Pricing Derivative Securities Using Cross-Entropy – An Economic Analysis

Branger Nicole

Abstract
This paper analyses two implied methods to determine the pricing function for derivatives when the market is incomplete. First, we consider the choice of an equivalent martingale measure with minimal cross-entropy relative to a given benchmark measure. We show that the choice of the numerative has an impact on the resulting pricing function, but that there is no sound economic answer to the question which numerative to choose. The ad-hoc choice of the numeric introduces an element of arbitrariness into the pricing function, thus contracting the motivation of this method as the least prejudiced way to choose the pricing operator. Second, we propose two new methods to select a pricing function: the choce of stochastic discount factor (SDF) with minimal extended cross-entropy relative to a given benchmark SDF, and the choice of the Arrow-Debreu (AD) prices with minimal extended cross-entropy relative to some set of benchmark AD prices. We show that these two methods are equivalent in that they generate identical pricing fuctions. They avoid the dependence on the numeraire and replace it by the dependence on the benchmark pricing function. The benchmark pricing function, however, can be chosen based on economic considerations, in contrast to the arbitrary choice of the numeraire.

Keywords
Equivalent Martingale Measure; Stochastic Discount factor; Cross-Entropy; Implied Distributions; Option Pricing