No-Arbitrage Principle and Risk-Neutral Valuation › Summary

This chapter established the theoretical foundations that underpin all derivative pricing:

The no-arbitrage principle asserts that markets should not permit risk-free profits from zero capital. This simple economic axiom, enforced by competitive trading, places powerful restrictions on relative prices of financial instruments.

The Law of One Price follows directly: portfolios with identical payoffs must have identical prices. This enables pricing by replication: find a portfolio of simple instruments that mimics the derivative, and its cost is the derivative's fair value.

The binomial model demonstrates these principles concretely. By constructing a hedging portfolio that replicates option payoffs, we derive prices without knowing or using real-world probabilities.

Risk-neutral probabilities emerge from the replication argument. These artificial probabilities make all assets earn the risk-free rate in expectation, allowing us to price derivatives as simply discounted expected payoffs. They incorporate risk adjustments implicitly, absorbing investor risk preferences into the probability measure.

The Fundamental Theorem of Asset Pricing formalizes the no-arbitrage/risk-neutral connection: absence of arbitrage is equivalent to existence of a risk-neutral measure under which discounted prices are martingales.

These ideas extend naturally to continuous time, where the change of measure from real-world to risk-neutral probabilities is governed by Girsanov's theorem. In the next chapter, we'll use these principles to derive the celebrated Black-Scholes-Merton partial differential equation, which opened the door to modern quantitative finance.