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 derivatives written on this assetfor 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 uniform conditional moment restrictions leads to an extension of the Generalized Method of Moments (GMM); indeed, GMM assumes that all restrictions are 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 estimating 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 efficient derivative prices, to measure the uncertainty of estimated prices, and to estimate the risk premium parameters.