# The Log-Asset Dynamic with Euler–Maruyama Scheme under Wishart Processes

Authors: Raphael Naryongo; Philip Ngare; Anthony Waititu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

This article deals with Wishart process which is defined as matrix generalization of a squared Bessel process. We consider a single risky asset pricing model whose volatility is described by Wishart affine diffusion processes. The multifactor volatility specification enables this model to be flexible enough to describe the market prices for short or long maturities. The aim of the study is to derive the log-asset returns dynamic under the double Wishart stochastic volatility model. The corrected Euler–Maruyama discretization technique is applied in order to obtain the numerical solution of the log-asset return dynamic under Bi-Wishart processes. The numerical examples show the effect of the model parameters on the asset returns under the double Wishart volatility model.


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## Body

## 1. Introduction

The introduction of the Heston stochastic volatility model was due to the Black and Scholes [1] model limitation of not accommodating the observable phenomenon that implied volatility of derivative products depending on strike and maturity. The Heston [2] model has been popular and widely applied in financial markets due to its flexibility, financial interpretation of parameters, and analytical tractability property since it belongs to the class of affine processes (see Filipovic and Mayerhofer [3]). The affine property allows the model to form a closed form solution of the characteristic function of the log-price, to obtain European call option price by Fourier transform inversion.However, despite the Heston model popularity, Da Fonseca et al. [4], Christoffersen et al. [5], Ahdida and Alfonsi and Alfonsi [6], Kang et al. [7], and Gouriéroux [8] have clearly stated that the biggest weakness of the model is that it does not generate the realistic term structure of the volatility smiles. Hence, the Heston model provides too flat implied volatility surface to attain reality, yet generally implied volatility has steep curve and convexity in short maturity and tends to be linear for long maturity. This indicates that the model is not flexible enough to describe the market prices. This problem can be handled through generalizing the Heston model into a multifactor form. Using two approaches, the first is by adding jump in the stock dynamic or volatility and secondly by investigating the multifactor nature of implied volatility as in the study by Benabid et al. [9], Da Fonseca et al. [4], and Kang and Kang [10].It is well accepted that the multifactor approach is the best one to solve the pricing problem of derivative products and volatility smile. This shows that among the multifactor models, the Wishart multidimensional stochastic volatility model (that is a matrix defined stochastic volatility model) is one of the most flexible model; this is because the term structure of the realized volatilities in this model is described by a positive semidefinite matrix-valued stochastic process. The Wishart process is defined in Bru [11] as a matrix generalization of a squared Bessel process. Da Fonseca et al. [12], Alfonsi [13], Da Fonseca et al. [4], and Benabid et al. [9] defined Wishart process as a stochastic process which is a positive semidefinite matrix-valued generalization of the square root process. The Wishart multidimensional stochastic volatility model found its application in finance by Gouriéroux and Sufana [14].The aim of this study is to derive the log-asset returns dynamic under double Wishart diffusion processes, through generalization of the Heston model into multifactor nature for a single asset pricing model. That is, the asset dynamic depends on two Wishart volatility diffusion processes, with two dependence matrices describing the correlations between the asset dynamic and Wishart processes. This enables the model to be flexible enough to describe the market prices or to match the term structure of implied volatilities. This study will be of importance to investors to analyze and predict the behavior of the asset price or asset return over a period of time. The numerical solution of the log-price asset returns dynamic under the double Wishart model is obtained through the application of the corrected Euler–Maruyama discretization technique. The numerical examples demonstrate the effect of model parameters on the asset return behavior under the double Wishart volatility model.The paper is organized as follows. In Section2, we give definition of Wishart process, uniqueness and existence of solution, change of probability measure, and Wishart volatility model with one dependence matrix. In Section 3, we present the double Wishart stochastic volatility model, correlation structure, log-asset returns dynamic, infinitesimal generator, and Euler–Maruyama discretization scheme for the double Wishart volatility model. In Section 4, we give the numerical illustrations for log-asset returns. In Section 5, we provide conclusion and recommendation for further research work.


## 2. The Wishart Process

Definition 1.
LetWt,t≥0 be a n×n matrix-valued Brownian motion under the probability measure ℚ. The Wishart process γt satisfies the equation(1)dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,where Q∈GLnℝ is the invertible matrix, M∈Mn is the nonpositive matrix, γ0∈Sn+ is the nonnegative symmetric matrix, and β is a real parameter. The condition β>n−1 is taken to ensure existence and uniqueness of the γt∈Sn+ solution for equation (1), and eigen values of the solution are nonnegative for all t≥0 a.s γt∈Sn+. Following Benabid et al. [9], the probability measure ℚ corresponds to a risk-neutral measure.
The infinitesimal generator of Wishart processγt has been already studied in Bru [11], formulated as follows:(2)ℒγ=TrβQTQ+Mγ+γMTD+2γDQTQD,where the matrix differential operator D is Di,j=∂/∂γi,j.


### 2.1. The Uniqueness and Existence of the Solution of Wishart Process

The Wishart processes are affine processes defined onSn+ℝ. Following Bru [11], La Bua and Marazzina [15], and Alfonsi [13], the results on weak and strong solution of the Wishart differential equation (1) can be shown.Lemma 1.
LetXt be affine process with continuous trajectories defined in Sn+ℝ, with stochastic differential equation as follows:(3)Xt=X0+∫0tα¯+DXsds+∫0tXsdWsQ+QTdWsTXs,where X0,α¯∈Sn+ℝ, Q∈Mnℝ, and D:Sn+ℝ⟶Sn+ℝ is a linear transformation. The process admits a unique weak solution inSn+ℝ if(i)
α¯−n−1QQT∈Sn+ℝ.(ii)
For allX1,X2∈Sn+ℝ such that TrX1X2=0⇒TrDX1X2≥0, where Tr. is a trace of a square matrix. So, X0 contains in a set of real positive-definite matrices Sn+ℝ, with condition (i) is replaced by a stronger requirement.(iii)
α¯−n+1QQT∈Sn+ℝ. Then, there exists a unique strong solution for equation (1) in Sn+ℝ. By observation α¯=ΩΩT and DX0=MX0+X0MT and when we assume restrictive parametrization for the deterministic part of the drift, ΩΩT=βQQT. Conditions (i) and (iii) are satisfied as long as β≥n−1 and β≥n+1, respectively; with direct comparison, we get Wishart SDE (1) from (3). The real positive parameter β also plays a role in Feller’s condition in the univariate case.Lemma 2.
Letℱtt≥0 represent the filtration generated by Wt,t≥0. We consider continuous ℱt-adapted processes att≥0,btt≥0,andctt≥0, respectively, valued in Mnℝ,Mnℝ, and Snℝ and a process Mtt≥0 that admits the following semimartingale decomposition:(4)dMt=ctdt+btdWtat+atTdWtTbtT.
Fori,j,m,n∈1,…,n, the quadratic covariation of Mti,j and Mtm,n is given as(5)dMti,j,dMtm,n=btbtTi,matTatj,n+btbtTi,natTatj,m+btbtTj,matTati,n+btbtTj,natTati,mdt.Proof.
Clearly, we can see that the quadratic covariation (4) depends on at and bt only through the matrices atT and btT as follows:(6)dMt=ctdt+btdWtat+atTdWtTbtT=ctdt+∑k,l=1nbti,katl,j+btj,katl,idWtk,l.
The quadratic covariation is calculated as follows:(7)dMti,j,dMtm,n=∑k,l=1nbti,katl,j+btj,katl,ibtm,katl,n+btn,katl,imdt=∑k,l=1nbti,kbtm,katl,natl,j+∑k,l=1nbti,kbtn,katl,matl,j+∑k,l=1nbtj,kbtm,katl,natl,i+∑k,l=1nbtj,kbtn,katl,matl,i=btbtTi,matTatTn,j+btbtTi,natTatTm,j+btbtTj,matTatTn,i+btbtTj,natTatTm,i.
ParticularlyMt can be written in form of trace as follows:(8)dTrMt=Trctdt+2TratTbtdWt.


### 2.2. Link between Wishart Processes and Ornstein–Uhlenbeck Processes

The best way to generate a Wishart process is to substitute the Gaussian vectorXi in the definition of a Wishart distribution with Ornstein–Uhlenbeck processes Xi,t; this implies that β is an integer. Considering β-independent n-dimensional Ornstein–Uhlenbeck processes, dXk,t=MXk,tdt+QTdWk,t and γt:=∑k=1βXk,tXk,tT (see Benabid et al. [9]).Here,dγt=γt+dt−γt; then, the process γt is a Wishart process with the following stochastic differential equation:(9)dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,where W a matrix-valued Brownian motion, obtained by(10)γtdWt=∑k=1βXk,tdWk,tT,where the matrix M is taken as the mean reversion parameter of the Wishart process and Q is the volatility parameter.


### 2.3. Change of the Probability Measure

From the mathematics point of view, it is necessary to describe the change of the probability measure to allow a change of the drift in the dynamics process. In financial application, especially in the practical aspects, Wishart process has to be simulated in its general form withβ≥n+1, such that β∈ℝ. The function permits to write K=β+2λ with β=K≥n+1 and λ real number with 0≤λ≤1/2.The goal is to find a change of probability measure in order to change the generalized Wishart diffusion process into the simple one, whereβ is an integer. Therefore, the new probability measure ℙ, following Benabid and Bjork can be expressed as follows.Theorem 1.
Letq=β+λ−n−1. If hTℚ,ℙ=dℚ/dℙ defines the Radon–Nikodym derivative of ℚ with respect to ℙ, then(11)hTℚ,ℙ=detγTdetγ0λ/2exp−λTTrMexp−λ2q∫0TTrγs−1QTQds.Proof.
The probability measureℙ can be specified through an exponential martingale.(12)dℚdℙ=expλ∫0TTrγs−1dWsQ−λ22∫0TTrγs−1QTQds.
We define the new process as follows:(13)Wt=Wt˜+λ∫0tγs−1QTds.
We check using Girsanov theorem thatW is a matrix-valued Brownian motion under the probability measure ℙ while the dynamics of Wishart diffusion process under the probability measure ℙ is as follows:(14)dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,where(15)Wt=Wt˜+λ∫0tγs−1QTds,dWt=dWt˜+λγt−1QTdt.
SubstitutingWt above in the Wishart process (14),(16)dγt=βQQT+Mγt+γtMTdt+γtdWt˜+λγt−1QTdtQ+QTdWt˜+λγt−1QTdtTγt.
Then,(17)dγt=β+2λQQT+Mγt+γtMTdt+γtdWt˜Q+QTdW˜tTγt.
The Radon–Nikodym derivative is computed using the determinant dynamics.(18)∫0Tdlogdetγs∫0Tβ−n−1Trγs−1QTQ+2TrMds+2∫0TTrγs−1dWsQlogdetγTdetγ0=2TTrM+∫0Tβ−n−1Trγs−1QTQds+2∫0TTrγs−1dWsQlogdetγTdetγ0λ/2=λTTrM+λ2∫0Tβ−n−1Trγs−1QTQds+λ∫0TTrγs−1dWsQ,expλ∫0TTrγs−1dWsQ=detγTdetγ0λ/2exp−λTTrMexp−λ2∫0Tβ−n−1Trγs−1QTQds.
From equation (12), the new measure with W˜ is given by(19)dℚdℙ=expλ∫0TTrγs−1dW˜sQexp−λ22∫0TTrγs−1QTQds.exp−λ2∫0Tβ+2λ−n−1Trγs−1QTQdsexp−λ22∫0TTrγs−1QTQds=detγTdetγ0λ/2exp−λTTrMexp−λ2β+2λ−n−1−λ22∫0TTrγs−1QTQds=detγTdetγ0λ/2exp−λTTrMexp−λ2β+λ−n−1∫0TTrγs−1QTQds.
The change of the probability measure is obtained.


### 2.4. Wishart Volatility Model

In the study by Da Fonseca et al. [4] and Benabid et al. [9], under the risk-neutral probability measure and arbitrage-free financial market, the risky asset price dynamic and its volatility process are as follows:(20)dStSt=μdt+TrγtdZt,S0=s,dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,γ0=γ,where μ represents the risk free interest rate, Tr is the trace operator, Z∈Mn is a matrix Brownian motion under the risk-neutral measure, and γt belongs to the set of symmetric n×n positive-definite matrices as well as its square root γt. We observe that volatility of the risky asset is the trace of the matrix γt, with Ω,M,Q∈Mn, and Wt∈Mn is a matrix Brownian motion.The dynamic of the Wishart process (see Bru [11]) denotes a matrix analogue of the square root mean-reverting process. To ensure strict positivity and the typical mean-reverting feature of the volatility, matrix M is considered to be negative semidefinite, with the real parameter β>n−1, for condition of uniqueness and existence of the solution of the dynamics of the Wishart processes.


## 2.1. The Uniqueness and Existence of the Solution of Wishart Process

The Wishart processes are affine processes defined onSn+ℝ. Following Bru [11], La Bua and Marazzina [15], and Alfonsi [13], the results on weak and strong solution of the Wishart differential equation (1) can be shown.Lemma 1.
LetXt be affine process with continuous trajectories defined in Sn+ℝ, with stochastic differential equation as follows:(3)Xt=X0+∫0tα¯+DXsds+∫0tXsdWsQ+QTdWsTXs,where X0,α¯∈Sn+ℝ, Q∈Mnℝ, and D:Sn+ℝ⟶Sn+ℝ is a linear transformation. The process admits a unique weak solution inSn+ℝ if(i)
α¯−n−1QQT∈Sn+ℝ.(ii)
For allX1,X2∈Sn+ℝ such that TrX1X2=0⇒TrDX1X2≥0, where Tr. is a trace of a square matrix. So, X0 contains in a set of real positive-definite matrices Sn+ℝ, with condition (i) is replaced by a stronger requirement.(iii)
α¯−n+1QQT∈Sn+ℝ. Then, there exists a unique strong solution for equation (1) in Sn+ℝ. By observation α¯=ΩΩT and DX0=MX0+X0MT and when we assume restrictive parametrization for the deterministic part of the drift, ΩΩT=βQQT. Conditions (i) and (iii) are satisfied as long as β≥n−1 and β≥n+1, respectively; with direct comparison, we get Wishart SDE (1) from (3). The real positive parameter β also plays a role in Feller’s condition in the univariate case.Lemma 2.
Letℱtt≥0 represent the filtration generated by Wt,t≥0. We consider continuous ℱt-adapted processes att≥0,btt≥0,andctt≥0, respectively, valued in Mnℝ,Mnℝ, and Snℝ and a process Mtt≥0 that admits the following semimartingale decomposition:(4)dMt=ctdt+btdWtat+atTdWtTbtT.
Fori,j,m,n∈1,…,n, the quadratic covariation of Mti,j and Mtm,n is given as(5)dMti,j,dMtm,n=btbtTi,matTatj,n+btbtTi,natTatj,m+btbtTj,matTati,n+btbtTj,natTati,mdt.Proof.
Clearly, we can see that the quadratic covariation (4) depends on at and bt only through the matrices atT and btT as follows:(6)dMt=ctdt+btdWtat+atTdWtTbtT=ctdt+∑k,l=1nbti,katl,j+btj,katl,idWtk,l.
The quadratic covariation is calculated as follows:(7)dMti,j,dMtm,n=∑k,l=1nbti,katl,j+btj,katl,ibtm,katl,n+btn,katl,imdt=∑k,l=1nbti,kbtm,katl,natl,j+∑k,l=1nbti,kbtn,katl,matl,j+∑k,l=1nbtj,kbtm,katl,natl,i+∑k,l=1nbtj,kbtn,katl,matl,i=btbtTi,matTatTn,j+btbtTi,natTatTm,j+btbtTj,matTatTn,i+btbtTj,natTatTm,i.
ParticularlyMt can be written in form of trace as follows:(8)dTrMt=Trctdt+2TratTbtdWt.


## 2.2. Link between Wishart Processes and Ornstein–Uhlenbeck Processes

The best way to generate a Wishart process is to substitute the Gaussian vectorXi in the definition of a Wishart distribution with Ornstein–Uhlenbeck processes Xi,t; this implies that β is an integer. Considering β-independent n-dimensional Ornstein–Uhlenbeck processes, dXk,t=MXk,tdt+QTdWk,t and γt:=∑k=1βXk,tXk,tT (see Benabid et al. [9]).Here,dγt=γt+dt−γt; then, the process γt is a Wishart process with the following stochastic differential equation:(9)dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,where W a matrix-valued Brownian motion, obtained by(10)γtdWt=∑k=1βXk,tdWk,tT,where the matrix M is taken as the mean reversion parameter of the Wishart process and Q is the volatility parameter.


## 2.3. Change of the Probability Measure

From the mathematics point of view, it is necessary to describe the change of the probability measure to allow a change of the drift in the dynamics process. In financial application, especially in the practical aspects, Wishart process has to be simulated in its general form withβ≥n+1, such that β∈ℝ. The function permits to write K=β+2λ with β=K≥n+1 and λ real number with 0≤λ≤1/2.The goal is to find a change of probability measure in order to change the generalized Wishart diffusion process into the simple one, whereβ is an integer. Therefore, the new probability measure ℙ, following Benabid and Bjork can be expressed as follows.Theorem 1.
Letq=β+λ−n−1. If hTℚ,ℙ=dℚ/dℙ defines the Radon–Nikodym derivative of ℚ with respect to ℙ, then(11)hTℚ,ℙ=detγTdetγ0λ/2exp−λTTrMexp−λ2q∫0TTrγs−1QTQds.Proof.
The probability measureℙ can be specified through an exponential martingale.(12)dℚdℙ=expλ∫0TTrγs−1dWsQ−λ22∫0TTrγs−1QTQds.
We define the new process as follows:(13)Wt=Wt˜+λ∫0tγs−1QTds.
We check using Girsanov theorem thatW is a matrix-valued Brownian motion under the probability measure ℙ while the dynamics of Wishart diffusion process under the probability measure ℙ is as follows:(14)dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,where(15)Wt=Wt˜+λ∫0tγs−1QTds,dWt=dWt˜+λγt−1QTdt.
SubstitutingWt above in the Wishart process (14),(16)dγt=βQQT+Mγt+γtMTdt+γtdWt˜+λγt−1QTdtQ+QTdWt˜+λγt−1QTdtTγt.
Then,(17)dγt=β+2λQQT+Mγt+γtMTdt+γtdWt˜Q+QTdW˜tTγt.
The Radon–Nikodym derivative is computed using the determinant dynamics.(18)∫0Tdlogdetγs∫0Tβ−n−1Trγs−1QTQ+2TrMds+2∫0TTrγs−1dWsQlogdetγTdetγ0=2TTrM+∫0Tβ−n−1Trγs−1QTQds+2∫0TTrγs−1dWsQlogdetγTdetγ0λ/2=λTTrM+λ2∫0Tβ−n−1Trγs−1QTQds+λ∫0TTrγs−1dWsQ,expλ∫0TTrγs−1dWsQ=detγTdetγ0λ/2exp−λTTrMexp−λ2∫0Tβ−n−1Trγs−1QTQds.
From equation (12), the new measure with W˜ is given by(19)dℚdℙ=expλ∫0TTrγs−1dW˜sQexp−λ22∫0TTrγs−1QTQds.exp−λ2∫0Tβ+2λ−n−1Trγs−1QTQdsexp−λ22∫0TTrγs−1QTQds=detγTdetγ0λ/2exp−λTTrMexp−λ2β+2λ−n−1−λ22∫0TTrγs−1QTQds=detγTdetγ0λ/2exp−λTTrMexp−λ2β+λ−n−1∫0TTrγs−1QTQds.
The change of the probability measure is obtained.


## 2.4. Wishart Volatility Model

In the study by Da Fonseca et al. [4] and Benabid et al. [9], under the risk-neutral probability measure and arbitrage-free financial market, the risky asset price dynamic and its volatility process are as follows:(20)dStSt=μdt+TrγtdZt,S0=s,dγt=βQQT+Mγt+γtMTdt+γtdWtQ+QTdWtTγt,γ0=γ,where μ represents the risk free interest rate, Tr is the trace operator, Z∈Mn is a matrix Brownian motion under the risk-neutral measure, and γt belongs to the set of symmetric n×n positive-definite matrices as well as its square root γt. We observe that volatility of the risky asset is the trace of the matrix γt, with Ω,M,Q∈Mn, and Wt∈Mn is a matrix Brownian motion.The dynamic of the Wishart process (see Bru [11]) denotes a matrix analogue of the square root mean-reverting process. To ensure strict positivity and the typical mean-reverting feature of the volatility, matrix M is considered to be negative semidefinite, with the real parameter β>n−1, for condition of uniqueness and existence of the solution of the dynamics of the Wishart processes.


## 3. Presentation of Double Wishart Model in Stock Market

This section introduces a proposed novel model, the multifactor model with two Wishart processes or double Wishart stochastic volatility model with two dependence matrices. The model takes two underlying volatility components defined as the trace of a Wishart process. However, following Naryongo et al. [17], the diagonal components of the Wishart matrices will be the factors guiding the dynamics of volatilities.Under arbitrage-free financial market and under the risk-neutral probability measure, we consider the following risky asset dynamic:(21)dStSt=μdt+Trγ1,tdZtγ1+γ2,tdZtγ2,where the quadratic variations are as follows:(22)dγ1,t=β1Q1Q1T+M1γ1,t+γ1,tM1Tdt+γ1,tdWtγ1Q1+Q1TdWtγ1Tγ1,t,γ1,0=γ1,dγ2,t=β2Q2Q2T+M2γ2,t+γ2,tM2Tdt+γ2,tdWtγ2Q2+Q2TdWtγ2Tγ2,t,γ2,0=γ2,where β1andβ2 are real parameters such that β1,β2>n−1, Q1,Q2M1,M2∈Mn, Q is invertible matrix, and Wtγ1,Wtγ2∈Mn are matrices Brownian motions, and also Ztγ1,Ztγ2∈Mn.Lemma 3.
The correlation between the Brownian matrices of the stock price dynamic and the Brownian matrices of the Wishart processes is given as(23)ρ1,t=TrR1TQ1γ1,tTrγ1,tTrQ1TQ1γ1,t,ρ2,t=TrR2TQ2γ2,tTrγ2,tTrQ2TQ2γ2,t.Proof.
(24)dStSt=rdt+Trγ1,tdZtγ1+γ2,tdZtγ2=rdt+Trγ1,tTrγ1,tdZtγ1Trγ1,t+Trγ2,tTrγ2,tdZtγ2Trγ2,t=rdt+Trγ1,tdXtγ1+γ2,tdXtγ2,where Xtγ1 and Xtγ2 are standard Brownian motions (see proof in Appendix), and also considering the trace of the dynamics of Wishart volatility process (21), we get(25)dTrγ1,t=β1TrQ1TQ1+2TrM1γ1,tdt+2TrQ1dWtγ1γ1,t,dTrγ2,t=β2TrQ2TQ2+2TrM2γ2,tdt+2TrQ2dWtγ2γ2,t.
These processes can still be presented in the form as follows:(26)dTrγ1,t=β1TrQ1TQ1+2TrM1γ1,tdt+2TrQ1TQ1γ1,tTrQ1dWtγ1γ1,tTrQ1TQ1γ1,t,dTrγ2,t=β2TrQ2TQ2+2TrM2γ2,tdt+2TrQ2TQ1γ2,tTrQ2dWtγ2γ2,tTrQ2TQ2γ2,t,where ξt and ηt are Brownian motions (see proof in Appendix), such that(27)dTrγ1,t=β1TrQ1TQ1+2TrM1γ1,tdt+2TrQ1TQ1γ1,tdξt,dTrγ2,t=β2TrQ2TQ2+2TrM2γ2,tdt+2TrQ2TQ2γ2,tdηt.
We now determine the covariation of the stock price and Wishart processes:(28)CovtdXtγ1,dξt=CovtTrγ1,tdZtγ1Trγ1,t,TrQ1dWtγ1γ1,tTrQ1TQ1γ1,t=EtTrγ1,tdWtγ1R1TTrγ1,tTrQ1dWtγ1γ1,tTrQ1TQ1γ1,t=∑ijCovteiTγ1,tdWtγ1R1Tei,ejTQ1dWtγ1γ1,tejTrγ1,tTrQ1TQ1γ1,t=∑ijEteiTγ1,tdWtγ1R1TeiejTQ1TdWtγ1Tγ1,tejTrγ1,tTrQ1TQ1γ1,t=∑ijeiTγ1,tTrR1TeiejTQ1Tγ1,tejdtTrγ1,tTrQ1TQ1γ1,t=∑ijTrQ1R1eiejTeiTγ1,tejdtTrγ1,tTrQ1TQ1γ1,t=∑ijejTQ1ReieiTγ1,tejdtTrγ1,tTrQ1TQ1γ1,t=∑ijejTQ1R1γ1,tejdtTrγ1,tTrQ1TQ1γ1,t=TrR1TQ1γ1,tTrγ1,tTrQ1TQ1γ1,t=ρ1,t.
The same procedures can be carried out on the second differential equation to obtain the covariation as follows:(29)CovtdXtγ2,dηt=TrR2TQ2γ2,tTrγ2,tTrQ2TQ2γ2,t=ρ2,t.
The two correlations between the Brownian motions of the asset and the Brownian motions of the Wishart processes are stochastic.


### 3.1. The Correlation Structure of the Model

The Brownian matricesWtγ1,Wtγ2,Ztγ1,andZtγ2 can be correlated in such a way that all the (scalar) Brownian motions belonging to i of Ztγ1andZtγ2 and corresponding Brownian motions of the column j of Wtγ1andWtγ2 have the same correlation, say Rijγ1andRijγ2. This provides constant matrices R1,R2∈Mn which describe the two correlation structures, in such a way that Ztγ1 and Ztγ2 presented as follows:(30)Ztγk:=WtγkRkT+BtγkI−RkRkT,fork=1,2,where I denotes the identity matrix, T is the transposition, and Btγ1 and Btγ2 are (matrices) Brownian motion independent of Wtγ1andWtγ2, respectively.Lemma 4.
The correlation structureZtγ1:=Wtγ1R1T+Btγ1I−R1R1T is a matrix Brownian motion.Proof.
We note thatZtγ1 is a matrix Brownian motion iff for α1,β1∈ℝn,(31)CovtdZtγ1,dZtγ1β1=EtdZtγ1α1dZtγ1β1T=α1Tβ1Idt.
Then,(32)CovtdZtγ1α1,dZtγ1β1=EtdWtγ1R1Tα1+dBtγ1I−R1R1Tα1dWtγ1R1Tβ1+dBtγ1I−R1R1Tβ1=CovtdWtγ1R1Tα1,dWtγ1R1Tβ1+CovtdBtγ1I−R1R1Tα1,dBtγ1I−R1R1Tβ1=α1TR1R1Tβ1Idt+α1TI−R1R1Tβ1Idt=α1Tβ1Idt,similarly for the second dynamic process as(33)CovtdZtγ2α2,dZtγ2β2=α2Tβ2Idt.


### 3.2. The Log-Asset Return Dynamic under Double Wishart Model

LetYt denote the log-price and the matrices R1 and R2 describe the correlations between the Brownian of the asset price process and those of the Wishart processes.Proposition 1.
The log-price processYt=logSt under the double Wishart volatility model is as follows:(34)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1+γ2,tdZtγ2.Proof.
LetYt=logSt, the asset dynamic is given as follows:(35)dStSt=rdt+Trγ1,tdZtγ1+γ2,tdZtγ2.
By applying Ito’s formula onYt (see Björk [16] and Shreve [18]), we get(36)dYt=dlogSt=dStSt−12dSt2St2.
Replacing the asset process (35) in the derivative equation (36) of Yt,(37)dYt=rdt+Trγ1,tdZtγ1+γ2,tdZtγ2−12Trγ1,t+γ2,tdt,dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1+γ2,tdZtγ2.
This can still be presented in the form as follows:(38)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdWtγ1R1T+dBtγ1I−R1R1T+γ2,tdWtγ2R2T+dBtγ2I−R2R2T,Y0=y.


#### 3.2.1. Infinitesimal Generator under Double Wishart Model

The log-price process and Wishart affine processes, combined with the corresponding pair of two correlated Brownian motions,Zsγ1,Y,Zsγ1,γ1 and Zsγ2,Y,Zsγ2,γ2, can be presented in the form as in Benabid et al. [9] for easy handling of the dynamics.(39)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1,Y+γ2,tdZtγ2,YdTrγ1,t=β1TrQ1TQ1+2TrM1γ1,tdt+2TrQ1TQ1γ1,tρ1,tdZtγ1,Y+I−ρ1,t2dZtγ1dTrγ2,t=β2TrQ2TQ2+2TrM2γ2,tdt+2TrQ2TQ2γ2,tρ2,tdZtγ2,Y+I−ρ2,t2dZtγ2,where(40)d<Zγ1,Y,Zγ1>t=ρ1,t=TrR1TQ1γ1,tTrγ1,tTrQ1TQ1γ1,t,d<Zγ2,Y,Zγ2>t=ρ2,t=TrR2TQ2γ2,tTrγ2,tTrQ2TQ2γ2,t.Lemma 5.
The infinitesimal generator under the double Wishart volatility model forYt,γ1,t,andγ2,t is as follows:(41)ℒY,γ1,γ2=r−Trγ1+γ22∂∂y+Trγ1+γ22∂2∂y2+β1TrQ1TQ1+2TrM1γ1∂∂γ1+2Trγ1∂∂γ1Q1TQ1∂∂γ1+β2TrQ2TQ2+2TrM2γ2∂∂γ2+2Trγ2∂∂γ2Q2TQ2∂∂γ2+2Trγ1R1Q1∂∂γ1∂∂y+2Trγ2R2Q2∂∂γ2∂∂y.Proof.
The infinitesimal generator has nontrivial term which arises from covariationd<γθij,Y> corresponding to the coefficients of the term(42)∂2∂xθ;ij∂y=∂∂xθ;ij∂∂y,i,j=1,…,n,θ=1,2.
LetCθ;t:=γθ;t be the square root matrix where(43)γθ;tij=∑t=1nCθ;tilCθ;tlj=∑t=1nCθ;tilCθ;tjl.
SinceCθ;t is symmetric, we determine the covariation terms corresponding to ∂2/∂xθ;ij∂y coefficients as follows:(44)dγθij,Y=Et∑i,k=1nCθ;tildWlkθQkjθ+∑l,k=1nCθ;tjldWlkθQkiθ∑l,k,h=1nCθ;tlkdWkhθRlhθ=∑l,k,h=1nCθ;tilQkjθ+Cθ;tjlQkiθCθ;thlRhkθdt=∑k,h=1n∑l=1nCθ;tilCθ;thlQkjθ+∑l=1nCθ;tjlCθ;thlQkiθRhkθdt=∑k,h=1nγθ;tihQkjθ+γθ;tjhQkiθRhkθdt.
This provides the corresponding coefficients of the term:(45)2TrγθRθQθDθ∂∂y=2∑i,j,k,h=1nDθijγθjhRhkθQkiθ∂∂y.
The notation whenθ=1,γ1=γ1,R1=R1,Q1=Q1,andD1=∂/∂xijγ1 while for θ=2,γ2=γ2,R2=R2,Q2=Q2andD2=∂/∂xijγ2.


### 3.3. The Euler–Maruyama Discretization Scheme

This section basically deals with the discretization scheme of the Bi-Wishart volatility model using Euler–Maruyama discretization technique.This is one of the best approximation methods to handle sophisticated stochastic differential equations as in the study by Ahdida and Alfonsi [6], Fadugba et al. [19], Dereich et al. [20], Berkaoui et al. [21], and Mao [22]. It is a time discrete approximation of an Ito process. Consider St as an Ito process on t∈t0,T satisfying differential equation as follows:(46)St=S0+∫0tμt,sds+∫0tγt,sdWs,t≥0.For discretizationt0<t1<t2⋯<tN=T or on a regular grid ti=iT/N, it is given as(47)S^ti+1=S^ti+μS^tiTN+γS^tiWti+1−Wti,1≤i≤N−1,and also considering a continuous case, we have(48)S^ti+1=S^ti+μS^titi+1−ti+γS^tiWti+1−Wti,t∈ti,ti+1.One can observe thatS^ the Euler scheme depends on N throughout the discretization grid, and it is similarly better to denote it as S^N.


### 3.4. The Corrected Euler–Maruyama Discretization Scheme for Double Wishart Affine Processes

This section discusses the discretization techniques for the Bi-variate Wishart model. The log-price dynamic and Wishart processes are discretized using corrected Euler–Maruyama method. We notice that the Euler scheme is not well defined; in fact, the Gaussian increment may lead the scheme to negative values with some positive probability, and the square root is then no long defined. Therefore, this calls in the need to apply the corrected Euler–Maruyama scheme to avoid negative values (see Alfonsi [13] and Gauthier and Possamaï [23]).The asset dynamic is as follows:(49)dStSt=μdt+Trγ1,tdZtγ1+γ2,tdZtγ2,S0=s,where the Wishart matrix processes are given as follows:(50)γj,t=γj,0+∫0tΩj+ηjγ,sds+∫0tγj,sdWsγjQj+∫0tQjTdWsγjTγj,sforj=1,2,where Ωj=βjQjQjT such that γj,0,Ωj∈Sn+ℝ while ηj∈ℒSn+ℝ and Mj∈Mnℝ.Then, discretize equations (50) and (37), by considering a time horizon T and regular time grid tiN=iT/N, for i=0,…,N.(51)γ^j,ti+1N=γ^j,tiN+Ωj+ηjγ^j,tiNTN+γ^j,tiN+Wti+1Nγj−WtiNγjQj+QjTWti+1Nγj−WtiNγjγ^j,tiN+,forj=1,2,where γ^j,t0N=γj,0=γj.Now, we proceed to discretize the log-price dynamics as follows:(52)Y^ti+1Ny=Y^tiNy+μ−12Trγ^1,tiN+γ^2,tiNTN+Trγ^1,tiN+Zti+1Nγ1−ZtiNγ1+γ^2,tiN+Zti+1Nγ2−ZtiNγ2.We now discretize the Brownian motions in equation (52) by considering the correlation structures in (30), that is, Ztγj=WtγjRjT+BtγjI−RjRjT,for,j=1,2:(53)Zti+1Nγj−ZtiNγj=Wti+1NγjRjT+Bti+1NγjI−RjRjT−WtiNγjRjT+BtiNγjI−RjRjT=Wti+1Nγj−WtiNγjRjT+Bti+1Nγj−BtiNγjI−RjRjT.These Brownian matricesZγj and Bγj follow normal distribution, that is,(54)Wti+1Nγj−WtiNγj∼N0,ti+1N−tiNI,Wti+1Nγj−WtiNγj∼ti+1N−tiNN0,I,and(55)Bti+1Nγj−BtiNγj∼ti+1N−tiNN0,I.Hence, simplifying the computation, we substitute equations (54) and (55) into equation (54)(56)Zti+1Nγj−ZtiNγj∼RjT+I−RjRjTti+1N−tiNN0,I.Equation (56) is reduced in the form as follows:(57)Zti+1Nγj−ZtiNγj∼RjT+I−RjRjTTNN0,I.Let the log-price dynamicsYt=logSt in equation (52) be Y^ti+1N=logS^ti+1N and logS^tiN as follows:(58)logS^ti+1N=logS^tiN+μ−12Trγ^1,tiN+γ^2,tiNTN+Trγ^1,tiN+Zti+1Nγ1−ZtiNγ1+γ^2,tiN+Zti+1Nγ2−ZtiNγ2.This log-price expression can be represented in the form as follows:(59)logS^ti+1N=logS^tiN+μTN−12∫tiNti+1NTrγ1s+γ2sds+Tr∫tiNti+1Nγ1+sdZsγ1+∫tiNti+1Nγ2+sdZsγ2.The incrementξk,i1≤k≤β allows to simulate Tr∫tiNti+1Nγ1sdWsγ1R1T, and for other terms, the standard Euler scheme is selected, that is,(60)∫tiNti+1Nγ1+sdWsγ1=∑k=1β∫tiNti+1NXk,sdWk,sγ1=Δt∑k=1βXk,tξk,iTTr∫tiNti+1Nγ1+sdBsγ1I−R1R1T=ΔtTrγ^1,tiN+I−R1R1Tζ1,i.We observe that the integral term in equation (59) needs to be simplified as follows:(61)∫tiNti+1Nγ1+sdZsγ1=∫tiNti+1Nγ1+sdWsγ1R1T+∫tiNti+1Nγ1+sdBsγ1I−R1R1T=ti+1N−tiN∑k=1lX^1k,tiNξk,iTR1T+ti+1N−tiNγ^1,tiN+I−R1R1Tζ1,i=TN∑k=1lX^1k,tiNξk,iTR1T+TNγ^1,tiN+I−R1R1Tζ1,i,where the Gaussian vector ζ1,i∼N0,I and the same for the second term are as follows:(62)∫tiNti+1Nγ2+sdZsγ2=TN∑k=1lX^2k,tiNξk,iTR2T+TNγ^2,tiN+I−R2R2Tζ2,i,where ζ2,i∼N0,I, and then we can substitute equations (61) and (62) in equation (59) as follows:(63)logS^ti+1NS^tiN=μTN−12∫tiNti+1NTrγ1s+γ2sds+TNTr∑k=1lX^1k,tiNξk,iTR1T+∑k=1lX^2k,tiNξk,iTR2T+TNTrγ^1,tiN+I−R1R1Tζ1,i+Trγ^2,tiN+I−R2R2Tζ2,i.Finally, we obtain the Euler–Maruyama discretization scheme for the double Wishart diffusion processes as follows:(64)logS^ti+1NS^tiN=μ−12Trγ^1,tiN+γ^2,tiNTN+TNTr∑k=1lX^1k,tiNξk,iTR1T+∑k=1lX^2k,tiNξk,iTR2T+TNTrγ^1,tiN+I−R1R1Tζ1,i+Trγ^2,tiN+I−R2R2Tζ2,i.


## 3.1. The Correlation Structure of the Model

The Brownian matricesWtγ1,Wtγ2,Ztγ1,andZtγ2 can be correlated in such a way that all the (scalar) Brownian motions belonging to i of Ztγ1andZtγ2 and corresponding Brownian motions of the column j of Wtγ1andWtγ2 have the same correlation, say Rijγ1andRijγ2. This provides constant matrices R1,R2∈Mn which describe the two correlation structures, in such a way that Ztγ1 and Ztγ2 presented as follows:(30)Ztγk:=WtγkRkT+BtγkI−RkRkT,fork=1,2,where I denotes the identity matrix, T is the transposition, and Btγ1 and Btγ2 are (matrices) Brownian motion independent of Wtγ1andWtγ2, respectively.Lemma 4.
The correlation structureZtγ1:=Wtγ1R1T+Btγ1I−R1R1T is a matrix Brownian motion.Proof.
We note thatZtγ1 is a matrix Brownian motion iff for α1,β1∈ℝn,(31)CovtdZtγ1,dZtγ1β1=EtdZtγ1α1dZtγ1β1T=α1Tβ1Idt.
Then,(32)CovtdZtγ1α1,dZtγ1β1=EtdWtγ1R1Tα1+dBtγ1I−R1R1Tα1dWtγ1R1Tβ1+dBtγ1I−R1R1Tβ1=CovtdWtγ1R1Tα1,dWtγ1R1Tβ1+CovtdBtγ1I−R1R1Tα1,dBtγ1I−R1R1Tβ1=α1TR1R1Tβ1Idt+α1TI−R1R1Tβ1Idt=α1Tβ1Idt,similarly for the second dynamic process as(33)CovtdZtγ2α2,dZtγ2β2=α2Tβ2Idt.


## 3.2. The Log-Asset Return Dynamic under Double Wishart Model

LetYt denote the log-price and the matrices R1 and R2 describe the correlations between the Brownian of the asset price process and those of the Wishart processes.Proposition 1.
The log-price processYt=logSt under the double Wishart volatility model is as follows:(34)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1+γ2,tdZtγ2.Proof.
LetYt=logSt, the asset dynamic is given as follows:(35)dStSt=rdt+Trγ1,tdZtγ1+γ2,tdZtγ2.
By applying Ito’s formula onYt (see Björk [16] and Shreve [18]), we get(36)dYt=dlogSt=dStSt−12dSt2St2.
Replacing the asset process (35) in the derivative equation (36) of Yt,(37)dYt=rdt+Trγ1,tdZtγ1+γ2,tdZtγ2−12Trγ1,t+γ2,tdt,dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1+γ2,tdZtγ2.
This can still be presented in the form as follows:(38)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdWtγ1R1T+dBtγ1I−R1R1T+γ2,tdWtγ2R2T+dBtγ2I−R2R2T,Y0=y.


### 3.2.1. Infinitesimal Generator under Double Wishart Model

The log-price process and Wishart affine processes, combined with the corresponding pair of two correlated Brownian motions,Zsγ1,Y,Zsγ1,γ1 and Zsγ2,Y,Zsγ2,γ2, can be presented in the form as in Benabid et al. [9] for easy handling of the dynamics.(39)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1,Y+γ2,tdZtγ2,YdTrγ1,t=β1TrQ1TQ1+2TrM1γ1,tdt+2TrQ1TQ1γ1,tρ1,tdZtγ1,Y+I−ρ1,t2dZtγ1dTrγ2,t=β2TrQ2TQ2+2TrM2γ2,tdt+2TrQ2TQ2γ2,tρ2,tdZtγ2,Y+I−ρ2,t2dZtγ2,where(40)d<Zγ1,Y,Zγ1>t=ρ1,t=TrR1TQ1γ1,tTrγ1,tTrQ1TQ1γ1,t,d<Zγ2,Y,Zγ2>t=ρ2,t=TrR2TQ2γ2,tTrγ2,tTrQ2TQ2γ2,t.Lemma 5.
The infinitesimal generator under the double Wishart volatility model forYt,γ1,t,andγ2,t is as follows:(41)ℒY,γ1,γ2=r−Trγ1+γ22∂∂y+Trγ1+γ22∂2∂y2+β1TrQ1TQ1+2TrM1γ1∂∂γ1+2Trγ1∂∂γ1Q1TQ1∂∂γ1+β2TrQ2TQ2+2TrM2γ2∂∂γ2+2Trγ2∂∂γ2Q2TQ2∂∂γ2+2Trγ1R1Q1∂∂γ1∂∂y+2Trγ2R2Q2∂∂γ2∂∂y.Proof.
The infinitesimal generator has nontrivial term which arises from covariationd<γθij,Y> corresponding to the coefficients of the term(42)∂2∂xθ;ij∂y=∂∂xθ;ij∂∂y,i,j=1,…,n,θ=1,2.
LetCθ;t:=γθ;t be the square root matrix where(43)γθ;tij=∑t=1nCθ;tilCθ;tlj=∑t=1nCθ;tilCθ;tjl.
SinceCθ;t is symmetric, we determine the covariation terms corresponding to ∂2/∂xθ;ij∂y coefficients as follows:(44)dγθij,Y=Et∑i,k=1nCθ;tildWlkθQkjθ+∑l,k=1nCθ;tjldWlkθQkiθ∑l,k,h=1nCθ;tlkdWkhθRlhθ=∑l,k,h=1nCθ;tilQkjθ+Cθ;tjlQkiθCθ;thlRhkθdt=∑k,h=1n∑l=1nCθ;tilCθ;thlQkjθ+∑l=1nCθ;tjlCθ;thlQkiθRhkθdt=∑k,h=1nγθ;tihQkjθ+γθ;tjhQkiθRhkθdt.
This provides the corresponding coefficients of the term:(45)2TrγθRθQθDθ∂∂y=2∑i,j,k,h=1nDθijγθjhRhkθQkiθ∂∂y.
The notation whenθ=1,γ1=γ1,R1=R1,Q1=Q1,andD1=∂/∂xijγ1 while for θ=2,γ2=γ2,R2=R2,Q2=Q2andD2=∂/∂xijγ2.


## 3.2.1. Infinitesimal Generator under Double Wishart Model

The log-price process and Wishart affine processes, combined with the corresponding pair of two correlated Brownian motions,Zsγ1,Y,Zsγ1,γ1 and Zsγ2,Y,Zsγ2,γ2, can be presented in the form as in Benabid et al. [9] for easy handling of the dynamics.(39)dYt=r−12Trγ1,t+γ2,tdt+Trγ1,tdZtγ1,Y+γ2,tdZtγ2,YdTrγ1,t=β1TrQ1TQ1+2TrM1γ1,tdt+2TrQ1TQ1γ1,tρ1,tdZtγ1,Y+I−ρ1,t2dZtγ1dTrγ2,t=β2TrQ2TQ2+2TrM2γ2,tdt+2TrQ2TQ2γ2,tρ2,tdZtγ2,Y+I−ρ2,t2dZtγ2,where(40)d<Zγ1,Y,Zγ1>t=ρ1,t=TrR1TQ1γ1,tTrγ1,tTrQ1TQ1γ1,t,d<Zγ2,Y,Zγ2>t=ρ2,t=TrR2TQ2γ2,tTrγ2,tTrQ2TQ2γ2,t.Lemma 5.
The infinitesimal generator under the double Wishart volatility model forYt,γ1,t,andγ2,t is as follows:(41)ℒY,γ1,γ2=r−Trγ1+γ22∂∂y+Trγ1+γ22∂2∂y2+β1TrQ1TQ1+2TrM1γ1∂∂γ1+2Trγ1∂∂γ1Q1TQ1∂∂γ1+β2TrQ2TQ2+2TrM2γ2∂∂γ2+2Trγ2∂∂γ2Q2TQ2∂∂γ2+2Trγ1R1Q1∂∂γ1∂∂y+2Trγ2R2Q2∂∂γ2∂∂y.Proof.
The infinitesimal generator has nontrivial term which arises from covariationd<γθij,Y> corresponding to the coefficients of the term(42)∂2∂xθ;ij∂y=∂∂xθ;ij∂∂y,i,j=1,…,n,θ=1,2.
LetCθ;t:=γθ;t be the square root matrix where(43)γθ;tij=∑t=1nCθ;tilCθ;tlj=∑t=1nCθ;tilCθ;tjl.
SinceCθ;t is symmetric, we determine the covariation terms corresponding to ∂2/∂xθ;ij∂y coefficients as follows:(44)dγθij,Y=Et∑i,k=1nCθ;tildWlkθQkjθ+∑l,k=1nCθ;tjldWlkθQkiθ∑l,k,h=1nCθ;tlkdWkhθRlhθ=∑l,k,h=1nCθ;tilQkjθ+Cθ;tjlQkiθCθ;thlRhkθdt=∑k,h=1n∑l=1nCθ;tilCθ;thlQkjθ+∑l=1nCθ;tjlCθ;thlQkiθRhkθdt=∑k,h=1nγθ;tihQkjθ+γθ;tjhQkiθRhkθdt.
This provides the corresponding coefficients of the term:(45)2TrγθRθQθDθ∂∂y=2∑i,j,k,h=1nDθijγθjhRhkθQkiθ∂∂y.
The notation whenθ=1,γ1=γ1,R1=R1,Q1=Q1,andD1=∂/∂xijγ1 while for θ=2,γ2=γ2,R2=R2,Q2=Q2andD2=∂/∂xijγ2.


## 3.3. The Euler–Maruyama Discretization Scheme

This section basically deals with the discretization scheme of the Bi-Wishart volatility model using Euler–Maruyama discretization technique.This is one of the best approximation methods to handle sophisticated stochastic differential equations as in the study by Ahdida and Alfonsi [6], Fadugba et al. [19], Dereich et al. [20], Berkaoui et al. [21], and Mao [22]. It is a time discrete approximation of an Ito process. Consider St as an Ito process on t∈t0,T satisfying differential equation as follows:(46)St=S0+∫0tμt,sds+∫0tγt,sdWs,t≥0.For discretizationt0<t1<t2⋯<tN=T or on a regular grid ti=iT/N, it is given as(47)S^ti+1=S^ti+μS^tiTN+γS^tiWti+1−Wti,1≤i≤N−1,and also considering a continuous case, we have(48)S^ti+1=S^ti+μS^titi+1−ti+γS^tiWti+1−Wti,t∈ti,ti+1.One can observe thatS^ the Euler scheme depends on N throughout the discretization grid, and it is similarly better to denote it as S^N.


## 3.4. The Corrected Euler–Maruyama Discretization Scheme for Double Wishart Affine Processes

This section discusses the discretization techniques for the Bi-variate Wishart model. The log-price dynamic and Wishart processes are discretized using corrected Euler–Maruyama method. We notice that the Euler scheme is not well defined; in fact, the Gaussian increment may lead the scheme to negative values with some positive probability, and the square root is then no long defined. Therefore, this calls in the need to apply the corrected Euler–Maruyama scheme to avoid negative values (see Alfonsi [13] and Gauthier and Possamaï [23]).The asset dynamic is as follows:(49)dStSt=μdt+Trγ1,tdZtγ1+γ2,tdZtγ2,S0=s,where the Wishart matrix processes are given as follows:(50)γj,t=γj,0+∫0tΩj+ηjγ,sds+∫0tγj,sdWsγjQj+∫0tQjTdWsγjTγj,sforj=1,2,where Ωj=βjQjQjT such that γj,0,Ωj∈Sn+ℝ while ηj∈ℒSn+ℝ and Mj∈Mnℝ.Then, discretize equations (50) and (37), by considering a time horizon T and regular time grid tiN=iT/N, for i=0,…,N.(51)γ^j,ti+1N=γ^j,tiN+Ωj+ηjγ^j,tiNTN+γ^j,tiN+Wti+1Nγj−WtiNγjQj+QjTWti+1Nγj−WtiNγjγ^j,tiN+,forj=1,2,where γ^j,t0N=γj,0=γj.Now, we proceed to discretize the log-price dynamics as follows:(52)Y^ti+1Ny=Y^tiNy+μ−12Trγ^1,tiN+γ^2,tiNTN+Trγ^1,tiN+Zti+1Nγ1−ZtiNγ1+γ^2,tiN+Zti+1Nγ2−ZtiNγ2.We now discretize the Brownian motions in equation (52) by considering the correlation structures in (30), that is, Ztγj=WtγjRjT+BtγjI−RjRjT,for,j=1,2:(53)Zti+1Nγj−ZtiNγj=Wti+1NγjRjT+Bti+1NγjI−RjRjT−WtiNγjRjT+BtiNγjI−RjRjT=Wti+1Nγj−WtiNγjRjT+Bti+1Nγj−BtiNγjI−RjRjT.These Brownian matricesZγj and Bγj follow normal distribution, that is,(54)Wti+1Nγj−WtiNγj∼N0,ti+1N−tiNI,Wti+1Nγj−WtiNγj∼ti+1N−tiNN0,I,and(55)Bti+1Nγj−BtiNγj∼ti+1N−tiNN0,I.Hence, simplifying the computation, we substitute equations (54) and (55) into equation (54)(56)Zti+1Nγj−ZtiNγj∼RjT+I−RjRjTti+1N−tiNN0,I.Equation (56) is reduced in the form as follows:(57)Zti+1Nγj−ZtiNγj∼RjT+I−RjRjTTNN0,I.Let the log-price dynamicsYt=logSt in equation (52) be Y^ti+1N=logS^ti+1N and logS^tiN as follows:(58)logS^ti+1N=logS^tiN+μ−12Trγ^1,tiN+γ^2,tiNTN+Trγ^1,tiN+Zti+1Nγ1−ZtiNγ1+γ^2,tiN+Zti+1Nγ2−ZtiNγ2.This log-price expression can be represented in the form as follows:(59)logS^ti+1N=logS^tiN+μTN−12∫tiNti+1NTrγ1s+γ2sds+Tr∫tiNti+1Nγ1+sdZsγ1+∫tiNti+1Nγ2+sdZsγ2.The incrementξk,i1≤k≤β allows to simulate Tr∫tiNti+1Nγ1sdWsγ1R1T, and for other terms, the standard Euler scheme is selected, that is,(60)∫tiNti+1Nγ1+sdWsγ1=∑k=1β∫tiNti+1NXk,sdWk,sγ1=Δt∑k=1βXk,tξk,iTTr∫tiNti+1Nγ1+sdBsγ1I−R1R1T=ΔtTrγ^1,tiN+I−R1R1Tζ1,i.We observe that the integral term in equation (59) needs to be simplified as follows:(61)∫tiNti+1Nγ1+sdZsγ1=∫tiNti+1Nγ1+sdWsγ1R1T+∫tiNti+1Nγ1+sdBsγ1I−R1R1T=ti+1N−tiN∑k=1lX^1k,tiNξk,iTR1T+ti+1N−tiNγ^1,tiN+I−R1R1Tζ1,i=TN∑k=1lX^1k,tiNξk,iTR1T+TNγ^1,tiN+I−R1R1Tζ1,i,where the Gaussian vector ζ1,i∼N0,I and the same for the second term are as follows:(62)∫tiNti+1Nγ2+sdZsγ2=TN∑k=1lX^2k,tiNξk,iTR2T+TNγ^2,tiN+I−R2R2Tζ2,i,where ζ2,i∼N0,I, and then we can substitute equations (61) and (62) in equation (59) as follows:(63)logS^ti+1NS^tiN=μTN−12∫tiNti+1NTrγ1s+γ2sds+TNTr∑k=1lX^1k,tiNξk,iTR1T+∑k=1lX^2k,tiNξk,iTR2T+TNTrγ^1,tiN+I−R1R1Tζ1,i+Trγ^2,tiN+I−R2R2Tζ2,i.Finally, we obtain the Euler–Maruyama discretization scheme for the double Wishart diffusion processes as follows:(64)logS^ti+1NS^tiN=μ−12Trγ^1,tiN+γ^2,tiNTN+TNTr∑k=1lX^1k,tiNξk,iTR1T+∑k=1lX^2k,tiNξk,iTR2T+TNTrγ^1,tiN+I−R1R1Tζ1,i+Trγ^2,tiN+I−R2R2Tζ2,i.


## 4. Numerical Illustrations

This section deals with numerical examples to examine the effects of volatility and other model parameters on the log-asset returns under the double Wishart volatility model.


### 4.1. The Log-Asset Returns under Double Wishart Volatility Model

The volatility specification under the double Wishart volatility model makes it mathematically flexible, to influence the asset return behavior. We quote market data drawn from QQQ (a fund by Invesco that tracks the performance of the stocks listed under the NASDAQ Index) options, April 2020, which has been used in Naryongo et al. [17]. We take the estimated parameter values from the market data, the variance matrices γ1 and γ2, with volatility matrices Q1 and Q2 are considered since they are very necessary parameters in a stochastic volatility model.(65)γ1,0=0.00008000.000095,γ2,0=0.00012000.0001,Q1=0.10000.23,Q2=0.20000.25,with strike price K=189 and interest rate r=0.05.Take the correlation values of(66)R1=−0.250.1−0.350.3,R2=−0.5−0.600.18−0.45.Figure1 demonstrates the effect of implied volatilities and correlation matrices on the log-asset returns under the double Wishart model. Given the parameter values, we observe that the log-asset returns under the double Wishart model exhibit interesting behaviors. This depends also on the choice of the model parameters, such as β1=4, β2=3, and the number of paths N=500. This example proves that the double Wishart volatility model has greater flexibility. The log-returns under the double Wishart model are more volatile towards the end of the trading period. The asset prices increase from the higher deviations with the number of paths under trading from the values of volatility matrices. It leads to higher returns in 3-month asset trading period under the double Wishart process.Figure 1
The log-asset return behaviors under the double Wishart volatility model at 3 months.Figure2 shows that a change in correlation matrices affects the behavior of the asset returns, that is, when(67)R1=0.00.85−0.050.75,R2=0.200.00.01−0.20,this provides different stock price behavior patterns due to flexibility in the model parameters. This can help investors to study and predict the behavior of stock price over time. The parameters β1=4andβ2=3 are maintained. The stock price trend for one year is under double Wishart model, when N=500; the asset price increases with time from the implied volatility as it approaches annual trading period from the volatility matrices.Figure 2
The log-asset returns trend under the double Wishart model whenN=500 at one year.Figure3 illustrates the effect of correlation matrices and N=500 on the log-asset returns trend under the double Wishart model while considering parameters β1=4andβ2=3, while taking the values of(68)R1=0.00.050.050.6,R2=0.80.50.450.65.Figure 3
The log-returns behavior under the double Wishart model at 3 months with different correlation matrices.The model shows interesting stock price trend predictions. We note that the model parameters influence the stock prices or log-returns greatly due to mathematical flexibility of the model. We can observe that there is higher asset returns at the end of the trading period.


## 4.1. The Log-Asset Returns under Double Wishart Volatility Model

The volatility specification under the double Wishart volatility model makes it mathematically flexible, to influence the asset return behavior. We quote market data drawn from QQQ (a fund by Invesco that tracks the performance of the stocks listed under the NASDAQ Index) options, April 2020, which has been used in Naryongo et al. [17]. We take the estimated parameter values from the market data, the variance matrices γ1 and γ2, with volatility matrices Q1 and Q2 are considered since they are very necessary parameters in a stochastic volatility model.(65)γ1,0=0.00008000.000095,γ2,0=0.00012000.0001,Q1=0.10000.23,Q2=0.20000.25,with strike price K=189 and interest rate r=0.05.Take the correlation values of(66)R1=−0.250.1−0.350.3,R2=−0.5−0.600.18−0.45.Figure1 demonstrates the effect of implied volatilities and correlation matrices on the log-asset returns under the double Wishart model. Given the parameter values, we observe that the log-asset returns under the double Wishart model exhibit interesting behaviors. This depends also on the choice of the model parameters, such as β1=4, β2=3, and the number of paths N=500. This example proves that the double Wishart volatility model has greater flexibility. The log-returns under the double Wishart model are more volatile towards the end of the trading period. The asset prices increase from the higher deviations with the number of paths under trading from the values of volatility matrices. It leads to higher returns in 3-month asset trading period under the double Wishart process.Figure 1
The log-asset return behaviors under the double Wishart volatility model at 3 months.Figure2 shows that a change in correlation matrices affects the behavior of the asset returns, that is, when(67)R1=0.00.85−0.050.75,R2=0.200.00.01−0.20,this provides different stock price behavior patterns due to flexibility in the model parameters. This can help investors to study and predict the behavior of stock price over time. The parameters β1=4andβ2=3 are maintained. The stock price trend for one year is under double Wishart model, when N=500; the asset price increases with time from the implied volatility as it approaches annual trading period from the volatility matrices.Figure 2
The log-asset returns trend under the double Wishart model whenN=500 at one year.Figure3 illustrates the effect of correlation matrices and N=500 on the log-asset returns trend under the double Wishart model while considering parameters β1=4andβ2=3, while taking the values of(68)R1=0.00.050.050.6,R2=0.80.50.450.65.Figure 3
The log-returns behavior under the double Wishart model at 3 months with different correlation matrices.The model shows interesting stock price trend predictions. We note that the model parameters influence the stock prices or log-returns greatly due to mathematical flexibility of the model. We can observe that there is higher asset returns at the end of the trading period.


## 5. Conclusion

The multifactor Heston model with two dependence matrices, that is, the double Wishart stochastic volatility model is flexible enough to describe the market prices. This solves the problem of pricing financial derivatives or assets in short or long maturities. The effect of the model parameter specification influences the behavior of the asset returns over time. This provides a theoretical framework for investors to study the market behavior over time. The numerical illustrations show the effect of model parameters on the asset return behavior under double Wishart model. We recommend future work on the exact simulation method and higher-order discretization techniques for the double Wishart volatility model.



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