Calibrating Volatility Surfaces Via Relative Entropy Minimization
Marco Avellaneda, Craig Friedman, Richard Holmes, Dominick Samperi

IJTAF, 1997

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  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.