Dynamic Evaluation of Contingent Claim Models (an Analysis of Model Error)

A Monte Carlo based Bayesian estimator of contingent claim models formally incorporating model error and parameter uncertainty is introduced. Estimate and prediction on the basis of a model extended by a regression of (functions of) variables and parameters or a non parametric expansion of the model is also proposed. For these models, we show how to make draws from the exact posterior distributions of the parameters or any function of interest such as hedge ratios or option values. It is crucial to obtain exact posterior or predictive densities in a Bayesian setup because (1) typical applications may involve small sample situations, (2) in an updating setup one will want to incorporate specific prior information, and (3) parameters and on linear functions thereof will be shown to be non-normal thus rendering standard asymptotic inference ineffective. The Bayesian Monte Carlo estimators derived allow the simulation of the exact posterior densities of parameters and deterministic functions of parameters, e.g., Delta, Gamma, Vega, Theta, model value, and the exact predictive densities of the contingent claim prices. We provide both within sample (residual analysis), and out of sample (predictive) specification tests which can be used in a dynamic trading system. Finally, the model is extended by allowing for the (small) probability of observation with a larger model error. This produces the posterior probability of each observation being an outlier, and may help distinguish market from model error