Calibrating Volatility Surfaces Via Relative-Entropy Minimization

We present a framework for calibrating a pricing model to a prescribed set of option prices quoted in the market. Our algorithm yields an arbitrage-free diffusion process that minimizes the relative entropy distance to a prior diffusion. We solve a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns in the optimization step is equal to the number of option prices that need to be matched, and is independent of the mesh-size used for the scheme. This results in an efficient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschemark over-the-counter options and computing interpolated implied-volatility curves.