Girsanov's theorem and change of measure

Girsanov's theorem says: by an appropriate change of probability measure, you can absorb the drift of a Brownian motion into the new measure. Equivalent in finance: real-world expected returns are different from risk-neutral expected returns, but the underlying randomness is the same. Girsanov is the deep theorem that lets us replace expectations under P with expectations under Q for pricing purposes.

Setup

Let W be a Brownian motion under measure P. Define a process θ(t) (the 'change-of-measure speed' or market price of risk), and let:

Z(t) = exp(-∫_0^t θ(s) dW(s) - (1/2) ∫_0^t θ²(s) ds)

Z is a positive martingale with E^P[Z(t)] = 1 (provided Novikov's condition E^P[exp((1/2) ∫θ²)] < ∞ holds). Define a new measure Q by dQ/dP = Z(T) on F_T. Q is equivalent to P (they assign zero probability to the same events) but different from P.

Girsanov's theorem

Under Q, the process W^Q(t) = W(t) + ∫_0^t θ(s) ds is a (standard) Brownian motion. Equivalently, dW^Q = dW + θ(t) dt: the new BM is the old BM plus a drift.

Application to GBM

Real-world dynamics: dS = μ S dt + σ S dW. Want a measure under which the discounted stock e^{-rt} S(t) is a martingale (no arbitrage). Need expected appreciation under the new measure to equal r:

Choose θ = (μ - r)/σ.
Under Q: dW^Q = dW + θ dt, so σ dW = σ dW^Q - σθ dt = σ dW^Q - (μ - r) dt.
Thus dS = μ S dt + σ S (dW^Q - θ dt) = μ S dt + σ S dW^Q - (μ - r) S dt = r S dt + σ S dW^Q.

Under Q, the stock has drift r — the risk-free rate. This is the risk-neutral measure that underpins all of derivatives pricing.

Equivalent martingale measure (EMM)

An EMM is a measure equivalent to P under which discounted asset prices are martingales. Girsanov shows how to construct one from a market-price-of-risk specification. In a single-asset world, the EMM is unique (one θ to choose); in multi-asset worlds, EMM uniqueness corresponds to market completeness.

Fundamental theorems of asset pricing

1. First FTAP: no arbitrage ⟺ existence of at least one EMM.
2. Second FTAP: completeness (every contingent claim hedgeable) ⟺ uniqueness of EMM.

These are the most important meta-results in continuous-time finance. They tell you that pricing-by-expectation under Q is not arbitrary — it's mathematically equivalent to the absence of arbitrage.

Why this is the deep theorem

Girsanov + FTAP says: to price an option, take its discounted payoff under Q, take the Q-expectation, and you get an arbitrage-free price. No utility functions, no risk aversion, no investor preferences — purely an algebraic consequence of no-arbitrage and the existence of a replicating portfolio. This is why option pricing is so clean: it reduces to integration under the right measure.

Computational implications

Monte Carlo pricing: simulate under Q (replace μ with r in the SDE), compute discounted payoff for each path, average. PDE pricing: derive the PDE under Q, solve numerically. Tree pricing: use risk-neutral probabilities (Module 9 of Numerical Methods).

Beyond constant drift

Girsanov works in much more generality: time-varying θ, multi-dimensional W, jump processes (with appropriate jump-measure changes). The mathematical structure remains: a Radon-Nikodym density Z transforms one probability measure into another while preserving the underlying source of randomness.