Research Article European Option Pricing under Wishart Processes

This study deals with a single risky asset pricing model whose volatility is described by Wishart aﬃne processes. This multifactor model with two dependency matrices describing the correlation between the asset dynamic and Wishart processes makes it more ﬂexible enough to ﬁt the market data for short or long maturities. The aim of the study is to derive and solve the call option pricing problem under the double Wishart stochastic volatility model. The Fourier transform techniques combined with perturbation methods are employed in order to price the European call options. The numerical illustrations on pricing predictions show similar behavior of price movements under the double Wishart model with respect to the market price.


Introduction
e limitation of the standard Black and Scholes [1] model of not accounting for the fact that implied volatility of derivative products varies by strike and maturity; this makes it less flexible to reproduce certain market conditions observed on derivative prices.
is caused the introduction of the Heston [2] model, which has been remarkably famous and heavily applied in financial markets due to its flexibility, financial interpretation of parameters, and analytical tractability property of the model since it belongs to the class of the affine process. is helps it to obtain the call price of a European option by inverting the Fourier transform and forming a closed-form solution of the log-price characteristic function.
However, despite the Heston model popularity, the existing literature documented the limitation of the model. Da Fonseca et al. [3], Christoffersen et al. [4], Ahdida and Alfonsi [5], Kang et al. [6], La Bua and Marazzina [7], and Gourieroux [8] pointed out that the major deficiency of the model does not generate the realistic term structure of the volatility smiles. e Heston model provides too flat implied volatility surface to attain reality. Yet, the implied volatility curve generally has steep slope, and it is convex in short maturity and tends to be linear for long maturity. e literature recommends that the problem of not capturing such stylized facts can be solved by generalizing the Heston model using two approaches: by adding jump in the stock dynamic or volatility and investigating the multifactor nature of the implied volatility as emphasized in the studies by Benabid et al. [9], Da Fonseca et al. [3], La Bua and Marazzina [7], and Kang and Kang [10]. It is well understood that the multifactor approach is capable of handling the pricing problem of derivative products and volatility skew.
is usher in the application of the stochastic matrix defined process, that is, the Wishart multidimensional stochastic volatility model. e Wishart process was introduced in Bru [11] and is defined as a positive semidefinite matrix-valued generalization of the square root process. e square root process is a Bessel process which is a generalization of the multiple chi-squared distribution [12,13].
e Wishart stochastic volatility model found its application in finance by Gourieroux and Sufana [14], and it has been applied in financial markets because of its nature of matrix specification which makes it flexible. e aim of this study is to price European option under two Wishart variance processes, through generalization of the Heston model into the multifactor nature of volatility modeling with two dependence matrices for single asset, that is, the asset dynamics depends on two Wishart volatility processes (double Wishart volatility processes). Following Kang et al. [6] and Benabid et al. [9], the matrix specification of the Wishart model makes it flexible enough to describe the market prices. e numerical illustrations show that price predictions under the double Wishart model have a similar behavior like the market prices.
e European call option pricing problem is handled through transform methods of Duffie et al. [15] and fast Fourier transform by Carr and Madan [16] combined with perturbation techniques of Benabid et al. [9] and Fouque et al. [17], in order to obtain the European call option price formula.
e study is structured as follows: Section 2 presents the definition of the Wishart process, and we introduce the Wishart stochastic volatility model and double Wishart model, correlation structures, and the infinitesimal generator of the Bi-Wishart model. Section 3 provides the solution for European call option pricing problem through Fourier transform methods combined with perturbation techniques. Section 4 presents the numerical illustrations to compare the price predictions under the double Wishart model with the market prices. Section 5 provides conclusion and further research work, together with technical proofs in the Appendix.

Wishart Process
Definition 1. Let W t , t ≥ 0 under the probability measure Q be a n × n matrix-valued Brownian motion. e Wishart matrix process Σ satisfies the following equation: where Q ∈ GL n (R) is the invertible matrix, M ∈ M n is the nonpositive matrix, and Σ 0 ∈ S + n is the nonnegative symmetric matrix, while β is a real parameter. e condition β > (n − 1) is considered to ensure existence and uniqueness of the Σ t ∈ S + n solution for equation (1), and Eigen values of the solution are nonnegative for all t ≥ 0 a.s Σ t ∈ S + n . Like in the study by Benabid et al. [9], the probability measure Q corresponds to a risk-neutral measure (Appendix: change of the probability measure).

Wishart Volatility Model in the Stock Market. In Da
Fonseca et al. [3], under the risk-neutral probability measure, the risky asset price dynamic and its quadratic variation are follows: where r denotes risk-less interest rate, Tr is the trace, Z ∈ M n is the Brownian matrix, and Σ t is in a set of symmetric n × n positive-definite matrices. We observe that asset volatility is a trace of Σ t matrix, which is a multidimensional process with M, Q ∈ M n , and W t ∈ M n is the Brownian matrix.
In the study by Bru [11], the Wishart process provides a matrix analogue of the square root mean-reverting process and M is considered negative to ensure positivity and meanreverting property of the volatility with parameter β > n − 1 for uniqueness and existence of the solution.

Correlation Structure.
e Brownian matrices W t and Z t are correlated, in such a way that it gives a constant correlated matrix R ∈ M n , like in the study by Da Fonseca et al. [3], which describes the correlation structure, in such a way that Z t can be presented as follows: where I denotes the identity matrix, T is the transpose, and B t is an independent matrix Brownian motion from W t . e correlation structure is a Brownian motion (Appendix).

Bivariate Wishart Stochastic Processes in Stock Market.
In this section, we present a proposed novel model, the multifactor model with two Wishart variance processes or double Wishart stochastic volatility model with two dependence matrices. e model takes two volatility components which is the trace of Wishart whose diagonal components will be the controlling factors for the dynamics of volatilities. Under arbitrage-free financial market and probability measure, we consider risky asset dynamic as follows: Proof. We derive the correlations as follows: where X t and X t are the standard Brownian motions (Appendix), and also taking the trace of the Wishart volatility dynamics (4), we have ese processes can still be written in the following form: en, where ξ t and η t are the Brownian motions (Appendix). We now determine the covariation of the stock price and Wishart processes: Similarly, for the second stochastic differential equation of the Wishart processes, the determination of covariation follows the same procedures, that is, □ Journal of Mathematics

e Correlation Structure of the Model.
e correlated Brownian motions W t , Z t and W t , Z t , respectively, result in constant correlations R, R ∈ M n , which describe the two respective correlation structures, such that Z t and Z t can be presented as follows: where I is the identity matrix, T is the transposition, and B t and B t are the Brownian matrices independent of W t and W t , respectively.

e Dynamic of Log-Price for the Double Wishart
Model. e matrices R and R describe the correlations between the Brownian of the asset price and those of the Wishart processes.

Lemma 2.
e log-price dynamic Y t � log(S t ) under the double Wishart model is given as Proof. Let Y t � log(S t ), and the asset dynamic is given as By applying Ito's formula on Y t , referring the studies by Björk [18] and Shreve [19], we get Substituting the asset process (14) in the derivative expression (15) which can be written as

Option Pricing Problem
is part deals with the European call option pricing problem, given its payoff as To handle this pricing problem, the infinitesimal generator of the Wishart processes should be obtained, in order to employ the conditional characteristic function on the log asset return. e Riccati ordinary differential equations are linearized to give the closed-form solution to the pricing problem through fast Fourier transforms of Duffie et al. [15].
is is possible because the Wishart processes preserve the property of analytical tractability, since it belongs to the class of affine [20].

Infinitesimal Generator.
e log-price process and Wishart processes, together with corresponding pair of two correlated Brownian motions, Z Y s , Z Σ s and Z Y s , Z Σ s , can be written in the form as in the study by Benabid et al. [9], for easy handling of the complexity of the dynamics:

Proposition 1. e infinitesimal generator of the double Wishart stochastic volatility model for
Due to the fact that V θ;t is symmetric, we find the covariation terms matching with z 2 /zx θ;ij zy coefficients: is gives the corresponding coefficients of the term since  [15] and Da Fonseca et al. [3], in order to solve the European options pricing problem, we consider the Laplace transform of the process (17). e Laplace transform of the Wishart process is exponentially affine as in the study by Bru [11]; hence, the conditional moment generating function of the log-asset returns is the exponential of an affine combination of Y and Wishart components, so we provide for the deterministic functions λ 1 (t), λ 2 (t) ∈ M n , and δ(t), ε(t) ∈ R, as parameters of the Laplace transform: where c ∈ R and E t denote the conditional expected value with respect to the probability measure.
Using Feynman-Kac argument helps us to get matrix Riccati equations: e Laplace transform of the asset returns where λ 1 , λ 2 , and ε are the solutions of the differential equations through linearization methods: with boundary conditions λ e solutions λ 1 , λ 2 , and ε are obtained as follows: with From equation (27), we have Journal of Mathematics Now, we need to identify the coefficients for above equations and obtain the matrix Riccati ordinary differential equations: For the constant ε, the matrix Riccati ordinary differential is obtained as follows: en, ε(τ) is obtained by integrating directly with λ 1 , λ 2 ∈ M n (R), and δ(τ), ε(τ) ∈ R. e matrix Riccati equations above are linearized to provide the closed-form solution, following the techniques of Benabid et al. [9] and Da Fonseca et al. [3]. So from equations (35) and (36), let where H 1 (τ) ∈ GL n (R), H 2 (τ) ∈ M n (R), and therefore, 8

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Since H 2 (τ) � H 1 (τ)λ 1 (τ), now we have is implies that en, we get expressions e solution of differential equation (46) above is obtained through exponentiation as follows: with conditions H 1 (0) � KI n and H − 1 1 (0) � K − 1 I n and for K � 1, H − 1 1 (0) � I n , such that where en, and since λ 1 (0) � 0, then In conclusion, we get is represents the closed-form solution of the Riccati equation (35). Now, we look at the solution of the second Riccati equation (36). Let then, Journal of Mathematics such that From equation (40), we go through the same procedures as above, with conditions I 2 (0) � I 1 (0)λ 2 (0) � λ 2 (0), therefore, since λ 2 (0) � 0, then us, it provides the expression for a closed-form solution of the Riccati equation: Let compute the last Riccati equation (37) for constant ε, From Riccati equations (35) and (36), we have From equation (63), we get and as well as from equation (64), By integrating (69), we obtain

Fast Fourier Transform Method and Characteristic
Function. In this section, we consider the fast Fourier transform (FFT) method in the study by Carr and Madan [16] to price European call option with α > 0, at time t, strike k � log(K), and time to maturity T, given as e modified price C α t in the study by Carr and Madan [16] is considered, with α � 1.1 as a good empirical value for the Heston model. e square integrable function is obtained through the modified price in order to use inverse Fourier transform: We introduce the Fourier transform of the modified price together with the application of the Fubini integration theorem: e call price can be provided through inversion of Fourier transform given ψ α t (T, θ) function having both odd imaginary and even real parts; so from equation (73), we have en, we have is is a Fourier transform: Journal of Mathematics 11 (77) Corollary 1. Let D be the symmetric matrix, and it is sufficient to find the conditional characteristic function of the double Wishart Σ t and Σ t given by (Appendix) where A 1 (τ), A 2 (τ) ∈ M n , and C ∈ C verify the following dynamics:

Proposition 3. e call price under double Wishart is given as
where Proof. Let ϕ(t, T) be the characteristic function of the logprice Y t . We have, where A j (t) is obtained from equation (79) with τ � t provided A j (0) � λ j (T − t), j � 1, 2. Hence, where

Perturbation Techniques of the Riccati Differential
Equations. is section deals with implementation of perturbation techniques since the system of the Riccati differential equations does not allow an analytical closed-form solution. is is due to noncommutativity of matrix multiplication, in order to approximate the call option price. e method procedures retain the affine properties and higher orders comparable to the standard perturbation scheme on partial differential equations that is complex after the first order [9,17]. Consider the differential Riccati equations of the double Wishart stochastic volatility model and take dimension n � 2. e two characteristics orders in perturbation p and q are considered. e solution of A(τ) is given in the following form: e developments in the perturbed differential equations are done by comparing the coefficients and identifying the terms in p and q to give expected approximations. Let p � M 1 and q � M 2 are small while ] i quantities remain constant.
en, we consider approximation at order one ( � � p √ , � q √ ) and order two (p, q) together with the following notations.
and also noting that Q is the volatility, which can be written as We can now rewrite the Riccati equations as in the form and for

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Development of the Riccati functions: by dealing with the development of the Riccati functions A 1 , A 2 , λ 1 , λ 2 , C, and ε, the perturbation of the Riccati equations at order two (order 2) is considered to obtain the European call option pricing formula as follows: o(max(p, q)), k � 1, 2.

16
Journal of Mathematics e variance processes Σ 0 and Σ 0 are given as Taking the trace, For functions C and ε, en, (p, q)).

(114)
is can still be written as where such that the call price is as follows: Finally, we obtain the approximated European call option price as follows:

Numerical Illustration
In this section, we deal with illustrative numerical examples to examine the implications of volatility in option pricing under the double Wishart stochastic volatility model.

Comparison between the Double Wishart Model Price and the Market Price.
e volatility specification of the double Wishart model remarkably makes it flexible to produce price predictions which exhibit a similar price behavior with the market price. 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.
Let us consider the estimated parameter values from the market data; the variance matrices Σ and Σ, with volatility of volatility matrices Q and Q, are considered since they are very important parameters in a stochastic volatility model.
While, the strike price is K � 189 and interest rate is r � 0.05. Figure 1 illustrates the effect of implied volatilities on the call price under the double Wishart model. Given the parameter values, we observe that the price predictions under the double Wishart model exhibits the same behavior with respect to the market price in short maturity time.
is depends also on the choice of the model parameters, such as c � 0.5, β � 4, β � 3, d 1 � 0.45, and d 2 � 0.479. is example proves that the double Wishart volatility model has greater flexibility. Figure 2 shows that a slight change in model parameters leads to the changes in the model call option price predictions. We observe that as time tends to maturity time of six months, (d 2 � 0.478), the model price movement behaves almost the same with the market price. e parameters c � 0.5, β � 4, β � 3, and d 1 � 0.45 are maintained. Figure 3 demonstrates the effect of parameters on the call European option price under the double Wishart model. e parameters c � 1, β � 4, β � 3, d 1 � 0.45, and d 2 � 0.4968. e model shows price predictions exhibiting the same market price behavior, in long maturity time of 1 year. We note that the model parameters influence the prices greatly and help long position holders not to experience arbitrage profits.

Conclusion
e generalization of the Heston model into a multifactor form with two dependence matrices efficiently solves the problem of pricing European call option and financial market data fitting for short or long maturities. e new Wishart affine model retains the remarkable property of analytical tractability, and this allows the model to form a closed-form solution of the conditional characteristic      function for the call option price expression, which is obtained through Fourier transforms together with perturbation methods.
e numerical results show that the double Wishart stochastic volatility model produces price predictions similar to the market prices. Notably the effect of parameters on our model makes it flexible enough for short or long maturities. We recommend future work on discretization schemes and also to investigate the behavior of nondiagonal matrix components in the model.

C. Brownian Motions
In Lemma 1, we show that the following processes X t , X t , ξ t , and η t are the Brownian motions.
Proof Similarly, for Brownian motion X t , its proof can be obtained through the same procedures as above.
en, we can also show that ξ t is a Brownian motion as follows. e same procedures can be followed to show that η t is a Brownian motion.