Using probability change techniques introduced by Drimus for Heston model, the paper derives a n^th order expansion formula of Wishart option price in terms of Black-Scholes price and Black-Scholes Greeks. Numerical results are given for the second-order case. Due to this new approximation, the smile implied by the Wishart model can be better understood. The sensitivity of Delta and Vega to the volatility (Vanna and Volga respectively) indeed appears explicitly in this formula. En route to this formula, the paper presents a number of new results on Laplace transforms and moments of the integrated Wishart processes.

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-of-the-money options are higher than Black-Scholes prices with at-the-money volatilities. This effect is known as the volatility smile: 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 to model correctly this behavior.

In order to model the smile efficiently, stochastic volatility models are a popular approach. They enable 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.

In the equity market, basket products have become very common since the appearance of Mountain Range options. These products may involve intricate dependencies on the variance-covariance structure. However, most stochastic volatility models handle one single asset at a time. Correlation then is introduced as an exogenous parameter through Brownian motions. Interest into Wishart model, studied by Da Fonseca et al. (2006a and 2006b) is rising rapidly as it proves to be an efficient way to model stochastic covariance behaviors. However, some practitioners may still be reluctant to adopt Wishart processes since they lack confidence into such complex processes. Getting an intuition on parameters seems an impossible task since the only existing call price formula, given in our previous paper (Gauthier and Possamaï, 2009), involves a Fourier integral of a matrix function. In order to better understand Wishart model, we propose to apply the expansion technique.

Computational Mathematics Journal, Wishart Model, Black-Scholes Greeks, Option Valuation Model, Black-Scholes Model, Equity Market, Stochastic Volatility Models, Expansion Techniques, Wishart Processes, Wishart Stochastic Model, Heston Model, Real Matrix Logarithm, Brownian Motions, Square Bessel Processes.