A linear goal programming method to recover risk neutral probabilities from options prices by maximum entropy

Abstract

We develop a new methodology to retrieve risk neutral probabilities (equivalent martingale measure) with maximum entropy from quoted option prices. We assume the no arbitrage hypothesis and model the efficient market hypothesis by means of a maximum entropic risk neutral distribution. The method is free of parametric assumption except for the simulation of the distribution support, for which purpose we can choose any stochastic model. Firstly, we innovate in the minimization of a different f-divergence than Kullback–Leibler’s relative entropy, resulting in the total variation distance. We minimize it by means of linear goal programming, thus guaranteeing a fast numerical resolution. The method values non-traded assets finding a RNP minimizing its f-divergence to the maximum entropy distribution over a simulated support—the uniform distribution—calibrated to the benchmarks prices constraints. Our second innovation is that in an incomplete market, we can increase the f-divergence from its minimum to obtain any asset price belonging to the interval satisfying the non-existence of an arbitrage portfolio, without presupposing any utility function for the decision maker. We exemplify our methodology by means of synthetic and real-world cases, showing that our methodology can either price non-traded assets or interpolate and extrapolate a volatility surface.