Asset Pricing with Second-Order Esscher Transforms

36 Pages Posted: 12 Sep 2012

See all articles by Alain Monfort

Alain Monfort

National Institute of Statistics and Economic Studies (INSEE) - Center for Research in Economics and Statistics (CREST); National Bureau of Economic Research (NBER); Maastricht University

Fulvio Pegoraro

Banque de France - Economics and Finance Research Center; CREST - Laboratoire de Finance et Assurance

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Date Written: September 1, 2012

Abstract

The purpose of the paper is to introduce, in a discrete-time no-arbitrage pricing context, a bridge between the historical and the risk-neutral state vector dynamics which is wider than the one implied by a classical exponential-affine stochastic discount factor (SDF) and to preserve, at the same time, the tractability and flexibility of the associated asset pricing model. This goal is achieved by introducing the notion of Exponential-Quadratic SDF or, equivalently, the notion of Second-Order Esscher Transform. The log-pricing kernel is specified as a quadratic function of the factor and the associated sources of risk are priced by means of possibly non-linear stochastic first-order and second-order risk-correction coefficients. Focusing on security market models, this approach is developed in the multivariate conditionally Gaussian framework and its usefulness is testified by the specification and calibration of what we name the Second-Order GARCH Option Pricing Model. The associated European Call option pricing formula generates a rich family of implied volatility smiles and skews able to match the typically observed ones.

Keywords: second-order Esscher transform, exponential-quadratic stochastic discount factor, non-linear stochastic risk-correction coefficients, variance-covariance spread, second-order GARCH option pricing model

JEL Classification: G12, G13

Suggested Citation

Monfort, Alain and Pegoraro, Fulvio, Asset Pricing with Second-Order Esscher Transforms (September 1, 2012). Banque de France Working Paper No. 397, Available at SSRN: https://ssrn.com/abstract=2144703 or http://dx.doi.org/10.2139/ssrn.2144703

Alain Monfort (Contact Author)

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Fulvio Pegoraro

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