Efficient Derivative Pricing by Extended Method of Moments

In this paper we consider an incomplete market framework and explainhow to use jointlyobserv ed prices of the underlying asset and of some derivativeswritten on this asset for an efficient pricing of other derivatives. Thisquestion involves two types of moment restrictions, which can be writteneither for a given value of the conditioning variable or can be uniform withrespect to this conditioning variable. This distinction between local and uniformconditional moment restrictions leads to an extension of the GeneralizedMethod of Moments (GMM), a method in which all restrictions are assumeduniform. The Extended Method of Moments (XMM) provides estimators ofthe parameters with different rates of convergence: the rate is the standardparametric one for the parameters which are identifiable from the uniformrestrictions, whereas the rate can be nonparametric for the risk premiumparameters. We derive the (kernel) nonparametric efficiencyb ounds for estimatinga conditional moment of interest and prove the asymptotic efficiencyof XMM. To avoid misleading arbitrage opportunities in estimated derivativeprices, an XMM estimator based on an information criterion is introduced.The general results are applied in a stochastic volatilitymo del to get efficientderivatice prices, to measure the uncertaintyof estimated prices and toestimate the risk premium parameters.