pwc-archive/papers-with-abstracts huggingface dataset:
Optimal control of the decumulation of a retirement portfolio with variable spending and dynamic asset allocation
Authors: Peter A. Forsyth, Kenneth R. Vetzal, Graham Westmacott
Date: 2021-01-07 00:00:00

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst five per cent of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.
Improved ACD-based financial trade durations prediction leveraging LSTM networks and Attention Mechanism
Authors: Yong Shi, Wei Dai, Wen Long, Bo Li
Date: 2021-01-07 00:00:00

The liquidity risk factor of security market plays an important role in the formulation of trading strategies. A more liquid stock market means that the securities can be bought or sold more easily. As a sound indicator of market liquidity, the transaction duration is the focus of this study. We concentrate on estimating the probability density function p({\Delta}t_(i+1) |G_i) where {\Delta}t_(i+1) represents the duration of the (i+1)-th transaction, G_i represents the historical information at the time when the (i+1)-th transaction occurs. In this paper, we propose a new ultra-high-frequency (UHF) duration modelling framework by utilizing long short-term memory (LSTM) networks to extend the conditional mean equation of classic autoregressive conditional duration (ACD) model while retaining the probabilistic inference ability. And then the attention mechanism is leveraged to unveil the internal mechanism of the constructed model. In order to minimize the impact of manual parameter tuning, we adopt fixed hyperparameters during the training process. The experiments applied to a large-scale dataset prove the superiority of the proposed hybrid models. In the input sequence, the temporal positions which are more important for predicting the next duration can be efficiently highlighted via the added attention mechanism layer.
Pricing spread option with liquidity adjustments
Authors: Kevin Shuai Zhang, Traian Pirvu
Date: 2021-01-01 00:00:00

We study the pricing and hedging of European spread options on correlated assets when, in contrast to the standard framework and consistent with imperfect liquidity markets, the trading in the stock market has a direct impact on stocks prices. We consider a partial-impact and a full-impact model in which the price impact is caused by every trading strategy in the market. The generalized Black-Scholes pricing partial differential equations (PDEs) are obtained and analysed. We perform a numerical analysis to exhibit the illiquidity effect on the replication strategy of the European spread option. Compared to the Black-Scholes model or a partial impact model, the trader in the full impact model buys more stock to replicate the option, and this leads to a higher option price.
A computational weighted finite difference method for American and barrier options in subdiffusive Black-Scholes model
Authors: Grzegorz Krzyżanowski, Marcin Magdziarz
Date: 2020-03-11 00:00:00

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black-Scholes (B-S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff-Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.
An approximate closed formula for European Mortgage Options
Authors: Manuel Lopez Galvan
Date: 2020-12-11 00:00:00

The aim of this paper is to investigate the use of close formula approximation for pricing European mortgage options. Under the assumption of logistic duration and normal mortgage rates the underlying price at the option expiry is approximated by shifted lognormal or regular lognormal distribution by matching moments. Once the price function is approximated by lognormal distributions, the option price can be computed directly as an integration of the distribution function over the payoff at the option expiry by using Black-Scholes-Merton close formula. We will see that lower curvature levels correspond to positively skewness price distributions and in this case lognormal approximation leads to close parametric formula representation in terms of all model parameters. The proposed methodologies are tested against Monte Carlo approach under different market and contract parameters and the tests confirmed that the close form approximation have a very good accuracy.
The Deep Parametric PDE Method: Application to Option Pricing
Authors: Kathrin Glau, Linus Wunderlich
Date: 2020-12-11 00:00:00
Tasks: BIG-bench Machine Learning

We propose the deep parametric PDE method to solve high-dimensional parametric partial differential equations. A single neural network approximates the solution of a whole family of PDEs after being trained without the need of sample solutions. As a practical application, we compute option prices in the multivariate Black-Scholes model. After a single training phase, the prices for different time, state and model parameters are available in milliseconds. We evaluate the accuracy in the price and a generalisation of the implied volatility with examples of up to 25 dimensions. A comparison with alternative machine learning approaches, confirms the effectiveness of the approach.
Data-Driven Option Pricing using Single and Multi-Asset Supervised Learning
Authors: Anindya Goswami, Sharan Rajani, Atharva Tanksale
Date: 2020-08-02 00:00:00
Tasks: BIG-bench Machine Learning

We propose three different data-driven approaches for pricing European-style call options using supervised machine-learning algorithms. These approaches yield models that give a range of fair prices instead of a single price point. The performance of the models are tested on two stock market indices: NIFTY$50$ and BANKNIFTY from the Indian equity market. Although neither historical nor implied volatility is used as an input, the results show that the trained models have been able to capture the option pricing mechanism better than or similar to the Black-Scholes formula for all the experiments. Our choice of scale free I/O allows us to train models using combined data of multiple different assets from a financial market. This not only allows the models to achieve far better generalization and predictive capability, but also solves the problem of paucity of data, the primary limitation of using machine learning techniques. We also illustrate the performance of the trained models in the period leading up to the 2020 Stock Market Crash (Jan 2019 to April 2020).
Nowcasting Networks
Authors: Marc Chataigner, Stephane Crepey, Jiang Pu
Date: 2020-11-27 00:00:00
Tasks: Outlier Detection, Time Series, Time Series Analysis

We devise a neural network based compression/completion methodology for financial nowcasting. The latter is meant in a broad sense encompassing completion of gridded values, interpolation, or outlier detection, in the context of financial time series of curves or surfaces (also applicable in higher dimensions, at least in theory). In particular, we introduce an original architecture amenable to the treatment of data defined at variable grid nodes (by far the most common situation in financial nowcasting applications, so that PCA or classical autoencoder methods are not applicable). This is illustrated by three case studies on real data sets. First, we introduce our approach on repo curves data (with moving time-to-maturity as calendar time passes). Second, we show that our approach outperforms elementary interpolation benchmarks on an equity derivative surfaces data set (with moving time-to-maturity again). We also obtain a satisfying performance for outlier detection and surface completion. Third, we benchmark our approach against PCA on at-the-money swaption surfaces redefined at constant expiry/tenor grid nodes. Our approach is then shown to perform as well as (even if not obviously better than) the PCA which, however, is not be applicable to the native, raw data defined on a moving time-to-expiry grid).
A bivariate Normal Inverse Gaussian process with stochastic delay: efficient simulations and applications to energy markets
Authors: Matteo Gardini, Piergiacomo Sabino, Emanuela Sasso
Date: 2020-11-09 00:00:00

Using the concept of self-decomposable subordinators introduced in Gardini et al. [11], we build a new bivariate Normal Inverse Gaussian process that can capture stochastic delays. In addition, we also develop a novel path simulation scheme that relies on the mathematical connection between self-decomposable Inverse Gaussian laws and L\'evy-driven Ornstein-Uhlenbeck processes with Inverse Gaussian stationary distribution. We show that our approach provides an improvement to the existing simulation scheme detailed in Zhang and Zhang [23] because it does not rely on an acceptance-rejection method. Eventually, these results are applied to the modelling of energy markets and to the pricing of spread options using the proposed Monte Carlo scheme and Fourier techniques
A Convergent Linear Regression Method for Forward-Backward Stochastic Differential Equations with Jumps
Authors: Tingting Ye, Liangliang Zhang
Date: 2018-05-15 00:00:00
Tasks: regression

In this paper, we introduce a large class of convergent numerical methods, based on (linear) basis function regression technique, to approximate the solution to a forward-backward stochastic differential equation with jumps (FBSDEJ hereafter). Numerical experiment shows good applicability of the proposed method.
Deep Reinforcement Learning for Asset Allocation in US Equities
Authors: Miquel Noguer i Alonso, Sonam Srivastava
Date: 2020-10-09 00:00:00
Tasks: Deep Reinforcement Learning, Management, reinforcement-learning, Reinforcement Learning, Reinforcement Learning (RL), Time Series, Time Series Analysis, Time Series Prediction

Reinforcement learning is a machine learning approach concerned with solving dynamic optimization problems in an almost model-free way by maximizing a reward function in state and action spaces. This property makes it an exciting area of research for financial problems. Asset allocation, where the goal is to obtain the weights of the assets that maximize the rewards in a given state of the market considering risk and transaction costs, is a problem easily framed using a reinforcement learning framework. It is first a prediction problem for expected returns and covariance matrix and then an optimization problem for returns, risk, and market impact. Investors and financial researchers have been working with approaches like mean-variance optimization, minimum variance, risk parity, and equally weighted and several methods to make expected returns and covariance matrices' predictions more robust. This paper demonstrates the application of reinforcement learning to create a financial model-free solution to the asset allocation problem, learning to solve the problem using time series and deep neural networks. We demonstrate this on daily data for the top 24 stocks in the US equities universe with daily rebalancing. We use a deep reinforcement model on US stocks using different architectures. We use Long Short Term Memory networks, Convolutional Neural Networks, and Recurrent Neural Networks and compare them with more traditional portfolio management. The Deep Reinforcement Learning approach shows better results than traditional approaches using a simple reward function and only being given the time series of stocks. In Finance, no training to test error generalization results come guaranteed. We can say that the modeling framework can deal with time series prediction and asset allocation, including transaction costs.
Volatility Forecasting with 1-dimensional CNNs via transfer learning
Authors: Bernadett Aradi, Gábor Petneházi, József Gáll
Date: 2020-09-07 00:00:00
Tasks: Model Selection, Transfer Learning

Volatility is a natural risk measure in finance as it quantifies the variation of stock prices. A frequently considered problem in mathematical finance is to forecast different estimates of volatility. What makes it promising to use deep learning methods for the prediction of volatility is the fact, that stock price returns satisfy some common properties, referred to as `stylized facts'. Also, the amount of data used can be high, favoring the application of neural networks. We used 10 years of daily prices for hundreds of frequently traded stocks, and compared different CNN architectures: some networks use only the considered stock, but we tried out a construction which, for training, uses much more series, but not the considered stocks. Essentially, this is an application of transfer learning, and its performance turns out to be much better in terms of prediction error. We also compare our dilated causal CNNs to the classical ARIMA method using an automatic model selection procedure.
Kernel-based collocation methods for Heath-Jarrow-Morton models with Musiela parametrization
Authors: Yuki Kinoshita, Yumiharu Nakano
Date: 2018-09-15 00:00:00

We propose kernel-based collocation methods for numerical solutions to Heath-Jarrow-Morton models with Musiela parametrization. The methods can be seen as the Euler-Maruyama approximation of some finite dimensional stochastic differential equations, and allow us to compute the derivative prices by the usual Monte Carlo methods. We derive a bound on the rate of convergence under some decay condition on the inverse of the interpolation matrix and some regularity conditions on the volatility functionals.
Multigrid Iterative Algorithm based on Compact Finite Difference Schemes and Hermite interpolation for Solving Regime Switching American Options
Authors: Chinonso Nwankwo, Weizhong Dai
Date: 2020-08-03 00:00:00

We present a multigrid iterative algorithm for solving a system of coupled free boundary problems for pricing American put options with regime-switching. The algorithm is based on our recently developed compact finite difference scheme coupled with Hermite interpolation for solving the coupled partial differential equations consisting of the asset option and the delta, gamma, and speed sensitivities. In the algorithm, we first use the Gauss-Seidel method as a smoother and then implement a multigrid strategy based on modified cycle (M-cycle) for solving our discretized equations. Hermite interpolation with Newton interpolatory divided difference (as the basis) is used in estimating the coupled asset, delta, gamma, and speed options in the set of equations. A numerical experiment is performed with the two- and four- regime examples and compared with other existing methods to validate the optimal strategy. Results show that this algorithm provides a fast and efficient tool for pricing American put options with regime-switching.
Weak error rates for option pricing under linear rough volatility
Authors: Christian Bayer, Eric Joseph Hall, Raúl Tempone
Date: 2020-09-02 00:00:00
Tasks: Time Series, Time Series Analysis

In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, $H < 1/2$, over reasonable timescales. Both time series of asset prices and option-derived price data indicate that $H$ often takes values close to $0.1$ or less, i.e., rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only $H$. We prove rate $H + 1/2$ for the weak convergence of the Euler method for the rough Stein-Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Our proof uses Talay-Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.
The SINC way: A fast and accurate approach to Fourier pricing
Authors: Fabio Baschetti, Giacomo Bormetti, Silvia Romagnoli, Pietro Rossi
Date: 2020-09-01 00:00:00

The goal of this paper is to investigate the method outlined by one of us (PR) in Cherubini et al. (2009) to compute option prices. We name it the SINC approach. While the COS method by Fang and Osterlee (2009) leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr and Madan (1999) and Lewis (2000); v) an extensive pricing exercise under the rough Heston model of Jaisson and Rosenbaum (2015); vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires to resort only to odd moments in the Fourier space. A previous version of this manuscript circulated with the title `Rough Heston: The SINC way'.
Stress testing and systemic risk measures using multivariate conditional probability
Authors: Tomaso Aste
Date: 2020-04-14 00:00:00

The multivariate conditional probability distribution models the effects of a set of variables onto the statistical properties of another set of variables. In the study of systemic risk in a financial system, the multivariate conditional probability distribution can be used for stress-testing by quantifying the propagation of losses from a set of `stressing' variables to another set of `stressed' variables. In this paper I describe how to compute such conditional probability distributions for the vast family of multivariate elliptical distributions, and in particular for the multivariate Student-t and the multivariate Normal distributions. Measures of stress impact and systemic risk are proposed. An application to the US equity market illustrates the potentials of this approach.
Optimal semi-static hedging in illiquid markets
Authors: Teemu Pennanen, Udomsak Rakwongwan
Date: 2020-08-04 00:00:00

We study indifference pricing of exotic derivatives by using hedging strategies that take static positions in quoted derivatives but trade the underlying and cash dynamically over time. We use real quotes that come with bid-ask spreads and finite quantities. Galerkin method and integration quadratures are used to approximate the hedging problem by a finite dimensional convex optimization problem which is solved by an interior point method. The techniques are extended also to situations where the underlying is subject to bid-ask spreads. As an illustration, we compute indifference prices of path-dependent options written on S&P500 index. Semi-static hedging improves considerably on the purely static options strategy as well as dynamic trading without options. The indifference prices make good economic sense even in the presence of arbitrage opportunities that are found when the underlying is assumed perfectly liquid. When transaction costs are introduced, the arbitrage opportunities vanish but the indifference prices remain almost unchanged.
Volatility model calibration with neural networks a comparison between direct and indirect methods
Authors: Dirk Roeder, Georgi Dimitroff
Date: 2020-07-07 00:00:00

In a recent paper "Deep Learning Volatility" a fast 2-step deep calibration algorithm for rough volatility models was proposed: in the first step the time consuming mapping from the model parameter to the implied volatilities is learned by a neural network and in the second step standard solver techniques are used to find the best model parameter. In our paper we compare these results with an alternative direct approach where the the mapping from market implied volatilities to model parameters is approximated by the neural network, without the need for an extra solver step. Using a whitening procedure and a projection of the target parameter to [0,1], in order to be able to use a sigmoid type output function we found that the direct approach outperforms the two-step one for the data sets and methods published in "Deep Learning Volatility". For our implementation we use the open source tensorflow 2 library. The paper should be understood as a technical comparison of neural network techniques and not as an methodically new Ansatz.
Deep Importance Sampling
Authors: Benjamin Virrion
Date: 2020-07-06 00:00:00

We present a generic path-dependent importance sampling algorithm where the Girsanov induced change of probability on the path space is represented by a sequence of neural networks taking the past of the trajectory as an input. At each learning step, the neural networks' parameters are trained so as to reduce the variance of the Monte Carlo estimator induced by this change of measure. This allows for a generic path dependent change of measure which can be used to reduce the variance of any path-dependent financial payoff. We show in our numerical experiments that for payoffs consisting of either a call, an asymmetric combination of calls and puts, a symmetric combination of calls and puts, a multi coupon autocall or a single coupon autocall, we are able to reduce the variance of the Monte Carlo estimators by factors between 2 and 9. The numerical experiments also show that the method is very robust to changes in the parameter values, which means that in practice, the training can be done offline and only updated on a weekly basis.