Recovering Nonlinear Dynamics from Option Prices

Using the wavelet-Galerkin method for solving partial integro-differential equations, we derive an implement computationally efficient formula for pricing European options on assets driven by multivariate jump-diffusions. This pricing formula is then used to solve the inverse problem of estimating the corresponding risk-neutral coefficient functions of the underlying jump-diffusions from observed option data. The ill-posedness of this estimation problem is proved, and a consistent estimation technique employing Tikhonov regularization is proposed. Using S&P 500 Index option data, it is shown that the coefficient functions in a stochastic volatility model with jumps are nonlinear, contrary to the affine specification widely used in the literature.