Asset Pricing with Second-Order Esscher Transforms

The purpose of the paper is to introduce, in the class of discrete time no-arbitrage asset pricing models, awider bridge between the historical and the risk-neutral state vector dynamics and to preserve, at the same time,its tractability and °exibility. This goal is achieved by introducing the notion of Exponential-Quadratic stochasticdiscount factor (SDF) or, equivalently, the notion of Second-Order Esscher Transform. Then, focusing on securitymarket models, this approach is developed in three important multivariate stochastic frameworks: the conditionallyGaussian 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 di®erent from the historical one. Theconditionally mixed-normal Gaussian case provides a ¯rst generalization of the Gaussian setting, in which the risk-neutral variance-covariance matrices and mixing weights of all components (in the ¯nite mixture) can be di®erentfrom the historical ones. The Gaussian switching regime case introduces further °exibility given the serial dependenceof regimes and the introduction of the regime indicator function in the exponential-quadratic SDF. We also developswitching regime models which include (in the factor's conditional mean and conditional variance) additive impactsof the present and past regimes and we stress their interpretation in terms of general "discrete-time jump-di®usion"models in which the risk included in the ¯rst 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 vectorand therefore this appr