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.