Efficient Derivative Pricing by Extended Method of Moments

In this paper, we consider an incomplete market framework and explain how to usejointly observed prices of the underlying asset and of some derivatives written on this assetfor an efficient pricing of other derivatives. This question involves two types of momentrestrictions, 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 betweenlocal and uniform conditional moment restrictions leads to an extension of the GeneralizedMethod of Moments (GMM); indeed, GMM assumes that all restrictions are uniform. TheExtended Method of Moments (XMM) provides estimators of the parameters with differentrates of convergence: the rate is the standard parametric one for the parameters, which areidentifiable from the uniform restrictions, whereas the rate can be nonparametric for the riskpremium parameters. We derive the kernel nonparametric efficiency bounds for estimatinga conditional moment of interest and prove the asymptotic efficiency of XMM. To avoidmisleading arbitrage opportunities in estimated derivative prices, an XMM estimator basedon an information criterion is introduced. The general results are applied in a stochasticvolatility model to get efficient derivative prices, to measure the uncertainty of estimatedprices, and to estimate the risk premium parameters.