month0.130.420.200.210.410.301.040.5159.00Month after next0.020.42−0.08−0.090.390.291.030.5054.50Current quarter−1.130.55−2.15−2.080.570.490.660.3828.00Next quarter−1.770.59−3.10−3.020.600.590.480.4228.50Quarter after next−0.230.53−0.54−0.510.550.320.620.5724.50Support vector machineCurrent month0.230.470.380.410.430.291.140.5059.00Next month0.230.400.460.480.380.261.180.4959.00Month after next−0.030.41−0.20−0.220.370.240.960.5054.50Current quarter−0.750.47−1.69−1.650.490.390.730.4128.00Next quarter−1.460.54−2.83−2.750.550.530.480.4628.50Quarter after next−0.280.54−0.62−0.590.570.330.590.5724.50NN (4)Current month0.150.510.200.220.480.320.980.5359.00Next month0.230.410.440.440.410.271.000.5359.50Month after next−0.010.43−0.15−0.160.400.300.970.5054.50Current quarter−0.740.47−1.67−1.610.490.370.690.4328.00Next quarter−1.490.54−2.85−2.790.550.530.500.4428.50Quarter after next−0.330.55−0.70−0.660.580.330.620.5525.00NN (8, 4)Current month−0.100.57−0.27−0.300.510.421.090.4759.00Next month−0.260.48−0.64−0.680.450.381.120.4459.00Month after next−0.170.45−0.50−0.550.400.311.040.4754.50Current quarter−0.990.53−1.98−1.910.550.450.680.3928.00Next quarter−2.070.60−3.53−3.410.620.650.430.4028.50Quarter after next−0.340.51−0.77−0.730.540.330.610.5524.50NN (16, 8, 4)Current month0.000.55−0.09−0.100.510.351.030.4959.00Next month0.030.45−0.03−0.040.420.271.130.4759.00Month after next−0.140.46−0.41−0.460.410.271.100.4654.50Current quarter−0.940.51−1.94−1.890.520.430.740.3828.00Next quarter−1.710.58−3.06−2.950.600.580.430.4628.50Quarter after next−0.350.55−0.72−0.690.580.340.610.5524.50NN (32, 16, 8, 4)Current month−0.580.71−0.89−0.940.670.481.040.4359.00Next month−0.810.59−1.45−1.510.570.481.030.4159.00Month after next−0.520.49−1.17−1.210.470.330.920.4654.50Current quarter−1.120.52−2.26−2.200.530.480.680.3628.00Next quarter−2.240.65−3.57−3.510.650.680.550.3328.50Quarter after next−0.700.64−1.18−1.130.670.430.610.4924.50Table 4
Option strategy for directly predicting the rise and fall of volatility (use 12 features M2).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.000.53−0.09−0.100.470.381.040.4959.00Next month−0.050.43−0.24−0.250.410.341.040.4859.00Month after next−0.020.42−0.16−0.170.380.301.050.4955.00Current quarter−1.040.53−2.06−1.990.550.460.640.3928.00Next quarter−1.850.60−3.16−3.090.620.610.510.4028.50Quarter after next−0.150.50−0.39−0.370.520.310.720.5524.50Support vector machineCurrent month−0.240.51−0.58−0.550.540.450.940.4859.00Next month−0.140.43−0.44−0.450.420.361.060.4759.00Month after next−0.230.41−0.68−0.690.400.350.770.5354.50Current quarter−0.670.47−1.54−1.480.480.380.680.4528.00Next quarter−1.250.51−2.54−2.510.520.490.530.4628.50Quarter after next−0.470.54−0.97−0.920.570.360.600.5324.50NN (4)Current month0.090.530.070.080.480.361.120.4859.00Next month0.090.430.100.100.420.321.020.5159.00Month after next−0.050.44−0.22−0.240.410.320.990.5054.50Current quarter−0.910.51−1.90−1.820.530.440.610.4328.00Next quarter−1.770.60−3.07−2.960.620.600.450.4428.50Quarter after next−0.440.57−0.87−0.820.600.380.570.5524.50NN (8, 4)Current month−0.350.66−0.60−0.660.610.451.020.4659.00Next month−0.260.50−0.62−0.630.490.360.880.5059.00Month after next−0.410.51−0.90−0.960.470.340.880.4954.50Current quarter−0.900.51−1.88−1.820.520.420.720.3928.00Next quarter−1.480.59−2.59−2.560.600.520.550.4428.50Quarter after next−0.440.58−0.85−0.800.610.340.530.5724.50NN (16, 8, 4)Current month0.220.470.370.420.410.291.170.4959.00Next month0.030.42−0.06−0.060.410.290.950.5259.50Month after next−0.010.40−0.16−0.170.370.270.910.5254.50Current quarter−0.740.47−1.68−1.610.490.370.640.4528.00Next quarter−1.340.52−2.67−2.630.530.490.520.4628.50Quarter after next−0.310.54−0.66−0.630.570.320.630.5525.00NN (32, 16, 8, 4)Current month−0.310.67−0.55−0.600.610.451.040.4659.00Next month−0.500.55−1.01−1.060.520.421.010.4459.00Month after next−0.260.47−0.65−0.700.440.320.970.4854.50Current quarter−1.150.57−2.12−2.050.590.500.620.3928.00Next quarter−2.460.74−3.40−3.380.740.710.580.3328.50Quarter after next−0.990.68−1.54−1.480.700.490.660.4324.50Table 5
Option strategy for directly predicting the rise and fall of volatility (use 21 features M3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.020.53−0.13−0.140.480.401.020.4959.00Next month−0.120.44−0.38−0.400.430.371.000.4859.00Month after next−0.080.43−0.31−0.340.390.301.010.4954.50Current quarter−1.130.55−2.15−2.080.570.490.660.3828.00Next quarter−1.730.58−3.08−2.990.590.580.490.4228.50Quarter after next−0.210.52−0.51−0.480.550.320.630.5724.50Support vector machineCurrent month0.050.490.000.000.530.390.990.5159.00Next month0.040.42−0.02−0.020.410.321.130.4759.00Month after next0.170.400.300.310.380.250.870.5654.50Current quarter−0.350.46−0.86−0.840.470.320.790.4828.00Next quarter−0.760.49−1.65−1.670.480.350.630.4928.50Quarter after next−0.500.54−1.02−0.980.560.380.630.5124.50NN (4)Current month−0.200.59−0.43−0.470.530.401.080.4659.00Next month−0.280.48−0.69−0.730.450.351.100.4459.00Month after next−0.050.43−0.24−0.250.410.280.850.5355.00Current quarter−0.880.50−1.87−1.820.510.410.720.3928.00Next quarter−1.760.59−3.09−3.030.600.590.520.4028.50Quarter after next−0.240.53−0.55−0.530.550.320.720.5324.50NN (8, 4)Current month0.180.480.280.310.430.391.190.4859.00Next month0.160.390.300.310.370.331.140.4959.00Month after next0.290.360.660.710.330.241.050.5354.50Current quarter−0.650.45−1.54−1.500.460.390.780.4128.00Next quarter−1.010.46−2.31−2.300.460.470.580.4628.50Quarter after next0.090.450.080.080.470.280.690.6124.50NN (16, 8, 4)Current month0.510.431.081.220.380.221.260.5259.00Next month0.460.361.141.220.340.221.200.5359.00Month after next0.340.380.780.880.330.211.070.5354.50Current quarter−0.170.38−0.57−0.550.390.270.740.5428.00Next quarter−0.590.42−1.54−1.540.420.360.550.5428.50Quarter after next0.100.450.100.090.480.250.530.6724.50NN (32, 16, 8, 4)Current month0.270.470.460.520.420.261.330.4759.00Next month0.420.380.981.030.360.201.290.5059.00Month after next0.340.360.800.860.340.171.010.5554.50Current quarter−0.430.42−1.14−1.110.430.320.720.4828.00Next quarter−1.260.51−2.58−2.540.520.470.540.4628.50Quarter after next−0.290.54−0.63−0.600.570.320.590.5724.50Table 6
Option strategy for directly predicting the rise and fall of volatility (use 30 features M4).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.050.52−0.01−0.010.460.381.100.4859.00Next month−0.050.43−0.24−0.250.410.341.080.4759.00Month after next0.000.42−0.12−0.130.380.291.020.5054.50Current quarter−0.980.52−1.99−1.910.540.440.620.4128.00Next quarter−1.730.58−3.07−2.990.590.580.490.4228.50Quarter after next−0.080.49−0.27−0.260.510.310.700.5725.00Support vector machineCurrent month−0.010.52−0.12−0.110.550.411.030.4959.00Next month0.020.44−0.06−0.060.430.331.120.4759.00Month after next0.220.400.430.440.390.250.830.5855.00Current quarter−0.420.48−0.98−0.960.490.330.800.4628.00Next quarter−0.740.49−1.63−1.640.480.340.590.5128.50Quarter after next−0.620.57−1.18−1.140.590.390.630.4924.50NN (4)Current month0.060.540.010.010.510.330.930.5359.00Next month−0.050.44−0.22−0.220.440.320.880.5359.50Month after next−0.350.49−0.81−0.850.470.330.930.4854.50Current quarter−0.880.50−1.85−1.790.520.440.720.3928.00Next quarter−1.900.57−3.43−3.280.590.630.390.4428.50Quarter after next−0.440.53−0.92−0.860.570.370.480.5925.00NN (8, 4)Current month0.150.520.200.200.500.280.910.5459.00Next month0.120.440.160.160.430.271.000.5259.00Month after next−0.050.44−0.23−0.240.420.270.860.5354.50Current quarter−0.620.45−1.48−1.430.470.340.710.4528.00Next quarter−1.330.51−2.68−2.590.530.500.440.4928.50Quarter after next−0.380.56−0.77−0.730.590.350.550.5724.50NN (16, 8, 4)Current month0.100.550.090.090.530.270.880.5459.00Next month0.170.440.270.280.420.231.090.5059.50Month after next−0.090.46−0.31−0.320.440.250.810.5454.50Current quarter−0.580.45−1.41−1.360.460.330.680.4628.00Next quarter−0.980.50−2.07−2.030.510.390.520.5128.50Quarter after next−0.330.55−0.70−0.660.580.340.570.5724.50NN (32, 16, 8, 4)Current month0.330.470.600.670.420.211.320.4759.00Next month0.390.400.850.950.360.171.500.4559.50Month after next0.310.380.670.750.340.151.130.5154.50Current quarter−0.940.51−1.93−1.890.520.430.770.3828.00Next quarter−0.940.47−2.13−2.140.460.410.560.4928.50Quarter after next−0.030.48−0.16−0.150.500.270.670.5924.50From Tables3–6, the evaluation results of 6 machine learning models under 6 different expiration month option contracts by 4 different feature sets are given respectively. A total of 9 evaluation indicators including annual return, annual risk, Sharpe ratio, Sortino ratio, downside risk, max drawdown, profit loss ratio, win ratio and trading times are recorded. It can be seen that in the annual return column, except for a few current month and next month option contracts of Random Forest and Neural Network, most machine learning algorithms are doing badly on the 4 data sets; The annualized standard deviation fluctuates between 0.36 and 0.69, which is far greater than the annualized rate of return; Sharpe ratio and Sortino ratio are basically negative numbers, which are highlighted in a bold font; Downside risk, max drawdown, profit-loss ratio and win ratio are also not optimistic; The lowest number of trading times is 24.5 and the highest is 59.The above 9 evaluation indicators can almost unanimously judge that the “option strategy of directly predicting the rise and fall of volatility” is a failed investment strategy, which requires rethinking the strategy design and further improving and redesigning the strategy.According to the investment experience in the industrial practice, most investment strategies with stable profitability have a certain tradeoff among profit-loss ratio, win ratio, trading times and data sampling frequency (this strategy is daily frequency). The trading frequency of the strategy, that is, the number of transactions, is often lower than the data sampling frequency. Taking the speculative strategy of daily frequency as an example, the position holding period is usually even longer at the weekly frequency. In contrast, the “option strategy of directly predicting the rise and fall of volatility” has a maximum of 59 transactions in half a year, which is obviously much higher than the practical experience of the industry. If the trading frequency of an investment strategy is high, it will put forward higher requirements for the trading win rate. Otherwise, under a short position holding period, the price change of the investment target is not large enough. Multiplying by the same win rate and profit-loss ratio, the low trading frequency strategy means that the expected return is higher than that of the high trading frequency strategy. In addition, higher trading frequency will bring higher trading costs, then the comprehensive result is that the strategy with high trading frequency will be inferior to the strategy with low trading frequency. Observe Tables3–6, according to the profit -loss ratio and win ratio, the “option strategy of directly predicting the rise and fall of volatility” does not dominate these two indicators, so it is obvious that the trading frequency is too high for the current strategy.This provides an idea for the improvement of investment strategy: reduce the trading times by adding some restrictions on trading conditions, so as to improve the profit loss ratio and win ratio of investment strategy.


## 3.3. Option Strategy of Predicting the Rise and Fall of Volatility by Threshold

It can be seen from the analysis in the previous section that it is necessary to reduce the trading times by adding some restrictions on trading conditions, so the direct solution is to set a threshold parameter: it will buy or sell option portfolio only when the rise and fall of the realized volatility predicted by the machine learning model relative to the real realized volatility of yesterday exceeds a certain threshold parameter, otherwise the strategy does not implement. Other strategy conditions are the same as the “option strategy of directly predicting the rise and fall of volatility.” In this way, the trading times can be significantly reduced by increasing the threshold parameter.Drawing on the design of confidence strategy 2 and confidence strategy 3 of Vrontos et al. [2] on VIX Index Investment. He set a probability interval based on the classification confidence probability of machine learning to trade selectively. The “option strategy of predicting the rise and fall of volatility by thresholds” in this section is essentially similar to that of confidence strategy 2 and confidence strategy 3 designed by Vrontos et al. [2]. We use the predicted volatility obtained by machine learning algorithm and set a threshold interval as the trading condition, only when predicted volatility exceeds threshold parameter, the strategy triggers.Tables7 to 18 summarizes the strategy evaluation results of “option strategy of predicting the rise and fall of volatility by thresholds” under the setting of different threshold parameters.Table 7
Option strategy of predicting the rise and fall of volatility by threshold (use 8 features M1, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.340.340.851.780.160.171.670.4435.00Next month0.300.280.881.590.150.121.630.4435.00Month after next0.250.290.701.130.180.141.270.4932.50Current quarter−0.180.18−1.32−1.620.140.141.100.3915.50Next quarter−0.950.31−3.24−3.360.300.380.480.4116.00Quarter after next0.120.180.400.550.130.111.090.5412.00Support vector machineCurrent month0.330.211.332.340.120.081.260.5625.00Next month0.390.211.612.910.120.091.440.5425.00Month after next0.190.190.731.170.120.121.110.5522.00Current quarter0.100.130.400.640.080.081.570.4612.00Next quarter−0.210.21−1.23−1.430.180.140.750.4812.50Quarter after next0.310.161.702.630.100.101.100.6710.50NN (4)Current month−0.300.59−0.59−0.700.490.431.130.4232.00Next month−0.180.45−0.52−0.580.400.331.130.4432.00Month after next−0.280.43−0.76−0.870.380.260.940.4729.00Current quarter−0.950.50−2.00−1.990.500.430.560.3515.50Next quarter−1.280.50−2.64−2.620.510.470.360.4716.00Quarter after next−0.470.56−0.93−0.910.570.330.410.5913.50NN (8, 4)Current month0.400.301.152.560.140.111.860.4530.00Next month0.370.251.302.310.140.081.530.4930.50Month after next0.330.251.161.860.150.081.150.5528.00Current quarter−0.070.17−0.69−0.880.130.101.070.4514.50Next quarter−0.510.26−2.17−2.370.240.260.580.4715.00Quarter after next0.280.151.522.480.090.061.150.6312.00NN (16, 8, 4)Current month0.540.311.613.690.130.121.730.4932.50Next month0.440.261.552.890.140.091.800.4632.50Month after next0.360.251.272.300.140.081.500.4929.50Current quarter−0.120.16−1.03−1.450.110.111.300.3814.50Next quarter−0.490.25−2.20−2.430.220.220.650.4315.00Quarter after next0.180.180.721.020.120.101.120.5612.50NN (32, 16, 8, 4)Current month−0.250.45−0.68−0.690.430.320.900.4830.50Next month−0.210.41−0.65−0.690.380.291.050.4430.50Month after next−0.210.37−0.69−0.730.350.240.800.5127.50Current quarter−0.740.46−1.72−1.710.460.360.550.3713.50Next quarter−1.720.56−3.16−3.140.560.580.410.3214.00Quarter after next−0.760.65−1.26−1.260.640.450.550.4511.00Table 8
Option strategy of predicting the rise and fall of volatility by threshold (use 12 features M2, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.320.340.791.640.160.181.560.4634.00Next month0.270.280.791.420.150.141.520.4634.50Month after next0.270.280.771.240.180.121.220.5131.50Current quarter−0.160.18−1.19−1.470.140.131.070.4116.00Next quarter−0.870.30−3.04−3.180.290.360.490.4216.50Quarter after next0.140.170.510.690.130.101.040.5612.50Support vector machineCurrent month0.190.170.851.270.110.061.010.5920.50Next month0.260.171.231.910.110.071.110.5920.50Month after next0.040.16−0.06−0.090.110.090.970.5318.00Current quarter0.020.12−0.28−0.410.080.071.300.4510.00Next quarter−0.250.20−1.48−1.660.180.150.550.5210.50Quarter after next0.250.141.402.040.100.100.950.689.50NN (4)Current month0.010.51−0.09−0.100.440.321.090.4836.50Next month−0.010.40−0.15−0.160.370.271.050.4936.00Month after next−0.140.40−0.48−0.520.360.250.870.5133.50Current quarter−0.650.45−1.56−1.540.450.350.540.4516.50Next quarter−1.010.47−2.25−2.250.470.400.500.4417.00Quarter after next−0.230.50−0.56−0.550.500.280.470.6214.50NN (8, 4)Current month−0.230.58−0.49−0.570.490.361.140.4337.00Next month−0.220.45−0.59−0.640.410.291.020.4637.50Month after next−0.280.43−0.77−0.830.390.280.850.4934.50Current quarter−1.000.52−2.03−2.020.520.450.660.3419.00Next quarter−1.740.59−3.03−3.000.600.580.450.3819.50Quarter after next−0.570.60−1.04−1.020.610.390.520.5316.00NN (16, 8, 4)Current month0.000.51−0.09−0.110.430.331.060.4937.00Next month−0.100.42−0.35−0.380.390.310.910.5136.50Month after next−0.240.42−0.69−0.740.390.290.730.5433.50Current quarter−0.760.47−1.73−1.710.470.370.620.4118.50Next quarter−1.280.48−2.78−2.740.490.480.380.4719.00Quarter after next−0.290.51−0.68−0.650.520.300.470.6116.50NN (32, 16, 8, 4)Current month0.430.331.172.370.160.131.630.4735.00Next month0.400.271.322.270.160.101.420.5134.50Month after next0.370.271.171.910.160.091.310.5232.00Current quarter−0.220.19−1.45−1.770.150.160.980.4116.00Next quarter−0.810.30−2.89−3.050.280.330.520.4216.50Quarter after next0.190.180.801.050.130.090.720.6713.50Table 9
Option strategy of predicting the rise and fall of volatility by threshold (use 21 features M3, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.290.340.711.490.160.181.600.4532.50Next month0.260.280.761.370.150.131.580.4532.50Month after next0.220.280.620.990.170.121.220.5030.00Current quarter−0.190.17−1.39−1.680.140.141.000.4015.00Next quarter−0.790.30−2.85−2.990.280.330.470.4515.50Quarter after next0.140.170.510.690.130.101.040.5612.50Support vector machineCurrent month0.240.161.141.720.110.050.980.6219.50Next month0.330.171.672.720.100.051.260.5919.50Month after next0.130.150.520.740.110.071.020.5617.00Current quarter0.060.120.080.120.080.071.220.5010.00Next quarter−0.120.19−0.90−1.040.170.110.580.5710.50Quarter after next0.250.141.382.000.100.100.950.689.50NN (4)Current month−0.070.54−0.22−0.250.460.331.220.4437.50Next month−0.110.44−0.38−0.410.400.251.150.4537.00Month after next0.010.38−0.11−0.120.360.210.690.5934.50Current quarter−0.660.45−1.56−1.540.460.340.600.4519.00Next quarter−1.310.49−2.78−2.720.500.490.410.4419.50Quarter after next−0.310.51−0.71−0.690.530.310.470.6116.50NN (8, 4)Current month0.100.490.100.120.420.341.110.4934.50Next month0.030.40−0.05−0.050.360.291.220.4634.00Month after next0.150.370.260.300.330.251.030.5232.50Current quarter−0.490.42−1.29−1.290.420.330.720.4318.50Next quarter−1.060.45−2.49−2.490.450.470.430.4719.00Quarter after next−0.240.49−0.59−0.580.490.300.550.5815.50NN (16, 8, 4)Current month0.590.291.864.320.120.091.920.4933.50Next month0.570.232.274.400.120.081.770.5233.50Month after next0.600.242.344.190.130.081.340.5931.50Current quarter0.200.141.071.480.100.071.200.5914.50Next quarter−0.270.22−1.45−1.550.210.220.430.6015.00Quarter after next0.380.122.635.400.060.051.100.7411.50NN (32, 16, 8, 4)Current month−0.170.44−0.51−0.530.420.310.840.5133.50Next month−0.030.39−0.20−0.210.370.251.010.4933.50Month after next−0.030.37−0.21−0.220.340.240.720.5829.50Current quarter−0.620.44−1.52−1.510.440.330.530.4514.50Next quarter−1.290.48−2.79−2.740.490.480.310.4715.00Quarter after next−0.450.54−0.92−0.900.550.330.370.6213.00Table 10
Option strategy of predicting the rise and fall of volatility by threshold (use 30 features M4, threshold = 0.3).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.350.330.911.890.160.151.590.4633.50Next month0.310.270.941.690.150.111.550.4633.50Month after next0.260.270.791.260.170.121.190.5231.00Current quarter−0.160.18−1.19−1.470.140.131.070.4116.00Next quarter−0.870.30−3.04−3.180.290.360.490.4216.50Quarter after next0.140.170.510.690.130.101.040.5612.50Support vector machineCurrent month−0.230.39−0.72−0.700.390.250.480.6119.00Next month−0.090.34−0.40−0.400.340.220.650.5819.00Month after next−0.260.33−0.96−0.950.330.220.540.5617.00Current quarter−0.430.39−1.23−1.220.390.250.440.5010.00Next quarter−0.550.37−1.62−1.620.370.300.310.5710.50Quarter after next−0.320.46−0.80−0.790.470.270.280.689.50NN (4)Current month0.360.320.991.970.160.121.320.5232.00Next month0.360.261.222.100.150.101.320.5231.50Month after next0.230.280.671.060.170.101.280.4930.00Current quarter−0.300.19−1.87−2.280.150.191.200.3215.50Next quarter−1.040.32−3.42−3.540.310.410.490.3816.00Quarter after next−0.070.17−0.74−0.930.130.100.790.5212.50NN (8, 4)Current month0.180.260.500.810.160.181.190.5036.00Next month0.230.250.711.210.150.141.510.4535.50Month after next−0.030.27−0.30−0.440.180.191.070.4832.50Current quarter−0.140.18−1.07−1.340.140.121.110.4117.00Next quarter−0.870.31−3.00−3.180.290.350.580.4017.50Quarter after next0.010.19−0.22−0.280.150.090.760.5714.00NN (16, 8, 4)Current month0.290.340.691.350.170.121.460.4736.50Next month0.270.290.781.360.160.131.620.4436.50Month after next0.170.290.430.660.190.171.120.5134.50Current quarter−0.280.19−1.74−2.100.160.180.870.4218.00Next quarter−0.830.30−2.95−3.070.290.340.510.4318.50Quarter after next0.080.180.160.210.140.081.010.5316.00NN (32, 16, 8, 4)Current month0.520.311.543.580.130.131.730.4934.00Next month0.610.262.144.280.130.111.730.5134.50Month after next0.620.242.384.660.120.131.700.5331.00Current quarter−0.190.15−1.56−2.120.110.161.330.3315.00Next quarter−0.680.27−2.73−2.990.240.290.750.3515.50Quarter after next0.080.160.170.240.110.121.090.5212.50Table 11
Option strategy of predicting the rise and fall of volatility by threshold (use 8 features M1, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.220.200.841.490.110.101.750.4716.00Next month0.420.182.004.390.080.052.420.5016.00Month after next0.440.192.033.440.110.050.870.7214.50Current quarter−0.010.13−0.45−0.580.100.090.960.507.00Next quarter−0.230.20−1.37−1.500.190.160.400.607.50Quarter after next0.240.101.985.070.040.031.590.735.50Support vector machineCurrent month0.050.09−0.03−0.050.070.060.880.598.50Next month0.080.090.290.400.070.061.300.538.50Month after next−0.060.08−1.31−1.570.070.050.630.537.50Current quarter0.120.071.002.110.030.032.280.564.50Next quarter−0.170.14−1.62−1.760.130.120.440.505.00Quarter after next0.160.081.423.410.030.021.220.754.00NN (4)Current month−0.400.48−0.94−1.030.440.381.100.3818.50Next month−0.290.41−0.82−0.900.370.311.280.3618.00Month after next−0.360.39−1.05−1.130.360.290.730.4917.50Current quarter−0.720.47−1.63−1.640.470.360.490.388.00Next quarter−1.190.48−2.57−2.550.490.450.270.418.50Quarter after next−0.750.59−1.35−1.360.590.390.400.477.50NN (8, 4)Current month−0.130.24−0.78−1.200.150.201.490.3621.00Next month0.010.22−0.16−0.260.140.141.660.3821.00Month after next0.160.220.480.720.150.100.960.5619.50Current quarter−0.260.16−1.96−2.290.130.170.790.379.50Next quarter−0.660.25−2.80−2.930.240.280.470.4010.00Quarter after next0.040.15−0.08−0.120.100.091.430.448.00NN (16, 8, 4)Current month0.640.262.337.330.080.062.150.5521.00Next month0.510.222.154.470.100.072.090.5021.00Month after next0.430.201.964.310.090.071.470.5818.00Current quarter0.100.120.450.760.070.041.270.539.50Next quarter0.020.16−0.17−0.210.140.090.700.6010.00Quarter after next0.400.122.989.110.040.022.330.718.50NN (32, 16, 8, 4)Current month−0.460.46−1.10−1.140.440.330.740.4519.00Next month−0.400.40−1.13−1.170.380.260.770.4519.00Month after next−0.310.36−0.99−1.040.340.220.630.5216.50Current quarter−0.660.44−1.63−1.630.440.330.540.339.00Next quarter−1.340.48−2.89−2.890.480.490.370.329.50Quarter after next−0.530.57−1.03−1.040.560.370.400.546.50Table 12
Option strategy of predicting the rise and fall of volatility by threshold (use 12 features M2, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.080.220.150.250.140.131.500.4318.50Next month0.300.201.262.440.100.071.960.4618.50Month after next0.400.211.732.750.130.070.970.6716.50Current quarter−0.150.16−1.30−1.590.130.121.010.399.00Next quarter−0.490.25−2.16−2.320.230.220.480.479.50Quarter after next0.040.14−0.07−0.090.100.110.990.537.50Support vector machineCurrent month0.020.09−0.34−0.480.060.060.980.538.50Next month0.060.090.130.180.070.061.200.538.50Month after next−0.040.09−1.00−1.270.070.060.820.508.00Current quarter0.090.070.621.330.030.022.380.505.00Next quarter−0.050.14−0.75−0.890.120.090.670.555.50Quarter after next0.230.091.985.600.030.021.390.784.50NN (4)Current month−0.150.52−0.39−0.470.430.311.130.4419.50Next month−0.010.39−0.15−0.170.340.221.160.4619.50Month after next−0.030.37−0.22−0.250.330.210.690.5818.00Current quarter−0.680.44−1.68−1.680.440.350.300.477.50Next quarter−0.710.38−2.00−1.980.390.340.270.508.00Quarter after next−0.390.47−0.95−0.940.470.280.220.676.00NN (8, 4)Current month−0.210.20−1.29−1.690.160.160.980.4320.00Next month−0.110.21−0.78−1.090.150.111.220.4119.50Month after next−0.220.22−1.25−1.600.170.170.950.4318.50Current quarter−0.350.16−2.53−2.770.140.170.760.3310.50Next quarter−0.790.29−2.87−3.060.270.330.480.4111.00Quarter after next−0.020.17−0.44−0.560.130.130.850.539.50NN (16, 8, 4)Current month−0.090.49−0.29−0.350.400.290.960.4921.50Next month−0.150.38−0.53−0.580.340.280.810.5121.50Month after next−0.080.35−0.38−0.410.320.240.660.5819.00Current quarter−0.470.39−1.34−1.340.390.290.470.4810.50Next quarter−0.690.36−2.04−2.030.360.370.300.5511.00Quarter after next−0.270.45−0.72−0.700.460.270.260.718.50NN (32, 16, 8, 4)Current month0.160.330.340.690.160.151.700.4120.50Next month0.150.230.420.690.140.101.360.4820.50Month after next0.130.240.310.470.160.081.110.5118.50Current quarter−0.170.16−1.37−1.630.140.140.870.429.50Next quarter−0.690.27−2.78−2.890.260.290.460.4010.00Quarter after next0.080.170.190.260.120.090.820.607.50Table 13
Option strategy of predicting the rise and fall of volatility by threshold (use 21 features M3, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.110.230.270.460.130.151.570.4318.50Next month0.260.201.062.050.100.082.040.4318.50Month after next0.350.201.482.420.120.081.020.6416.50Current quarter−0.130.15−1.22−1.520.120.121.050.399.00Next quarter−0.490.22−2.42−2.560.210.230.500.429.50Quarter after next0.160.120.981.910.060.051.650.537.50Support vector machineCurrent month−0.010.09−0.61−0.850.060.061.110.477.50Next month0.040.09−0.16−0.220.070.061.370.477.50Month after next−0.080.08−1.53−1.820.070.050.640.507.00Current quarter0.090.070.651.400.030.022.500.504.00Next quarter−0.060.14−0.82−0.970.110.090.600.564.50Quarter after next0.220.091.795.050.030.021.520.754.00NN (4)Current month0.030.49−0.04−0.050.400.301.250.4525.50Next month0.060.370.030.030.330.221.150.4825.50Month after next0.150.330.310.340.300.190.660.6423.50Current quarter−0.460.40−1.29−1.280.400.310.420.5413.00Next quarter−0.660.37−1.93−1.900.380.370.330.5613.50Quarter after next−0.170.46−0.48−0.470.470.290.340.7011.50NN (8, 4)Current month0.080.270.100.200.130.111.400.4418.00Next month0.030.20−0.11−0.170.130.091.240.4618.00Month after next−0.020.21−0.34−0.470.150.080.910.5216.50Current quarter−0.170.13−1.68−1.880.120.100.770.418.50Next quarter−0.410.23−2.03−2.140.210.220.350.569.00Quarter after next0.010.11−0.34−0.440.090.050.700.607.50NN (16, 8, 4)Current month0.460.251.674.320.090.051.930.5419.50Next month0.420.182.134.780.080.032.000.5519.00Month after next0.300.201.272.100.120.081.170.5818.00Current quarter0.110.110.520.680.080.061.300.568.00Next quarter−0.220.18−1.50−1.630.170.190.490.538.50Quarter after next0.280.092.566.710.030.021.110.836.00NN (32, 16, 8, 4)Current month0.030.38−0.04−0.040.350.221.100.4920.50Next month0.090.350.120.130.320.201.070.5120.50Month after next0.220.310.530.570.290.170.610.6819.00Current quarter−0.390.40−1.10−1.100.390.240.520.478.50Next quarter−0.700.39−1.96−1.930.390.340.240.569.00Quarter after next−0.380.48−0.90−0.890.480.280.340.607.50Table 14
Option strategy of predicting the rise and fall of volatility by threshold (use 30 features M4, threshold = 0.5).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.120.23−0.73−1.090.150.191.240.4017.50Next month0.110.200.310.520.120.101.610.4317.50Month after next0.240.210.891.320.140.090.810.6515.50Current quarter−0.180.16−1.48−1.770.130.140.830.418.50Next quarter−0.640.25−2.77−2.890.240.280.450.399.00Quarter after next0.070.150.140.200.100.101.260.507.00Support vector machineCurrent month−0.010.09−0.61−0.850.060.061.110.477.50Next month0.040.09−0.16−0.220.070.061.370.477.50Month after next−0.080.08−1.53−1.820.070.050.640.507.00Current quarter0.090.070.651.400.030.022.500.504.00Next quarter−0.060.14−0.82−0.970.110.090.600.564.50Quarter after next0.220.091.795.050.030.021.520.754.00NN (4)Current month0.230.310.591.280.140.111.630.4520.00Next month0.150.250.410.740.140.111.650.4320.00Month after next0.100.270.170.270.170.121.250.4719.00Current quarter−0.190.16−1.49−1.790.130.130.980.3810.50Next quarter−0.700.28−2.74−2.830.270.300.390.4511.00Quarter after next−0.050.16−0.61−0.770.120.080.780.538.50NN (8, 4)Current month0.000.22−0.24−0.380.140.141.310.4322.00Next month0.060.210.060.100.130.101.580.4122.00Month after next0.040.22−0.06−0.090.150.111.010.5119.50Current quarter−0.100.15−0.96−1.190.120.101.230.3810.50Next quarter−0.510.24−2.31−2.460.230.230.580.4111.00Quarter after next0.090.160.230.320.110.091.000.569.00NN (16, 8, 4)Current month0.230.280.631.300.130.101.560.4721.50Next month0.310.231.142.030.130.081.880.4521.00Month after next0.250.230.881.320.150.080.980.5919.50Current quarter−0.310.15−2.33−2.570.140.140.730.3611.00Next quarter−0.680.27−2.70−2.790.260.290.370.4811.50Quarter after next−0.120.14−1.18−1.320.130.100.630.539.50NN (32, 16, 8, 4)Current month−0.100.23−0.68−1.210.130.151.710.3318.00Next month0.040.21−0.04−0.060.130.111.490.4218.00Month after next0.230.190.991.670.110.071.090.5816.50Current quarter−0.120.14−1.22−1.670.100.121.430.339.00Next quarter−0.250.19−1.58−1.850.160.130.730.449.00Quarter after next0.080.140.190.260.100.090.990.568.00Table 15
Option strategy of predicting the rise and fall of volatility by threshold (use 8 features M1, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.020.06−1.31−1.820.040.041.210.405.00Next month0.070.080.190.260.060.041.750.505.00Month after next0.180.071.723.950.030.020.630.894.50Current quarter0.000.06−0.86−1.140.050.041.460.402.50Next quarter−0.120.10−1.70−1.800.100.070.230.602.50Quarter after next0.030.05−0.39−0.550.030.020.550.752.00Support vector machineCurrent month−0.010.07−0.88−1.200.050.031.360.402.50Next month−0.010.06−0.96−1.100.050.040.590.602.50Month after next−0.010.03−1.93−2.390.030.020.770.502.00Current quarter0.000.000.000.000.000.000.000.001.00Next quarter0.000.000.000.000.000.000.000.001.50Quarter after next0.000.000.000.000.000.000.000.001.00NN (4)Current month−0.380.42−1.02−1.080.400.270.540.477.50Next month−0.310.37−0.98−1.030.350.240.630.477.50Month after next−0.250.34−0.89−0.940.320.200.370.626.50Current quarter−0.620.43−1.56−1.580.420.310.490.334.50Next quarter−0.750.40−2.02−2.010.400.330.160.444.50Quarter after next−0.820.57−1.52−1.530.570.380.310.384.00NN (8, 4)Current month−0.120.19−0.93−1.470.120.161.950.269.50Next month0.010.15−0.27−0.480.080.091.770.379.50Month after next−0.070.18−0.70−0.940.130.110.720.538.50Current quarter−0.050.12−0.84−1.060.100.071.270.384.00Next quarter−0.360.19−2.18−2.290.180.170.370.444.50Quarter after next0.120.090.771.810.040.031.740.573.50NN (16, 8, 4)Current month0.350.241.284.260.070.052.080.5211.50Next month0.310.181.462.970.090.072.260.4811.50Month after next0.360.161.934.450.070.061.160.7010.00Current quarter0.100.090.520.880.050.031.170.605.00Next quarter0.140.120.791.190.080.051.360.605.00Quarter after next0.120.080.952.150.030.022.430.573.50NN (32, 16, 8, 4)Current month0.090.160.240.550.070.092.480.367.00Next month0.050.140.010.010.080.071.210.507.00Month after next0.080.130.210.420.060.051.410.506.00Current quarter0.110.070.904.190.010.014.900.503.00Next quarter−0.330.16−2.43−2.490.150.160.480.293.50Quarter after next0.170.091.437.950.020.014.850.602.50Table 16
Option strategy of predicting the rise and fall of volatility by threshold (use 12 features M2, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.010.05−1.16−1.470.040.040.940.505.00Next month0.070.080.230.290.060.051.160.605.00Month after next0.180.081.743.880.030.020.610.894.50Current quarter0.000.06−0.80−1.050.050.041.000.503.00Next quarter−0.100.11−1.39−1.500.100.060.250.673.00Quarter after next0.030.05−0.37−0.530.040.020.550.752.00Support vector machineCurrent month−0.010.07−0.88−1.200.050.031.360.402.50Next month−0.010.06−0.96−1.100.050.040.590.602.50Month after next−0.010.03−1.93−2.390.030.020.770.502.00Current quarter0.000.000.000.000.000.000.000.001.00Next quarter0.000.000.000.000.000.000.000.001.50Quarter after next0.000.000.000.000.000.000.000.001.00NN (4)Current month−0.100.11−1.35−1.470.100.080.700.477.50Next month0.030.10−0.23−0.300.070.051.280.477.50Month after next0.010.12−0.35−0.410.100.060.570.647.00Current quarter−0.100.10−1.45−1.590.090.070.510.504.00Next quarter−0.030.11−0.76−0.860.100.060.490.634.00Quarter after next0.090.050.802.080.020.011.150.802.50NN (8, 4)Current month−0.260.14−2.25−2.430.130.130.510.449.00Next month−0.100.13−1.21−1.470.100.071.120.399.00Month after next−0.130.14−1.28−1.410.130.090.480.568.00Current quarter−0.230.12−2.31−2.450.120.110.640.305.00Next quarter−0.220.16−1.64−1.790.150.140.450.505.00Quarter after next−0.030.11−0.70−0.840.090.090.520.634.00NN (16, 8, 4)Current month0.040.07−0.09−0.130.050.041.300.5010.00Next month0.120.080.931.500.050.031.600.5510.00Month after next0.200.081.793.840.040.030.930.768.50Current quarter0.090.060.811.870.020.021.930.634.00Next quarter−0.070.13−0.91−0.990.120.110.350.674.50Quarter after next0.140.051.775.860.020.012.430.754.00NN (32, 16, 8, 4)Current month0.130.120.631.020.070.051.400.5212.50Next month0.170.130.921.420.090.051.090.6012.50Month after next0.100.140.350.500.090.061.090.5511.00Current quarter−0.050.09−1.10−1.330.080.071.030.426.00Next quarter−0.400.20−2.26−2.340.190.180.380.466.50Quarter after next0.000.13−0.40−0.520.100.090.830.555.50Table 17
Option strategy of predicting the rise and fall of volatility by threshold (use 21 features M3, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.040.05−1.96−2.360.040.040.750.455.50Next month0.020.08−0.41−0.500.060.051.390.455.50Month after next0.020.08−0.35−0.460.060.040.600.674.50Current quarter0.000.06−0.78−1.040.050.041.030.502.00Next quarter−0.200.14−1.78−1.820.140.100.170.602.50Quarter after next−0.030.04−1.97−2.190.040.040.530.502.00Support vector machineCurrent month0.000.07−0.79−1.080.050.030.980.503.00Next month−0.020.06−1.06−1.210.060.040.820.503.00Month after next−0.050.04−2.37−2.650.040.020.650.402.50Current quarter0.000.000.000.000.000.000.000.001.50Next quarter−0.150.12−1.65−1.710.110.090.260.502.00Quarter after next0.000.000.000.000.000.000.000.001.50NN (4)Current month0.080.180.150.270.100.081.740.4113.50Next month0.120.180.400.630.110.081.400.4813.50Month after next0.280.161.492.500.090.061.020.6712.00Current quarter0.020.11−0.30−0.440.070.051.090.507.00Next quarter−0.200.18−1.38−1.500.170.150.510.537.50Quarter after next0.130.120.691.050.080.051.070.626.50NN (8, 4)Current month0.150.091.152.360.040.031.260.677.50Next month0.240.092.057.420.030.024.850.537.50Month after next0.220.082.035.500.030.021.260.797.00Current quarter0.100.060.882.250.020.012.290.564.50Next quarter0.110.130.440.540.100.070.420.805.00Quarter after next0.170.071.845.340.020.011.390.784.50NN (16, 8, 4)Current month0.390.211.617.670.040.032.230.6510.00Next month0.310.112.3813.350.020.024.520.639.50Month after next0.230.141.352.890.060.051.310.639.50Current quarter0.110.051.104.380.010.013.020.605.00Next quarter−0.100.14−1.10−1.200.130.130.370.645.50Quarter after next0.250.082.5813.390.010.011.670.894.50NN (32, 16, 8, 4)Current month0.030.12−0.14−0.300.050.052.450.3211.00Next month0.090.140.260.450.080.051.890.4111.00Month after next0.220.131.282.310.070.070.870.7010.00Current quarter−0.020.06−1.11−1.450.050.051.590.333.00Next quarter−0.320.16−2.29−2.330.160.160.230.433.50Quarter after next0.000.05−0.92−1.250.040.031.020.503.00Table 18
Option strategy of predicting the rise and fall of volatility by threshold (use 30 features M4, threshold = 0.7).
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month−0.010.04−1.46−2.010.030.031.130.444.50Next month0.030.07−0.23−0.290.060.041.700.444.50Month after next0.080.060.570.990.030.020.810.754.00Current quarter0.000.06−0.78−1.040.050.041.030.502.00Next quarter−0.200.14−1.78−1.820.140.100.170.602.50Quarter after next−0.030.04−1.97−2.190.040.040.530.502.00Support vector machineCurrent month0.000.07−0.79−1.080.050.030.980.503.00Next month−0.020.06−1.06−1.210.060.040.820.503.00Month after next−0.050.04−2.37−2.650.040.020.650.402.50Current quarter0.000.000.000.000.000.000.000.001.50Next quarter−0.150.12−1.65−1.710.110.090.260.502.00Quarter after next0.000.000.000.000.000.000.000.001.50NN (4)Current month−0.050.09−1.14−1.430.070.040.700.538.50Next month0.000.12−0.44−0.570.090.050.880.538.50Month after next−0.040.13−0.72−0.950.100.060.870.508.00Current quarter−0.040.09−1.01−1.170.070.060.300.713.50Next quarter−0.030.14−0.53−0.610.120.100.360.713.50Quarter after next−0.080.10−1.32−1.470.090.080.340.602.50NN (8, 4)Current month−0.210.14−1.86−2.150.120.130.810.3812.00Next month−0.070.14−0.85−1.090.110.091.140.4212.00Month after next−0.080.15−0.88−1.050.120.090.530.6010.00Current quarter−0.150.11−1.79−1.960.100.071.030.275.50Next quarter−0.310.18−1.95−2.020.180.160.350.506.00Quarter after next−0.060.08−1.27−1.530.070.060.710.505.00NN (16, 8, 4)Current month0.350.211.397.520.040.033.510.5010.00Next month0.350.132.3012.250.020.036.200.479.50Month after next0.320.132.086.780.040.031.710.718.50Current quarter−0.010.05−1.23−2.010.030.031.710.334.50Next quarter−0.100.13−1.15−1.250.120.070.400.605.00Quarter after next0.060.060.150.260.030.021.410.564.50NN (32, 16, 8, 4)Current month0.170.180.671.510.080.082.760.379.50Next month0.240.171.102.140.090.091.710.539.50Month after next0.250.151.302.520.080.050.870.718.50Current quarter0.050.10−0.02−0.040.060.062.240.365.50Next quarter−0.120.14−1.23−1.440.120.110.540.555.50Quarter after next0.180.091.352.850.040.020.870.784.50Tables7 to 9 shows the trading results with the threshold parameter of 0.3 under the four feature sets of M1, M2, M3 and M4. As you can see, compared with Tables 19–22, the minimum number of trading times decreased from 24.5 to 10, and the maximum number of trading times decreased from 59 to 37.5. With the decrease of the number of trading times, the annual return, annual standard deviation, Sharpe ratio, Sortino ratio and other indicators have also improved.Table 19
Investment evaluation on train data set with parameters [0.05, −0.2].
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.410.251.463.150.110.111.790.4516.50Next month0.590.252.174.870.110.072.780.3916.50Month after next0.540.251.993.870.130.091.480.5315.00Current quarter−0.190.13−1.85−2.330.100.091.280.337.50Next quarter−0.150.17−1.17−1.440.140.100.870.477.50Quarter after next0.340.142.163.780.080.041.260.647.00Support vector machineCurrent month0.850.243.3010.670.070.042.580.5413.00Next month0.650.252.425.590.110.062.460.4613.00Month after next0.500.251.813.900.120.071.740.5012.00Current quarter−0.060.11−1.01−1.360.080.041.890.305.00Next quarter−0.150.15−1.35−1.560.130.050.660.505.00Quarter after next0.050.100.020.030.080.040.800.605.00NN (4)Current month0.420.251.462.980.120.101.860.4417.00Next month0.500.261.743.780.120.072.610.3817.00Month after next0.690.242.715.240.120.051.340.5916.00Current quarter−0.290.13−2.57−3.200.100.091.270.297.00Next quarter−0.240.17−1.70−2.030.140.100.840.437.00Quarter after next0.160.110.981.560.070.031.360.546.50NN (8, 4)Current month0.780.243.026.340.120.091.720.5417.50Next month0.950.243.829.020.100.072.770.4917.50Month after next0.960.233.998.120.110.041.500.6416.50Current quarter0.070.160.130.190.100.081.690.4010.00Next quarter0.210.220.761.180.140.081.080.5510.00Quarter after next0.500.172.585.220.090.041.320.639.50NN (16, 8, 4)Current month0.930.263.469.930.090.062.120.5516.50Next month0.960.253.658.700.100.052.800.4816.50Month after next0.970.253.668.160.110.061.710.6015.00Current quarter0.050.12−0.01−0.010.080.041.840.386.50Next quarter0.080.170.200.260.130.050.750.626.50Quarter after next0.320.132.083.450.080.041.000.696.50NN (32, 16, 8, 4)Current month0.550.242.065.320.090.062.400.4315.00Next month0.600.262.144.930.110.052.830.4015.00Month after next0.790.253.016.350.120.071.710.5714.00Current quarter−0.070.12−0.99−1.320.090.051.640.336.00Next quarter−0.190.17−1.39−1.630.150.060.680.506.00Quarter after next0.070.120.190.270.080.050.870.586.00Table 20
Investment evaluation on test data set with parameters [0.05, −0.2].
Machine learningContractAnnual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesRandom forestCurrent month0.320.400.671.620.160.102.750.3310.50Next month0.200.250.611.190.130.081.590.4510.00Month after next0.080.290.120.200.170.091.510.429.50Current quarter−0.150.16−1.23−1.510.130.101.500.305.00Next quarter−0.450.21−2.40−2.620.190.140.610.405.00Quarter after next0.480.123.5529.130.010.013.010.863.50Support vector machineCurrent month0.170.220.520.970.120.101.580.477.50Next month0.150.180.531.040.090.081.560.477.50Month after next−0.060.17−0.61−1.160.090.071.410.386.50Current quarter0.020.12−0.25−0.440.060.061.800.384.00Next quarter−0.620.21−3.25−3.310.200.160.420.334.50Quarter after next0.320.132.147.730.040.023.690.504.00NN (4)Current month−0.250.47−0.64−1.410.210.212.620.2414.50Next month−0.480.33−1.60−2.580.200.221.490.3114.50Month after next−0.350.32−1.25−1.870.210.161.180.3813.00Current quarter−0.510.19−2.99−3.450.160.161.070.217.00Next quarter−0.860.26−3.55−3.730.240.220.510.367.00Quarter after next0.270.171.301.990.110.081.050.645.50NN (8, 4)Current month0.410.380.962.550.140.072.730.3611.00Next month0.250.250.841.750.120.071.540.4811.00Month after next0.180.280.460.760.170.071.220.5010.00Current quarter−0.180.16−1.40−1.720.130.102.350.205.00Next quarter−0.410.21−2.16−2.350.190.140.470.505.00Quarter after next0.450.133.2513.050.030.011.450.863.50NN (16, 8, 4)Current month0.390.380.892.380.140.073.250.3212.50Next month0.310.241.122.470.110.052.620.3812.50Month after next0.310.270.951.530.170.071.100.5711.50Current quarter−0.020.15−0.49−0.610.120.071.100.455.50Next quarter−0.250.20−1.50−1.640.180.120.510.555.50Quarter after next0.580.124.49NaN0.000.001.004.00NN (32, 16, 8, 4)Current month0.210.400.411.040.150.103.740.2510.00Next month0.280.240.941.990.110.051.800.4510.00Month after next0.230.240.741.230.140.071.110.568.00Current quarter−0.150.16−1.25−1.540.130.091.690.275.50Next quarter−0.370.21−2.04−2.230.190.130.590.455.50Quarter after next0.470.123.4520.960.020.014.270.754.00Table 21
Realized volatility prediction evaluation of the Wavenet in test set.
RMSEMAEpr_square0.4443180.3463831.20672−0.20431Table 22
Error evaluation of three volatility prediction models and their integrated models.
ModelRMSEMAEpr_squarefBm0.3522210.2704880.6777280.323525185arma0.350530.2692830.6712350.33000595BSSwithGammakernel0.3509990.2674310.6730350.328208943Integrated model0.3483460.2665680.6628980.338326951Specifically, Table23 which presents “Option strategy of predicting the rise and fall of volatility by thresholds” with 21 features performance better than other data sets. Neural Networks with different structures are better than Random Forest and SVM. Go through Tables 7 to 10, the more features, the Neural Networks with more complex structure performance better. For example, NN (8, 4) and NN (16, 8, 4) are better than NN (32, 16, 8, 4) in Table 23 while NN (32, 16, 8, 4) is better in Table 24.Table 23
The values of the optimal lower bound and upper bound for grid optimization and the values in the corresponding RV distribution.
LBUBQuantile0.010.68RV1.9522.8982Table 24
Option strategy of volatility prediction with dynamic thresholds based on quantile neural network and model integration algorithm.
Annual returnAnnual riskSharpe ratioSortino ratioDownside riskMax draw downProfit loss ratioWin ratioTrading timesQuantile neural network and ensemble modelCurrent month1.050.511.996.790.150.072.680.4718.00Next month1.120.522.076.820.160.082.890.4718.00Month after next0.710.381.764.200.160.091.460.5618.00Current quarter−0.030.51−0.15−0.180.430.190.800.5510.00Next quarter0.690.341.914.990.130.061.340.6510.00Quarter after next0.510.281.633.840.120.041.090.6510.00NN (8, 4)Current month0.410.380.962.550.140.072.730.3611.00Next month0.250.250.841.750.120.071.540.4811.00Month after next0.180.280.460.760.170.071.220.5010.00Current quarter−0.180.16−1.40−1.720.130.102.350.205.00Next quarter−0.410.21−2.16−2.350.190.140.470.505.00Quarter after next0.450.133.2513.050.030.011.450.863.50Next, we adjust the threshold parameter, increase the threshold from 0.3 to 0.5, and observe the performance of the strategy. From Tables11 to 14, the relevant results show.Tables11 to 14 shows the investment evaluation with the threshold parameter set to 0.5 under four different feature sets. Compared with the results from Tables 7 to 10 it can be seen that the investment evaluation with the threshold parameter set at 0.5 is better than the investment evaluation with the threshold parameter set at 0.3. For example, the number of trading times has been reduced to 4–25.5, which is more in line with the empirically profitable daily frequency strategy; The Sortino ratio fluctuates from −3.05 to 9.11 under the threshold parameter of 0.5 which is higher than that under the threshold parameter of 0.3 floating from −3.36 to 5.19; In addition, most other indicators are also better under the threshold parameter of 0.5.Specifically, Tables11–13 show Random Forest is better than other models except for NN (16, 8, 4). This result is different from the result with parameter set at 0.3 where Neural Networks are better than Random Forest.Next, increase the threshold from 0.5 to 0.7 again and observe strategy performance from Tables15 to 18.Tables15 to 18 shows the evaluation of investment results with the threshold parameter set to 0.7. Because the threshold parameters are set too strictly and the number of trading times is too low, many favorable trading opportunities are missed. Therefore, most evaluation indicators are relatively lower than those from Tables 11 to 14.Although result with parameter set to 0.7 deteriorates, we still find that NN (8, 4) and NN (16, 8, 4) in Table17 which shows the result with 21 features (feature set M3) performance good. This indicates that M3 maybe the best feature set, and the Neural Network with proper structures is more robust than other models.Comprehensive comparison Tables7 to 18, the setting of threshold parameters can indeed greatly affect the performance of investment strategy. From the trading conditions never set the threshold parameter, then the threshold parameters are 0.3, 0.5, and finally 0.7. The trading conditions are gradually strict, and the investment result evaluation shows a trend of first rising and then falling. This shows that too loose and too strict trading restrictions are unfavorable to the performance of the strategy. Only by selecting an appropriate threshold parameter can we get the optimal investment result. This provides ideas for further improvement.


## 3.4. Option Strategy of Volatility Prediction with Dynamic Thresholds

This section further obtains clues to strategy improvement by observing the characteristics and the performance of the predicted variables. In most cases, predictive research is an inference process from known independent variables to unknown dependent variables, which can be compared with the mapping of functionsX⟶fY. In this cognitive framework, if you want to improve the final prediction accuracy of Y, there are only three breakthroughs to improve, either X, Y, or prediction method f. In this study, the research topic is the application of machine learning method in option investment, so the prediction method f has been given, then only X or Y can be improved.First, further observe the variable Y to be predicted, that is, the realized volatility. Figure1 is the histogram of realized volatility.Figure 1
Histogram of logarithmic realized volatility of CSI 300 index.From Figure1, it can be observed that the logarithmic realized volatility distribution of the real CSI 300 index is not symmetrical, and its right tail is longer than the left tail, which provides the first idea for the improvement of investment strategy: the setting of threshold parameter should be consistent with the logarithmic realized volatility distribution of the real CSI 300 index, and be set as positive threshold parameter and negative threshold parameter, the threshold parameter on both sides should also be set asymmetrically.Secondly, starting from the independent variableX, although the three feature screening algorithms have different evaluation on the feature, they all give high scores on the evaluation of RVD. Figure 2 is the histogram of feature important score of random forest.Figure 2
Feature important score of random forest.From Figure2 it can be seen that the feature important score of RVD is much higher than that of other features, and the feature important score of RVD alone is close to 0.3. The RVW ranked second is also the weekly lag term of logarithmic realized volatility. This makes it worthwhile for us to further observe the relationship between the logarithmic realized volatility of the prediction target CSI 300 index and its own lag term.Figure3 shows the partial autocorrelation coefficient diagram of logarithmic realized volatility of CSI 300 index. The partial autocorrelation coefficient is the correlation coefficient of the dependent variable with respect to its own k-order lag term. It can be seen from the figure that the correlation between the logarithmic realized volatility of CSI 300 index and the first-order lag term is still strongest. The correlation of the second-order lag term is more than half that of the first-order lag term, and the correlation shows a power-law decline with the increase of the lag order.Figure 3
Partial autocorrelation coefficient of logarithmic realized volatility of CSI 300 index.From the above analysis, it can be seen that considering the significant importance of RVD, it is a reliable idea to establish an investment strategy improvement starting from the first-order lag term RVD. Specifically, we set the threshold parameter of the strategy to some form associated with the first-order lag term of the logarithmic realized volatility of the CSI 300 index, so that the relationship between the value of the logarithmic realized volatility on dayT and the value of the logarithmic realized volatility on day T − 1 can reproduce the relationship between the logarithmic realized volatility in the partial autocorrelation coefficient and its first-order lag term. Therefore, the threshold parameter is set as a certain percentage of the logarithmic realized volatility of T − 1 day to make the threshold dynamic.After the above two steps of improvement, the two parameters of positive percentile threshold parameter and negative percentile threshold parameter are obtained. Next, the reasonable threshold parameters need to be determined through parameter optimization.The feature set is divided into training feature set, test feature set, training label and test label according to the ratio of 50%: 50%. Figures4 and 5 is the visualization of the results of grid optimization algorithm obtained by using training feature set and training label. The positive percentile threshold parameter is Positive Threshold axis, the negative percentile threshold parameter is negative threshold axis, and Obv. Is the optimization goal of grid optimization algorithm. In this study, it is set as the sum of the Sortino ratio of the investment results on the all 6 option contracts.Figure 4
Visualization of results of grid optimization algorithm 1.Figure 5
Visualization of results of grid optimization algorithm 2.Figures4 and 5 is the visual result of grid optimization of training data. The result surface of parameter optimization is observed from two different angles. The values of positive threshold and negative threshold corresponding to the high point of the surface in the figure are the optimized values of positive opening threshold and negative opening threshold. It can be observed from the figure that the positive opening threshold is about 0.05 and the negative opening threshold is about −0.2. Two threshold parameters are asymmetric, which confirms the conclusion of the histogram of logarithmic realized volatility of CSI 300 index. Let’s look at Tables 19 and 20. These two tables record the investment strategy evaluation on the training set and the test set.Table19 is the result evaluation of the investment strategy with the threshold parameter [0.05, −0.2] under the training data set. It can be seen that after grid optimization, in the evaluation of investment strategy results, the Sortino ratio ranges from 2.98 to 10.97, which is higher than the previous static threshold.Table20 is the result evaluation of the investment strategy with the threshold parameter [0.05, −0.2] under the test data set. In the test set, the Sortino ratio of NN (8, 4) and NN (16, 8, 4) are generally better, and NN (32, 16, 8, 4) on quarter after next contract reaches 20.96. The results of the test set are not inferior to those of the training set, which means that the strategy model constructed in this study has no over fitting problem in parameter optimization.Figure6 shows the net investment value curve of each algorithm on the option contract of each month. The parameters used are still [0.05, −0.2]. The curve of the solid line part is the net investment value curve on the training set, and the dotted line is the net investment value curve on the test set. As can be seen from the figure, although some curves are loss, the strategy is profitable in most cases. Especially NN (8, 4) and NN (16, 8, 4), better investment results are obtained.Figure 6
Net value curve of 6 machine learning algorithms on 6 option contracts under [0.05, −0.2] parameters.From the frameworkX⟶fY, we have improved the investment strategy from the perspective of X and Y. In the following section, we consider improving the prediction method f.


## 3.5. Benchmark: Wavenet

Since Ribeiro et al. [11] and Ribeiro et al. [12] proves the performance of the Wavenet learners in load forecasting. In this section, we test the Wavenet learners in realized volatility prediction. Take daily realized volatility of CSI 300 index from 2005 to 04-11 to 2021-11-16 to have a try. We split the data set to get train set and leave last 500 observations as test set. Train Wavenet for 100 epochs until the loss function converges, show as Figure 7.Figure 7
Loss function converges with time of epochs.Then, we use the test set to test the performance of the Wavenet in the realized volatility prediction, Figure8 show the result of prediction performance, we can see that prediction series fit the target test set to a certain extent, but it is not as good as we presume.Figure 8
Performance of the Wavenet in the test set.Table21 gives the evaluation matrix, RMSE, MAE, P ratio, R square all show that the Wavenet does not do good job in realized volatility prediction.


## 3.6. Quantile Based Neural Network and Model Integration Algorithm

Machine learning has some advantages in volatility prediction that the traditional financial economics model does not have, but the traditional financial economics model also has its irreplaceable value. This section attempts to combine two models with different theoretical backgrounds to build an integrated combination of quantile based neural network and model for volatility prediction, and then apply it to the “option strategy of volatility prediction with dynamic thresholds” in Section3.4.We choose three classical volatility prediction models, namely fractional Brownian motion (fBm), autoregressive moving average model (ARMA) and Brownian semi-stationary with gamma kernel (BSSwithGammakernel). Then, integrate the three models, that is, calculate the average prediction value of the three models as the final prediction. The prediction errors of the four models are shown in Table22.It can be seen from Table22 that the utility brought by the integration model is positive. The RMSE and other indicators of the integrated model are improved compared with the three basic models.Next, continue to observe the performance of model prediction error under different quantiles of realized volatility. Figure9 describes the prediction error of three volatility prediction models and the integrated model in 10 quantiles from low to high.Figure 9
Quantile error of realized volatility of three volatility prediction models and their integrated model.The three volatility prediction models and the integrated model have similar performance. The RMSE is low in the middle and high on both sides, which means that the prediction efficiency of the distribution tail of realized volatility by the traditional financial and economic model is not high. It is understandable that most of the traditional financial and economic models are linear structures, so the nonlinear mapping ability is poor. However, at the tail of the volatility distribution, the change of volatility is nonlinear, which can be improved by combining neural network.The calculation process of the quantile based neural network and model integration algorithm is as follows:Training process: first, set the grid range of the quantile of the left tail and the quantile of the right tail, and then train the model under the parameters of different grids. When the two grid parameters are selected, the neural network is trained separately with the data of the tail from the two quantiles, while the traditional financial and economic model is trained with full sample data in the way of expanding, and the one-step prediction sequence of realized volatility is obtained. Then, minimize the RMSE, the two optimal grid parameters are selected.Prediction process: judge whether the volatility value is within or outside the quantile parameters on both sides according to the realized volatility of the previous day. If it is outside the two quantile parameters, the neural network is called for training and prediction. If the realized volatility of the previous day is within the two quantiles, the prediction sequence of the integrated model of the traditional financial and economic model is called for prediction.The pseudo code of the above algorithm is as follows:Training process:Setting the starting data window paddingFor: the date series of realized volatility from paddingTrain fBm, ARMA and BSSwithGammakernel models day by day in the way of expanding in the set window, and make one-step prediction, and then take the mean value of the predicted values of the three models as the predicted value of the integrated modelEndSet the value range of quantile parameters on both sides to build a two-dimensional parameter gridFor: each grid pointSelect two quantile parameters of a grid pointFor: the date series of realized volatility from paddingJudge whether the realized volatility of the previous day falls between two quantile parameters. If yes, skip. If no, train the neural network with data other than the two quantiles in the realized volatility from the first day to the previous day. Then the trained neural network is used to predict the realized volatility of next one dayEndEndFor: each grid pointSelect the one-step prediction sequence of neural network under each grid point parameter and fill the skipped realized volatility prediction value with the prediction of the integrated model on that day, so as to obtain a complete Realized Volatility Prediction sequenceEndCompare the RMSE of all realized volatility prediction sequences, and finally select the grid parameter with the smallest RMSEForecast process:The grid parameter with the smallest RMSE is selected as the quantile parameters on both sides of the distributionFor: the date series of realized volatility from paddingJudge whether the realized volatility of the previous day falls between two quantiles. If yes, the prediction of the integrated model is used for next day prediction. If not, the neural network is trained with the data other than the two quantiles in the distribution of realized volatility from the first day to the previous day. Then the trained neural network is used to predict the realized volatility for next dayEndFigure10 shows the results of grid optimization. The color bar on the right side of the figure shows the colors corresponding to different RMSEs. It can be seen that the darker areas corresponding to Lower Bound tend to have smaller values, while the darker areas corresponding to Upper Bound tend to have larger trends. From Tables 2 and 19, two optimal parameters with the smallest RMSE are selected.Figure 10
Grid optimization results of quantile parameters on both sides.Table23 shows that the value of the optimal Lower Bound quantile is 0.01, the value of the optimal Upper Bound quantile is 0.68, and the value of the realized volatility distribution is 1.952 and 2.8982, respectively.Then, according to Table23, the realized volatility is predicted by using the quantile based neural network and model integration algorithm, the prediction sequence of volatility is obtained, and the prediction error is measured, as shown in Table 25.Table 25
Prediction error of quantile based neural network and model integration algorithm.
RMSEMAEpr_square0.3197240.2606460.4913090.515611042From Table25, it can be seen that the “Quantile based neural network and model integration algorithm” can be compared with the RMSE and other indicators of the Integrated model in Table 22. All measurements in Table 25 are improved compared with those of integrated model in Table 22, which means that it is of positive significance to combine the neural network with the integrated model by using the quantile parameters to segment the distribution of volatility. Besides, the performance of “Quantile based neural network and model integration algorithm” is much better than that of the Wavenet showed in Table 21.Although Table25 shows that the prediction error of quantile based neural network and model integration algorithm has a positive improvement compared with the integrated model, whether the improvement is significant still needs to be verified by a statistical test. Therefore, Table 26 is given to show the results of analysis of ANOVA.Table 26
ANOVA analysis of quantile based neural network and model integration algorithm and integrated model.
SourceSSdfMSFp value (F)Group0.13032910.1303292.8425878759.40E-02Error6.4188241400.045849[][]Sum up6.549153141[][][]From the test results of ANOVA, thep value is significant at the 10% confidence level. The statistical significance of the improvement is not too high, which can also be explained. This is because the integrated model itself is an improvement on the single model. Therefore, although the superimposed neural network does improve the final result, the significance of the improvement will decrease.Table24 shows the investment results of the option strategy that brings the quantile neural network and model integration algorithm proposed in this section into the option strategy of volatility prediction with dynamic thresholds in the previous section, and the strategy parameters still use [0.05, −0.2]. Compared with using neural NN (8, 4) alone as the volatility prediction for the option strategy, we can see that Sortino ratio, annual return, max drawdown and other indicators have improved.Finally, Figure11, the net value curve of the investment results is visualized. It can be seen from Figure 11 that the investment results are good in most option contracts, especially in the contracts of the current month and the next month. This is because the option contract month is closer, and the expiration time has less impact on the option price, so the exposed volatility risk has a greater impact on the investment strategy. In this way, the quantile neural network and model integration algorithm for the purpose of predicting volatility has greater advantages.Figure 11
The investment results of option strategy of volatility prediction with dynamic thresholds in different option contracts.


## 4. Discussion

In the discussion section, we compare our work with previous works to show the advantages and disadvantages of this study.This study is a comprehensive study of traditional financial and economic volatility prediction model, machine learning and option investment. For the purpose of volatility investment of the option portfolio at the money, this paper tries a variety of volatility prediction methods. Meligkotsidou et al. [23] finds that no single feature can provide more information in a long time perspective. That’s why our study tries to improve the volatility prediction from a model perspective. Our work extends and verifies Meligkotsidou et al. [23]. Degiannakis and Livada [24] prove that RV is based on a leptokurtic and asymmetrical distribution. They provide better forecast error measurements making use of the leptokurtic and asymmetrical distribution. Our paper constructs the strategy by double side asymmetrical thresholds which are consistent with the conclusions of Degiannakis and Livada [24]. That is to say, the volatility investment strategy of this study makes use of the leptokurtic and asymmetrical RV distribution in the investment perspective which is another kind of contributions that is different from the forecast measurement of Degiannakis and Livada [24]. Wang et al. [25] forecasts the 5 min-RV of China 50ETF volatility prediction and proposes Realized-GARCH-Kernel-type models which have a prediction error of RMSE ranging from 0.004 to 0.013. Take 2.77 as a daily mean value of the LogRV. Roughly transfer the 5 min to one day, the relative error of Wang et al. [25] is around 6.9% to 22.5%. Our work does the forecasting of the LogRV of CIS 300 index in a daily perspective, the relative error is 11.5%. As a result, this paper is not bad than the work of Wang et al. [25]. Louzis et al. [26] uses quantile methods which account for the fat tails and the asymmetry of the innovation distribution which gives evidence in favor of the skewed student distribution and the Extreme Value Theory (EVT) method to improve the Basel II accuracy and the Regulatory accuracy. Essentially speaking, Louzis et al. [26] also finds the asymmetry of the RV distribution and apply this properties into regulatory problems. This verifies our work which applies the asymmetry of the RV distribution into option investment strategy.However, there still are many further works to be done. Firstly, Although Neural Network is a good volatility prediction model in the tail of volatility prediction distribution compared with classical Volatility Prediction Models, the RMSE tells that the prediction error is still as high as 0.4. We are interesting that how to get a higher performance in the tail prediction. Secondly, the performance of investment strategy does not incorporate transaction cost. Thirdly, because of the lack of liquidity in the CSI 300 index option, the performance of investment strategy does not consider if the trading is success or failed and assume all the transactions occur at historical price are success.


## 5. Conclusion

This section summarizes the contribution of this paper. Since this study is comprehensive, we simplify the contributions in following aspects:Firstly, set up a generalized machine learning procedure including feature selection, model comparison, forecasting performance evaluation. This study collects 30 financial and economic features, and uses three different feature screening algorithms to segment four feature sets M1, M2, M3 and M4. Six models of support vector machine, four neural networks with different structures and random forest are compared on different feature sets. It is found that 21 feature sets M3 are the best feature sets, and random forest and neural network are better than support vector machine.Secondly, set up a generalized framework to improve an investment strategy which is under developing. To develop a strategy is a long-time work which will go on with the changing of the capital market. However, no matter how the capital market changes, how the investment strategy fails, the frameworkX⟶fY for improvement of investment strategy always works. “Option strategy of volatility prediction with dynamic thresholds” proposed by this paper is just one example.Thirdly, this paper proposes a realized volatility prediction algorithm “Quantile based neural network and model integration algorithm.” This algorithm makes comprehensive use of the theoretical advantages of traditional financial and economic models and the nonlinear advantages of neural networks, combines them in different quantiles of volatility distribution, the RMSE performance of this algorithm is 0.319724 which is better than that of the integrated model (RMSE:0.348346) and the Wavenet (RMSE:0.444318).Fourthly, after several times improvement, this study constructs the “Option strategy of volatility prediction with dynamic thresholds,” this investment strategy achieves Sharpe ratio 1.99 in Current month option contracts and 2.07 in Next month option contracts, this performance is superior to single machine learning algorithm and classical Volatility Prediction Models.For the next step, since there are still some works left as presented in the Discussion section, further study need to be done in problems of volatility tail prediction, transaction cost, market liquidity. [27–35].



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*Source: 8952996-2022-04-07.xml*