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Prices Expansion in the Wishart Model

Using probability change techniques introduced by Drimus for Heston model, we derive 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. Thanks to this new approximation, the smile implied by Wishart model can be better understood. The sensitivity of Delta and Vega to the volatility (respectively Vanna and Volga) indeed appear explicitly in this formula. En route to our formula, we present a number of new - to our knowledge - results on Laplace transforms and moments of the integrated Wishart processes.