Extracting implied correlation matrices from index option prices: a statistical approach

We propose a new approach to the model calibration problem, which takes into account the multiplicity of solutions. Starting from a prior distribution on model parameters and a set of observed option prices, we propose a probabilistic construction which yields an arbitrage free pricing rule consistent with these observed option prices. Our approach yields a simple Monte Carlo algorithm for simulating from this posterior distribution, taking into account the value of liquidly traded (”vanilla”) options. We also compute the sensitivity of our pricing model to the liquid option prices and provide a hedging portfolio that minimizes the risk of mispricing any exotic payoff. This algorithm can be seen as an arbitrage-free version of the Weighted Monte Carlo algorithm in Avellaneda et al [1], applicable to a wide range of pricing models and products. We illustrate our method by calibrating the correlation matrix of stocks in basket options given a set of observed call options on the basket.