Option pricing accuracy for estimated stochastic volatility models

Stochastic differential equations (SDEs) with space dependent coefficients are used to model asset price and volatility dynamics in the mathematical study of financial markets. The valuation of option contracts based on a given asset is then derived by numerically solving associated partial differential equations (PDEs), derived themselves from the SDEs driving the underlying asset and volatility. The parameters of these SDEs have to be estimated from observed data recording daily asset price and volatility. We quantify how the unavoidable errors in the estimation of model parameters impact the valuation of option contracts, generating errors in option pricing.;We have developed and numerically implemented a fast and efficient approximate maximum likelihood approach to estimate the 5 parameters of the widely used Heston model which involves two interacting SDEs from realistic numbers of recorded asset price and volatility data. We study and prove the asymptotic consistency of these estimators and evaluate explicitly their speed of convergence as the number of observations increases. We evaluate the sensitivity of option pricing to parameter estimation errors by solving numerically the 6 PDEs satisfied by the option price and by its derivatives with respect to the 5 underlying parameters. We study the size of the option pricing errors by intensive simulations of SDEs for realistic bench test groups of 5 parameters. We successfully apply our methods to the computation of option pricing errors for actual stock market data: the daily S&P 500 index and its approximate volatility (namely the VIX index).