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

Naryongo, Raphael; Ngare, Philip; Waititu, Anthony

doi:https://doi.org/10.1155/2021/4050722

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 4050722 | https://doi.org/10.1155/2021/4050722

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

Raphael Naryongo,¹Philip Ngare,²and Anthony Waititu¹

Academic Editor: Remi Léandre

Received27 Aug 2021

Accepted28 Oct 2021

Published25 Nov 2021

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.

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 Section 2, 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. Let be a matrix-valued Brownian motion under the probability measure . The Wishart process satisfies the equationwhere is the invertible matrix, is the nonpositive matrix, is the nonnegative symmetric matrix, and is a real parameter. The condition is taken to ensure existence and uniqueness of the solution for equation (1), and eigen values of the solution are nonnegative for all a.s . Following Benabid et al. [9], the probability measure corresponds to a risk-neutral measure.
The infinitesimal generator of Wishart process has been already studied in Bru [11], formulated as follows:where the matrix differential operator is .

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

The Wishart processes are affine processes defined on . 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. Let be affine process with continuous trajectories defined in , with stochastic differential equation as follows:where , , and is a linear transformation. The process admits a unique weak solution in if(i).(ii)For all such that , where is a trace of a square matrix. So, contains in a set of real positive-definite matrices , with condition (i) is replaced by a stronger requirement.(iii). Then, there exists a unique strong solution for equation (1) in . By observation and and when we assume restrictive parametrization for the deterministic part of the drift, . Conditions (i) and (iii) are satisfied as long as and , respectively; with direct comparison, we get Wishart SDE (1) from (3). The real positive parameter also plays a role in Feller’s condition in the univariate case.

Lemma 2. Let represent the filtration generated by . We consider continuous -adapted processes , respectively, valued in , and and a process that admits the following semimartingale decomposition:For , the quadratic covariation of and is given as

Proof. Clearly, we can see that the quadratic covariation (4) depends on and only through the matrices and as follows:The quadratic covariation is calculated as follows:Particularly can be written in form of trace as follows:

2.2. Link between Wishart Processes and Ornstein–Uhlenbeck Processes

The best way to generate a Wishart process is to substitute the Gaussian vector in the definition of a Wishart distribution with Ornstein–Uhlenbeck processes ; this implies that is an integer. Considering -independent n-dimensional Ornstein–Uhlenbeck processes, and (see Benabid et al. [9]).

Here, ; then, the process is a Wishart process with the following stochastic differential equation:where a matrix-valued Brownian motion, obtained bywhere the matrix is taken as the mean reversion parameter of the Wishart process and 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 with , such that . The function permits to write with and real number with .

The goal is to find a change of probability measure in order to change the generalized Wishart diffusion process into the simple one, where is an integer. Therefore, the new probability measure , following Benabid and Bjork can be expressed as follows.

Theorem 1. Let . If defines the Radon–Nikodym derivative of with respect to , then

Proof. The probability measure can be specified through an exponential martingale.We define the new process as follows:We check using Girsanov theorem that is a matrix-valued Brownian motion under the probability measure while the dynamics of Wishart diffusion process under the probability measure is as follows:whereSubstituting above in the Wishart process (14),Then,The Radon–Nikodym derivative is computed using the determinant dynamics.From equation (12), the new measure with is given byThe 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:where represents the risk free interest rate, is the trace operator, is a matrix Brownian motion under the risk-neutral measure, and belongs to the set of symmetric positive-definite matrices as well as its square root . We observe that volatility of the risky asset is the trace of the matrix , with , and 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 is considered to be negative semidefinite, with the real parameter , 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:where the quadratic variations are as follows:where are real parameters such that , , Q is invertible matrix, and are matrices Brownian motions, and also .

Lemma 3. The correlation between the Brownian matrices of the stock price dynamic and the Brownian matrices of the Wishart processes is given as

Proof. where and are standard Brownian motions (see proof in Appendix), and also considering the trace of the dynamics of Wishart volatility process (21), we getThese processes can still be presented in the form as follows:where and are Brownian motions (see proof in Appendix), such thatWe now determine the covariation of the stock price and Wishart processes:The same procedures can be carried out on the second differential equation to obtain the covariation as follows: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 matrices can be correlated in such a way that all the (scalar) Brownian motions belonging to of and corresponding Brownian motions of the column of have the same correlation, say . This provides constant matrices which describe the two correlation structures, in such a way that and presented as follows:where denotes the identity matrix, is the transposition, and and are (matrices) Brownian motion independent of , respectively.

Lemma 4. The correlation structure is a matrix Brownian motion.

Proof. We note that is a matrix Brownian motion iff for ,Then,similarly for the second dynamic process as

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

Let denote the log-price and the matrices and describe the correlations between the Brownian of the asset price process and those of the Wishart processes.

Proposition 1. The log-price process under the double Wishart volatility model is as follows:

Proof. Let , the asset dynamic is given as follows:By applying Ito’s formula on (see Björk [16] and Shreve [18]), we getReplacing the asset process (35) in the derivative equation (36) of ,This can still be presented in the form as follows:

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, and , can be presented in the form as in Benabid et al. [9] for easy handling of the dynamics.where

Lemma 5. The infinitesimal generator under the double Wishart volatility model for is as follows:

Proof. The infinitesimal generator has nontrivial term which arises from covariation corresponding to the coefficients of the termLet be the square root matrix whereSince is symmetric, we determine the covariation terms corresponding to coefficients as follows:This provides the corresponding coefficients of the term:The notation when while for .

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 as an Ito process on satisfying differential equation as follows:

For discretization or on a regular grid , it is given asand also considering a continuous case, we have

One can observe that the Euler scheme depends on throughout the discretization grid, and it is similarly better to denote it as .

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:where the Wishart matrix processes are given as follows:where such that while and .

Then, discretize equations (50) and (37), by considering a time horizon and regular time grid , for .where .

Now, we proceed to discretize the log-price dynamics as follows:

We now discretize the Brownian motions in equation (52) by considering the correlation structures in (30), that is, :

These Brownian matrices and follow normal distribution, that is,and

Hence, simplifying the computation, we substitute equations (54) and (55) into equation (54)

Equation (56) is reduced in the form as follows:

Let the log-price dynamics in equation (52) be and as follows:

This log-price expression can be represented in the form as follows:

The increment allows to simulate , and for other terms, the standard Euler scheme is selected, that is,

We observe that the integral term in equation (59) needs to be simplified as follows:where the Gaussian vector and the same for the second term are as follows:where , and then we can substitute equations (61) and (62) in equation (59) as follows:

Finally, we obtain the Euler–Maruyama discretization scheme for the double Wishart diffusion processes as follows:

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 and , with volatility matrices and are considered since they are very necessary parameters in a stochastic volatility model.with strike price and interest rate .

Take the correlation values of

Figure 1 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 , , and the number of paths . 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 2 shows that a change in correlation matrices affects the behavior of the asset returns, that is, whenthis 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 are maintained. The stock price trend for one year is under double Wishart model, when ; the asset price increases with time from the implied volatility as it approaches annual trading period from the volatility matrices.

Figure 3 illustrates the effect of correlation matrices and on the log-asset returns trend under the double Wishart model while considering parameters , while taking the values of

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.

Appendix

Verification of Brownian Motions in Lemma 3

The following processes are Brownian motions.

Proof. Similarly, for Brownian motion , its proof can be obtained through the same procedures as above.

Then, the authors show that is a Brownian motion as follows.

Proof. Similarly, the same procedures can be followed to show that is a Brownian motion.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors in this paper declare that they have no conflicts of interest with regard to the publication of this paper.

Acknowledgments

The authors extend their gratitude to the Pan African University Institute for Basic sciences, Technology and Innovation (Pausti) at Jomo Kenyatta University (JKUAT).

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Copyright © 2021 Raphael Naryongo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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