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

Raphael Naryongo,Anthony Waititu,Philip Ngare,Remi Léandre

doi:10.1155/2021/4050722

Raphael Naryongo, Anthony Waititu + Show 2 more

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

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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.

Highlights

Introduction e 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. e 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. e 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
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 Gourieroux [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
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. is indicates that the model is not flexible enough to describe the market prices. is problem can be handled through generalizing the Heston model into a multifactor form

Summary

Mn are

Lemma 3. e correlation between the Brownian matrices of the stock price dynamic and the Brownian matrices of the Wishart processes is given as ρ1,t. Where Xct 1 and Xct 2 are standard Brownian motions (see proof in Appendix), and considering the trace of the dynamics of Wishart volatility process (21), we get dTr􏼐c1,t􏼑. E log-price process Yt log(St) under the double Wishart volatility model is as follows: dYt 􏼒r −. Is 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. E Corrected Euler–Maruyama Discretization Scheme for Double Wishart Affine Processes. E log-price dynamic and Wishart processes are discretized using corrected Euler–Maruyama method. En, discretize equations (50) and (37), by considering a time horizon T and regular time grid tNi iT/N, for i 0, . . . , N

Wcj tNi

Log Return