Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework
Résumé
In this paper, a study of a stochastic volatility model for asset pricing is described. Originally presented by J. Da Fonseca, M. Grasselli and C. Tebaldi, the Wishart volatility model identifies the volatility of the asset as the trace of a Wishart process. Contrary to a classic multifactor Heston model, this model allows to add degrees of freedom with regard to the stochastic correlation. Thanks to its flexibility, this model enables a better fit of market data than the Heston model. Besides, the Wishart volatility model keeps a clear interpretation of its parameters and conserves an efficient tractability. Firstly, we recall the Wishart volatility model and we present a Monte Carlo simulation method in sight of the evaluation of complex options. Regarding stochastic volatility models, implied volatility surfaces of vanilla options have to be obtained for a short time. The aim of this article is to provide an accurate approximation method to deal with asymptotic smiles and to apply this procedure to the Wishart volatility model in order to well understand it and to make its calibration easier. Inspired by the singular perturbations method introduced by J.P Fouque, G. Papanicolaou, R. Sircar and K. Solna, we suggest an efficient procedure of perturbation for affine models that provides an approximation of the asymptotic smile (for short maturities and for a two-scale volatility). Thanks to the affine properties of the Wishart volatility model, the perturbation of the Riccati equations furnishes the expected approximations. The convergence and the robustness of the procedure are analyzed in practice but not in theory. The resulting approximations allow a study of the parameters influence and can also be used as a calibration tool for a range of parameters.
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