Bayesian Analysis of Contingent Claim Models

Eric Jacquier
Carroll School of Management, Boston College

Robert Jarrow
Graduate School of Management, Cornell University

This paper formally incorporates parameter uncertainty and model error into the implementation of contingent claim models. We make hypotheses for the distribution of errors to allow the use of likelihood based estimators consistent with parameter uncertainty and model error. We then write a Bayesian estimator which does not rely on large sample properties but allows exact inference on the relevant functions of the parameters (option value, hedge ratios) and forecasts. This is crucial because the common practice of frequently updating the model parameters leads to small samples. Even for simple error structures and the Black-Scholes model, the Bayesian estimator does not have an analytical solution. Markov Chain Monte Carlo estimators help solve this problem. We show how they extend to some generalizations of the error structure. We apply these estimators to the Black-Scholes. Given recent work using non-parametric function to price options, we nest the B-S in a polynomial expansion of its inputs. Despite improved in-sample fit, the expansions do not yield any out-of-sample improvement over the B-S. Also, the out-of-sample errors, though larger than in-sample, are of the same magnitude. This contrasts with the performance of the popular implied tree methods which produce outstanding in-sample but disastrous out-of-sample fit as Dumas, Fleming and Whaley~(1997) show. This means that the estimation method is as crucial as the model itself.