The Entropic Measure Transform

TL;DR

This paper introduces the entropic measure transform (EMT), a new mathematical framework for pricing defaultable bonds and other financial derivatives, extending existing stochastic control methods and providing economic insights.

Contribution

The paper develops the EMT for general processes, characterizes the optimal measure via BSDEs, and applies it to pricing in jump diffusion models, extending prior stochastic control approaches.

Findings

Unique optimal measure exists for EMT.

Provides a new formula for conditional expectations under affine models.

Numerical example demonstrates EMT's application to defaultable bonds.

Abstract

We introduce the entropic measure transform (EMT) problem for a general process and prove the existence of a unique optimal measure characterizing the solution. The density process of the optimal measure is characterized using a semimartingale BSDE under general conditions. The EMT is used to reinterpret the conditional entropic risk-measure and to obtain a convenient formula for the conditional expectation of a process which admits an affine representation under a related measure. The entropic measure transform is then used provide a new characterization of defaultable bond prices, forward prices, and futures prices when the asset is driven by a jump diffusion. The characterization of these pricing problems in terms of the EMT provides economic interpretations as a maximization of returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy. The EMT is shown to extend the optimal stochastic control characterization of default-free bond prices of Gombani and Runggaldier (Math. Financ. 23(4):659-686, 2013). These methods are illustrated numerically with an example in the defaultable bond setting.