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The IUP Journal of Computational Mathematics
Efficient Simulation of the Wishart Model
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In financial mathematics, Wishart processes have emerged as efficient tools to model stochastic covariance structures. Their numerical simulation may be quite challenging since they involve matrix processes. In this article, we propose an extensive study of financial applications of Wishart processes. First, we derive closed-form formulas for option prices in the single-asset case. Then, we show the relationship between Wishart processes and Wishart law. Finally, we review the existing discretization schemes (Euler and Ornstein-Uhlenbeck) and propose a new scheme, adapted from Hestonís QEM discretization scheme. Extensive numerical results support our comparison of these three schemes.

 
 
 

Since Black and Scholes introduced their option valuation model, an extensive literature focuses on pointing out its limitations. More precisely, two major features of the equity index options market cannot be captured through Black-Scholes model. First of all, observed market prices for both in-the-money and out-the-money options are higher than Black-Scholes prices with at-the-money volatilities. This effect is known as the volatility smile, where the volatility depends both on the option expiry and the option strike. Second, there exists a term structure of implied volatilities. As a matter of fact, a constant volatility parameter does not enable us to model this behavior correctly.

In order to model the volatility smile efficiently, stochastic volatility models are a popular approach. They enable us to have distinct processes for the stock return and its variance. Thus, they may generate volatility smiles. Moreover, if the variance process embeds a mean reversion term, these models can capture the term-structure in the variance dynamics. Popular stochastic volatility models include Heston (1993), SABR and square-root models to name but a few. Efficient calibration of stochastic volatility models requires an analytical formula for option prices. For numerous models, including Heston, this is achieved through the Fourier-transform technique described in Carr and Madan (1999). Further refinements have been discussed in Lewis (2001) and Lord and Kahl (2008).

In order to precisely match the market implied volatility surface, it turns out that Heston model does not have enough parameters. Therefore, we may wish to add degrees of freedom (i.e., additional parameters) while retaining analytical tractability. Many academicians and practitioners have tackled this challenge by considering time-dependent extensions of the original Heston model. Another direction is to model the variance by a variable of higher dimension. Such a path has been followed by Gouriéroux et al. (2006) and Da Fonseca et al. (2006a, 2006b, 2007 and 2008), who replaced the Cox-Ingersoll-Ross variance process by a Wishart process. Their model enables us to have a tighter control of the covariance dynamics as it is represented by a matrix of size n (2 is often enough in practice). In this paper, we follow their approach, taking a closer look at numerical issues.

 
 
 

Computational Mathematics Journal, Logistic Regression Models, Business Management, Decision Theory, Logistic Regression Programs, SPSS Nonlinear Program, Proportional Reductions, Statistical Aanalysis Software Package, Statistical Software SPSS, Squared Pearson Correlation .