Efficient Derivative Pricing by Extended Method of Moments

In this paper we consider an incomplete market framework and explain how to use jointly observed prices of the underlying asset and of some deriv- atives written on this asset for an efficient pricing of other derivatives. This question involves two types of moment restrictions, which can be written either for a given value of the conditioning variable, or can be uniform with respect to this conditioning variable. This distinction between local and uni- form conditional moment restrictions leads to an extension of the Generalized Method of Moments (GMM), a method in which all restrictions are assumed uniform. The Extended Method of Moments (XMM) provides estimators of the parameters with different rates of convergence: the rate is the standard parametric one for the parameters which are identifiable from the uniform restrictions, whereas the rate can be nonparametric for the risk premium parameters. We derive the (kernel) nonparametric efficiency bounds for esti- mating a conditional moment of interest and prove the asymptotic efficiency of XMM. To avoid misleading arbitrage opportunities in estimated derivative prices, an XMM estimator based on an information criterion is introduced. The general results are applied in a stochastic volatility model to get effi- cient derivative prices, to measure the uncertainty of estimated prices and to estimate the risk premium parameters.