Market Completeness and Measure Changes — The Esscher Transform

In the Black-Scholes model, the unique equivalent martingale measure (EMM) is pinned down by the market price of Brownian risk: no additional degrees of freedom. Add a compound Poisson jump component, and the EMM is no longer unique: any choice of risk premium for the jump component (affecting the jump intensity and/or jump size distribution under $\mathbb{Q}$) produces a valid pricing measure, as long as no static arbitrage results.

The set of all valid pricing measures is a convex family, and any point in this family corresponds to a different exotic option price. The desk must choose — and the choice is a model assumption, not a market observable. Two canonical choices are the Esscher transform (which shifts the log-moment generating function and is the unique choice under exponential utility) and the minimal entropy martingale measure (which minimises the relative entropy from the historical measure, the "least distorted" view of the world).

Learning objectives. By the end of this note you should be able to: