The Relative Entropy of Expectation and Price

TL;DR

This paper introduces a novel entropic risk metric to quantify market incompleteness and default risks, deriving a log-martingale pricing condition that adjusts derivative prices accordingly.

Contribution

It proposes a new pricing framework based on minimum entropy relative to expectations, extending market models to incorporate default and funding risks.

Findings

Quantifies convexities from bid-offer spreads and embedded options.

Derives a log-martingale pricing condition for incomplete markets.

Applies the framework to model risk, deep hedging, and quantum information.

Abstract

As operators acting on the undetermined final settlement of a derivative security, expectation is linear but price is non-linear. When the market of underlying securities is incomplete, non-linearity emerges from the bid-offer around the mid price that accounts for the residual risks of the optimal funding and hedging strategy. At the extremes, non-linearity also arises from the embedded options on capital that are exercised upon default. In this essay, these convexities are quantified in an entropic risk metric that evaluates the strategic risks, which is realised as a cost with the introduction of bilateral margin. Price is then adjusted for market incompleteness and the risk of default caused by the exhaustion of capital. In the complete market theory, price is derived from a martingale condition. In the incomplete market theory presented here, price is instead derived from a log-martingale condition: \begin{equation} p=-\frac{1}{\alpha}\log\mathbb{E}\exp[-\alpha P] \notag \end{equation} for the price $p$ and payoff $P$ of a funded and hedged derivative security, where the price measure $\mathbb{E}$ has minimum entropy relative to economic expectations, and the parameter $\alpha$ matches the risk aversion of the investor. This price principle is easily applied to standard models for market evolution, with applications considered here in model risk analysis, deep hedging and quantum information.