The Minimal Entropy Martingale Measure in a Market of Traded Financial and Actuarial Risks

Abstract

In arbitrage-free but incomplete markets, the equivalent martingale measure Q for pricing traded assets is not uniquely determined. A possible approach when it comes to choosing a particular pricing measure is to consider the one that is 'closest' to the physical probability measure P, where closeness is measured in terms of relative entropy.

In this paper, we determine the minimal entropy martingale measure in a market where securities are traded with payoffs depending on two types of risks, which we will call financial and actuarial risks, respectively. In case only purely financial and purely actuarial securities are traded, we prove that financial and actuarial risks are independent under the physical measure if and only if these risks are independent under the entropy measure. Moreover, in such a market the entropy measure of the combined financial-actuarial world is the product measure of the entropy measures of the financial and the actuarial subworlds, respectively.

Keywords: Minimal entropy martingale measure, relative entropy, financial risks, actuarial risks, independence, incomplete markets.