Risk-Neutral Pricing and the Fundamental Theorem of Asset Pricing

Black-Scholes prices a European call. But when can any claim — a barrier option, an Asian average, a variance swap — be priced and hedged? The Fundamental Theorem of Asset Pricing (FTAP) gives the answer: no-arbitrage is equivalent to the existence of an equivalent martingale measure (EMM), and market completeness is equivalent to the EMM being unique. Under the EMM the discounted stock price is a martingale, and the arbitrage-free price of any replicable claim is just a discounted expectation under $\mathbb{Q}$.

This is not an abstraction — the martingale condition is exactly what constrains the drift to $r$ (the Girsanov shift), and uniqueness of the EMM is exactly what makes a replicating portfolio constructible from the Brownian filtration. Both properties are consequences of Girsanov's theorem and the martingale representation theorem.

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