Asset Pricing with Second-Order Esscher Transforms

Abstract

The purpose of the paper is to introduce, in the class of discrete time no-arbitrage asset pricing models, a wider bridge between the historical and the risk-neutral state vector dynamics and to preserve, at the same time, its tractability and flexibility. This goal is achieved by introducing the notion of Exponential-Quadratic stochastic discount factor (SDF) or, equivalently, the notion of Second-Order Esscher Transform. Then, focusing on security market models, this approach is developed in three important multivariate stochastic frameworks: the conditionally Gaussian framework, the conditionally Mixed-Normal and the conditionally Gaussian Switching Regimes framework.

In the conditionally multivariate Gaussian case, our approach determines a risk-neutral mean as a function of (the short rate and of) the risk-neutral variance-covariance matrix which is different from the historical one. The conditionally mixed-normal Gaussian case provides a first generalization of the Gaussian setting, in which the risk-neutral variance-covariance matrices and mixing weights of all components (in the finite mixture) can be different from the historical ones. The Gaussian switching regime case introduces further flexibility given the serial dependence of regimes and the introduction of the regime indicator function in the exponential-quadratic SDF. We also develop switching regime models which include (in the factor's conditional mean and conditional variance) additive impacts of the present and past regimes and we stress their interpretation in terms of general "discrete-time jump-diffusion'' models in which the risk included in the first and second moment of jumps is priced.

Even if we focus on security market models, we do not make any particular assumption about the state vector and therefore this approach could be used not only in option pricing models, but also for instance in interest rate and credit risk models.

Keywords: Second-Order Esscher Transform, Exponential-Quadratic Stochastic Discount Factor, No-Arbitrage Asset Pricing Models, Security Market Economies.

JEL Classification: G12, G13