# Stochastic Volatility Effects on Correlated Log-Normal Random Variables

Authors: Yong-Ki Ma
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

The transition density function plays an important role in understanding and explaining the dynamics of the stochastic process. In this paper, we incorporate an ergodic process displaying fast moving fluctuation into constant volatility models to express volatility clustering over time. We obtain an analytic approximation of the transition density function under our stochastic process model. Using perturbation theory based on Lie–Trotter operator splitting method, we compute the leading-order term and the first-order correction term and then present the left and right skew scenarios through numerical study.


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## Body

## 1. Introduction

The multivariate log-normal distribution is a widely used stochastic model in social sciences. What is the probability of the sum or difference of log-normal random variables? The solution to this question has wide applications in many fields such as finance [1, 2], actuarial science [3, 4], and physics [5]. Especially, in physics, examples include wave transmittance in random media, dissipation rate of turbulence energy, and temporal fluctuations of some nonlinear systems. However, to the best of the author of this paper’s knowledge, almost nothing is known about the question yet. So, finding analytical approximations to it is as important as an alternative approach (see [6] for details). The main advantage of the analytical approximation approaches compared to other numerical methods is that in general the first ones are much faster and precise at least under certain model parameter regime. In addition, analytic approximation formulas retain qualitative model information and preserve an explicit dependence of the results on the underlying parameters. Many approximate methods can be categorized by their scope into three classes: generally correlated [7–10], independent [11–14], and independent and identically distributed [15, 16]. The probability density function of the correlated log-normal especially is carried out to propose various approximations of the sum distribution. These approximations can be divided into two categories. The first method is approximated by another log-normal random variable. For example, the log-normal parameters are obtained by moment matching [7], the mean and the variance are computed recursively [17], and the shifted log-normal distribution [18]. A second method of these methods is introduced to construct efficient numerical integrations techniques such as a steepest-descent integration [19] and a Smolyak’s algorithm [20].In option pricing theory, it is well-known that constant volatility for the stock price in the Black-Scholes model [21] can hardly capture the accumulated empirical evidence in financial markets since the parameters contained in the model actually change over time. A main drawback in the assumption of this model lies in the flat implied volatility surface, which is contradictory to empirical results that the implied volatilities of the equity options exhibit the smile or skew curve, in particular, from the 1987 crash onward. Among those several methods of overcoming the above drawback and relaxing the assumption of the Black-Scholes constant volatility model, a variety of stochastic volatility models have been suggested to incorporate the volatility skew. Fast mean reversion especially is one of the notable features of volatility. For example, Fouque et al. [22] estimated that its volatility reverts to the mean with characteristic time about 1.5 days in high-frequency S&P 500 index data. Also, Engle [23] and Bollerslev [24] introduced the family of autoregressive conditional heteroskedasticity and generalized autoregressive conditional heteroskedasticity, respectively, to describe the evolution of the volatility of the stock price in discrete time setting and showed that econometric tests of these models reject the assumption of constant volatility and find evidence of volatility clustering over time.Based on these observations together with the renowned contribution of stochastic volatility formula to option pricing, we propose the following fast mean-reverting (FMR) model in which asset prices are conditionally log-normal and the volatility process is a positive increasing function of a FMR process:(1)dX1±X2=f1Y±f2YX1±X2dW1±W2,(2)dY=αm-Ydt+βgYdWY, where X1 and X2 are log-normal random variables and W1, W2, and WY are standard Brownian motions correlated as follows: (3)dW1,W2t=ρ12dt,dW1,WYt=ρ1Ydt,dW2,WYt=ρ2Ydt with -1≤ρ12,ρ1Y,ρ2Y≤1. Here, the parameter α measures the rate at which the ergodic Markov process Y reaches its long-term mean value m, g is a function, β is a constant, and the correlations ρ1Y and ρ2Y control the slope of the skew of each underlying. We assume that α is large so that volatility is sufficiently fast mean-reversion when looked over the time scale of options Fouque et al. [22]. We do not specify the concrete forms of f1, f2, and g since they will not play an essential role in the perturbation theory performed in this paper but the functions must satisfy a sufficient growth condition to avoid some kind of bad cases such as nonexistence of the moments of Y. This generalization is a clear advantage of this model in that the FMR model encompasses the models which reflect stylized facts such as a feedback effect between volatility and volatility of volatility.


## 2. Asymptotic Analysis

### 2.1. Problem Formulation

Now we will denotex1±x2 and x10±x20 as x± and x0±, respectively, for simplicity. The probability distribution of the sum or difference of the two correlated log-normal distributions can be obtained by calculating the integral (4)u±t,x±;t0,x0±=∫0∞∫0∞dx1dx2ut,x1,x2;t0,x10,x20δx±-x0±, where u is the joint probability distribution of the two log-normal random variables and δ(x±-x0±) is the Dirac delta function. Unfortunately, a closed-form representation for this probability distribution still does not exist.FMR volatility enables us to make good use of perturbation theory. To end this, we use the small positive parameterϵ, which denotes the inverse of the rate of mean-reversion α (which is assumed to be large). We suppose β=ν2/ϵ, where the variance ν2 of the invariant distribution of Y is a constant with respect to ϵ. If rewriting (2) in terms of ϵ, we have (5)dY=1ϵm-Ydt+ν2ϵgYdWY. This gives a singular perturbation problem with respect to the small parameter ϵ, which furnishes us with analytic tractability. We define the scaled FMR (SFMR) model as the FMR model used with an first-order asymptotic method in this paper.If we compare the Heston model [25], one of representative stochastic volatility models, with the SFMR model, we can get some strengths. The Heston model has five parameters required to be estimated from market data while the SFMR model has two. In addition, the Heston model has certain restrictions on the volatility functions f1, f2, and g whereas the SFMR model does not have any.


### 2.2. Perturbation Theory Based on Lie–Trotter Operator Splitting Method

Perturbation theory as developed by Fouque et al. [22] is a methodology utilized to find an approximated solution when the original problem is difficult to solve by separating it into more easily solvable, simple parts. If we apply the Feynman-Kac formula, we find that u±ϵ satisfies the following Kolmogorov backward equation:(6)Lϵu±ϵt,x1,x2,y1;t0,x10,x20,y0=0,t<T,(7)Lϵ≔1ϵL0+1ϵL1+L2,u±ϵt,x1,x2,y1;t0⟶t,x10,x20,y0=δx±-x0±, where(8)L0=m-y0∂∂y0+ν2g2y0∂2∂y02,L1=ν2f1y0x10gy0ρ1Y∂2∂x10∂y0+ν2f2y0x20gy0ρ2Y∂2∂x20∂y0,(9)L2=∂∂t0+12x102f12y0∂2∂x102+12x202f22y0∂2∂x202+x10x20ρ12f1y0f2y0∂2∂x10∂x20. Here, L0 is the infinitesimal generator of the SFMR process Y. L1 contains the mixed partial derivatives due to the correlations of the two Brownian motions W1 and WY and W2 and WY, respectively. L2 is the operator of a generalized version of the two-dimensional standard Brownian motion at the volatility levels f1(y0) and f2(y0) instead of constant volatilities, respectively.Before we solve problem (6), we write a useful lemma about the centering (or solvability) condition on the Poisson equation related to the operator L0 as follows.Lemma 1.
Ifψ(y)∈C2(R) which is a solution to the Poisson equation(10)L0χy+ψy=0exists, then the centering condition 〈ψ〉=∫ψy1/2πν2exp⁡-(y-m)2/2ν2=0 must be satisfied, where the notation · is the average (or expectation) with respect to the invariant distribution (namely, N(m,ν2)) of Y. Then, solutions of (10) are given by the form (11)χy=∫0∞EyψY∣Y=ydt.Proof.
See Fouque et al. [22].Using the resultant partial differential equation (PDE) (6) and expanding u±ϵ in powers of ϵ, one can approximate u±ϵ to the sum of the leading term u±(0) and the first correction term ϵu±(1) as follows:(12)u±ϵ≈u±0+ϵu±1. Since we focus on the first-order correction term for u±ϵ, we reset (12) with respect to ϵu±(1) and denote it by u~±(1). Using perturbation theory based on Lie–Trotter operator splitting method, u±(0) and u~±(1) must satisfy the following PDEs with boundary conditions, respectively:(13)L2u±0t,x1,x2;t0,x10,x20=0(14)witht,x1,x2;t0⟶t,x10,x20=δx±-x0±,(15)L2u~±1t,x1,x2;t0,x10,x20=L1L0-1L2-L2u±0(16)withu~±1t,x1,x2;t0⟶t,x10,x20=0 and then one obtains the solutions u±(0) and u~±(1) of the PDEs, respectively, (17)u±0=1x±2σ~±2πt-t0exp⁡-ln⁡x±/x0±+1/2σ~±2t-t022σ~±2t-t0,u~±1=-T-tV1∂3∂x103+V2∂3∂x203+V12∂3∂x10x202+V21∂3∂x102x20u±0,where (18)L2=∂∂t0+12x102σ¯1∂2∂x102+12x202σ¯2∂2∂x202+σ¯1σ¯2ρ¯x10x20∂2∂x10∂x20,σ¯1≔f12,σ¯2≔f22,ρ¯≔ρ12f1f2σ¯1σ¯2,σ~+≔σ¯12+σ¯22+2σ¯1σ¯2ρ¯2,σ~-≔σ¯12-σ¯222σ¯12+σ¯22-2σ¯1σ¯2ρ¯ and the constant parameters V1,V2,V12, and V21 are defined as follows: (19)V1=νρ1Yϵx1032f1gθ1′,V2=νρ2Yϵx2032f2gθ2′,V12=νρ1Yϵx10x2022f1gθ2′+ν2ϵρ12ρ2Yx10x202f2gθ12′,V21=νρ2Yϵx102x202f2gθ1′+ν2ϵρ12ρ1Yx102x20f1gθ12′.See the Appendix for the relevant technique.As a result, the distributions of the sum and difference of correlated log-normal random variables under SFMR model are shown to follow combining a shifted log-normal distribution with mixed partial derivatives of it. Also, all the original parameters are absorbed in the group parametersV1, V2, V12, and V21 and the present level y of the hidden process Y driving the fast time-scale volatility needs not be specified in the present approximation.


### 2.3. Numerical Experiment

In this subsection, we illustrate the effectiveness of our result (12) by showing numerical results.Several parameter sets are obtained from Lo except for group parameters. Figure1(a) depicts log-normal distribution (leading-order term or Lo’s result) and log-normal distributions under SFMR model with positive or negative group parameters (leading-order term plus first-order correction term), respectively. The solid line denotes the shifted log-normal distribution and the dash lines show combining the shifted log-normal distribution and the first correction term, respectively. Figure 1(b) depicts the errors calculated by subtracting the log-normal distributions with positive or negative group parameters from the log-normal distribution, respectively. We also apply the same result to the difference of the probability density function in Figure 2. Our numerical results show that the left and right skew scenarios are presented through first-order correction term and the major discrepancies appear around the peak of the probability density function. These pictures are sensitive to the choice of the involved parameters and give a lot of flexibility to the shape of the transition densities.Figure 1
Sum of probability density function:x10=110,  x20=100,  σ¯1=0.25,  σ¯2=0.15,  ρ¯=-0.5, and t-t0=1.
(a)
(b)Figure 2
Difference of probability density function:x10=110,  x20=100,  σ¯1=0.25,  σ¯2=0.15,  ρ¯=-0.5, and  t-t0=1.


## 2.1. Problem Formulation

Now we will denotex1±x2 and x10±x20 as x± and x0±, respectively, for simplicity. The probability distribution of the sum or difference of the two correlated log-normal distributions can be obtained by calculating the integral (4)u±t,x±;t0,x0±=∫0∞∫0∞dx1dx2ut,x1,x2;t0,x10,x20δx±-x0±, where u is the joint probability distribution of the two log-normal random variables and δ(x±-x0±) is the Dirac delta function. Unfortunately, a closed-form representation for this probability distribution still does not exist.FMR volatility enables us to make good use of perturbation theory. To end this, we use the small positive parameterϵ, which denotes the inverse of the rate of mean-reversion α (which is assumed to be large). We suppose β=ν2/ϵ, where the variance ν2 of the invariant distribution of Y is a constant with respect to ϵ. If rewriting (2) in terms of ϵ, we have (5)dY=1ϵm-Ydt+ν2ϵgYdWY. This gives a singular perturbation problem with respect to the small parameter ϵ, which furnishes us with analytic tractability. We define the scaled FMR (SFMR) model as the FMR model used with an first-order asymptotic method in this paper.If we compare the Heston model [25], one of representative stochastic volatility models, with the SFMR model, we can get some strengths. The Heston model has five parameters required to be estimated from market data while the SFMR model has two. In addition, the Heston model has certain restrictions on the volatility functions f1, f2, and g whereas the SFMR model does not have any.


## 2.2. Perturbation Theory Based on Lie–Trotter Operator Splitting Method

Perturbation theory as developed by Fouque et al. [22] is a methodology utilized to find an approximated solution when the original problem is difficult to solve by separating it into more easily solvable, simple parts. If we apply the Feynman-Kac formula, we find that u±ϵ satisfies the following Kolmogorov backward equation:(6)Lϵu±ϵt,x1,x2,y1;t0,x10,x20,y0=0,t<T,(7)Lϵ≔1ϵL0+1ϵL1+L2,u±ϵt,x1,x2,y1;t0⟶t,x10,x20,y0=δx±-x0±, where(8)L0=m-y0∂∂y0+ν2g2y0∂2∂y02,L1=ν2f1y0x10gy0ρ1Y∂2∂x10∂y0+ν2f2y0x20gy0ρ2Y∂2∂x20∂y0,(9)L2=∂∂t0+12x102f12y0∂2∂x102+12x202f22y0∂2∂x202+x10x20ρ12f1y0f2y0∂2∂x10∂x20. Here, L0 is the infinitesimal generator of the SFMR process Y. L1 contains the mixed partial derivatives due to the correlations of the two Brownian motions W1 and WY and W2 and WY, respectively. L2 is the operator of a generalized version of the two-dimensional standard Brownian motion at the volatility levels f1(y0) and f2(y0) instead of constant volatilities, respectively.Before we solve problem (6), we write a useful lemma about the centering (or solvability) condition on the Poisson equation related to the operator L0 as follows.Lemma 1.
Ifψ(y)∈C2(R) which is a solution to the Poisson equation(10)L0χy+ψy=0exists, then the centering condition 〈ψ〉=∫ψy1/2πν2exp⁡-(y-m)2/2ν2=0 must be satisfied, where the notation · is the average (or expectation) with respect to the invariant distribution (namely, N(m,ν2)) of Y. Then, solutions of (10) are given by the form (11)χy=∫0∞EyψY∣Y=ydt.Proof.
See Fouque et al. [22].Using the resultant partial differential equation (PDE) (6) and expanding u±ϵ in powers of ϵ, one can approximate u±ϵ to the sum of the leading term u±(0) and the first correction term ϵu±(1) as follows:(12)u±ϵ≈u±0+ϵu±1. Since we focus on the first-order correction term for u±ϵ, we reset (12) with respect to ϵu±(1) and denote it by u~±(1). Using perturbation theory based on Lie–Trotter operator splitting method, u±(0) and u~±(1) must satisfy the following PDEs with boundary conditions, respectively:(13)L2u±0t,x1,x2;t0,x10,x20=0(14)witht,x1,x2;t0⟶t,x10,x20=δx±-x0±,(15)L2u~±1t,x1,x2;t0,x10,x20=L1L0-1L2-L2u±0(16)withu~±1t,x1,x2;t0⟶t,x10,x20=0 and then one obtains the solutions u±(0) and u~±(1) of the PDEs, respectively, (17)u±0=1x±2σ~±2πt-t0exp⁡-ln⁡x±/x0±+1/2σ~±2t-t022σ~±2t-t0,u~±1=-T-tV1∂3∂x103+V2∂3∂x203+V12∂3∂x10x202+V21∂3∂x102x20u±0,where (18)L2=∂∂t0+12x102σ¯1∂2∂x102+12x202σ¯2∂2∂x202+σ¯1σ¯2ρ¯x10x20∂2∂x10∂x20,σ¯1≔f12,σ¯2≔f22,ρ¯≔ρ12f1f2σ¯1σ¯2,σ~+≔σ¯12+σ¯22+2σ¯1σ¯2ρ¯2,σ~-≔σ¯12-σ¯222σ¯12+σ¯22-2σ¯1σ¯2ρ¯ and the constant parameters V1,V2,V12, and V21 are defined as follows: (19)V1=νρ1Yϵx1032f1gθ1′,V2=νρ2Yϵx2032f2gθ2′,V12=νρ1Yϵx10x2022f1gθ2′+ν2ϵρ12ρ2Yx10x202f2gθ12′,V21=νρ2Yϵx102x202f2gθ1′+ν2ϵρ12ρ1Yx102x20f1gθ12′.See the Appendix for the relevant technique.As a result, the distributions of the sum and difference of correlated log-normal random variables under SFMR model are shown to follow combining a shifted log-normal distribution with mixed partial derivatives of it. Also, all the original parameters are absorbed in the group parametersV1, V2, V12, and V21 and the present level y of the hidden process Y driving the fast time-scale volatility needs not be specified in the present approximation.


## 2.3. Numerical Experiment

In this subsection, we illustrate the effectiveness of our result (12) by showing numerical results.Several parameter sets are obtained from Lo except for group parameters. Figure1(a) depicts log-normal distribution (leading-order term or Lo’s result) and log-normal distributions under SFMR model with positive or negative group parameters (leading-order term plus first-order correction term), respectively. The solid line denotes the shifted log-normal distribution and the dash lines show combining the shifted log-normal distribution and the first correction term, respectively. Figure 1(b) depicts the errors calculated by subtracting the log-normal distributions with positive or negative group parameters from the log-normal distribution, respectively. We also apply the same result to the difference of the probability density function in Figure 2. Our numerical results show that the left and right skew scenarios are presented through first-order correction term and the major discrepancies appear around the peak of the probability density function. These pictures are sensitive to the choice of the involved parameters and give a lot of flexibility to the shape of the transition densities.Figure 1
Sum of probability density function:x10=110,  x20=100,  σ¯1=0.25,  σ¯2=0.15,  ρ¯=-0.5, and t-t0=1.
(a)
(b)Figure 2
Difference of probability density function:x10=110,  x20=100,  σ¯1=0.25,  σ¯2=0.15,  ρ¯=-0.5, and  t-t0=1.


## 3. Final Remarks

Stochastic processes are popular in modeling various economics and financial variables. The transition density function especially plays a key role in the analysis of continuous-time diffusion models. In this paper, we obtained an analytic approximation of correlated log-normal random variables under SFMR model.This paper offers various possible directions for further development. Our result can be applied to pricing and hedging spread options and is to incorporate a slowly varying volatility-driving process into the SFMR model. Also, this result can provide a very useful guide for credit risk management (see [26, 27]). We leave these issues as future research topics.



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