Model Error in Contingent Claim Models (Dynamic Evaluation)

Abstract

This paper formally incorporates parameter uncertainty and model error into the estimation of contingent claim models and the formulation of forecasts. This allows inference on functions of interest (option values, bias functions, hedge ratios) consistent with uncertainty in both parameters and models. We show how to recover the exact posterior distributions of the parameters or any function of the parameters. Exact posterior or predictive densities are crucial because a frequent updating setup results in small samples and requires the incorporation of specific prior information. Markov Chain Monte Carlo estimators are developed to solve this estimation problem. Within sample and predictive model specifications tests are provided which can be used in dynamic testing (or trading systems) making use of cross-sectional and time series options data. Finally, we discuss several generalizations of the error structure. These new techniques are applied to equity options using the Black-Scholes model. When model error is taken into account, Black-Scholes appears very robust, in contrast with previous studies which at best only incorporated parameter uncertainty. We extend the Black-Scholes model by adding polynomial functions of its inputs. This allows for intuitive specification tests. Although these simple extended models improve the in-sample error properties of the Black-Scholes, they do not result in major improvements in out of sample predictions. The differences between these models are important, however, because they produce different hedge ratios and posterior probabilities of mispricing.

JEL Classification: G1, G13, C1, C5