Efficient Derivative Pricing by the Extended Method of Moments

Authors

  • P. Gagliardini,

    1. Faculty of Economics, Università della Svizzera Italiana (USI), Via Buffi 13, CH-6900 Lugano, Switzerland and Swiss Finance Institute; patrick.gagliardini@usi.ch
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  • C. Gourieroux,

    1. Laboratoire de Finance Assurance, 15, Boulevard Gabriel Péri, 92245 Malakoff Cedex, France, Dept. of Economics, University of Toronto, 150 St. George St., Toronto, Ontario M5S 3G7, Canada, and CEPREMAP; Christian.Gourieroux@ensae.fr
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  • E. Renault

    1. Dept. of Economics, University of North Carolina at Chapel Hill, 107 Gardner Hall, CB 3305, NC 27599-3305, U.S.A. and CIRANO-CIREQ (Montreal); renault@email.unc.edu
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    • We thank D. Andrews, X. Chen, F. Corsi, R. Davidson, J. Fan, R. Garcia, J. Jackwerth, J. Jasiak, O. Linton, L. Mancini, A. Melino, C. Ortelli, O. Scaillet, M. Stutzer, F. Trojani, A. Vedolin, B. Werker, E. Zivot, a co-editor, and anonymous referees for helpful comments. The first author gratefully acknowledges financial support of the Swiss National Science Foundation through the NCCR FINRISK network. The second author gratefully acknowledges financial support of NSERC Canada and of the chair AXA/Risk Foundation: “Large Risks in Insurance.”


Abstract

In this paper, we introduce the extended method of moments (XMM) estimator. This estimator accommodates a more general set of moment restrictions than the standard generalized method of moments (GMM) estimator. More specifically, the XMM differs from the GMM in that it can handle not only uniform conditional moment restrictions (i.e., valid for any value of the conditioning variable), but also local conditional moment restrictions valid for a given fixed value of the conditioning variable. The local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns. The estimated derivative prices are consistent for a large time series dimension, but a fixed number of cross sectionally observed derivative prices. The asymptotic properties of the XMM estimator are nonstandard, since the combination of uniform and local conditional moment restrictions induces different rates of convergence (parametric and nonparametric) for the parameters.

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