The Breeden–Litzenberger Distribution

Abstract

These notes derive and explore the Breeden–Litzenberger formula, which extracts the risk-neutral probability density of a future asset price directly from observed European option prices via a second derivative with respect to strike. The derivation is presented in two ways: first through a butterfly spread replication argument, then rigorously via Leibniz’s rule applied to the call pricing integral. We examine the no-arbitrage conditions encoded in the density, and draw a sharp distinction between the risk-neutral measure ℚ and the real-world measure ℙ, connecting them through the stochastic discount factor. The CRRA pricing kernel and Esscher transform are introduced as practical tools for converting between measures given a view on expected returns or tail probabilities. The variance risk premium is derived as a consequence of the covariance between the discount factor and realised variance. The final section discusses how the density shape varies across asset classes — energy commodities and equity indices — and how discrepancies between one’s real-world density and the market’s risk-neutral density identify over- and under-priced options.