Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data

This article investigates use of the Principle of Maximum Entropy for approximation of the risk-neutral probability density on the price of a financial asset as inferred from market prices on associated options. The usual strict convexity assumption on the market-price to strike-price function is relaxed, provided one is willing to accept a partially supported risk-neutral density. This provides a natural and useful extension of the standard theory. We present a rigorous analysis of the related optimization problem via convex duality and constraint qualification on both bounded and unbounded price domains. The relevance of this work for applications is in explaining precisely the consequences of any gap between convexity and strict convexity in the price function. The computational feasibility of the method and analytic consequences arising from non-strictly-convex price functions are illustrated with a numerical example.