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How Girsanov’s Theorem Changes the Drift for Derivative Pricing

Original question

How does Girsanov's theorem allow us to change probability measures when pricing financial derivatives?
Girsanov's theorem changes the probability weights on paths so that a Brownian motion with a drift adjustment becomes Brownian under the new measure. In a no-dividend Black–Scholes model with a constant interest rate, this changes the stock's drift from its physical expected return μ\mu to the risk-free rate rr. The volatility coefficient stays the same, and the discounted stock is a martingale under the new measure.
Under the physical measure P\mathbb P, the stock model describes how frequently different price paths occur. Under the risk-neutral measure Q\mathbb Q, those same paths are reweighted for pricing. The reweighting makes the expected return of the stock, in units of the money-market account, zero over each horizon. This lets a replicable option be valued as a discounted expected payoff under Q\mathbb Q even when the stock's physical expected return is uncertain.

Choose the sign by substituting back into the SDE

Assume dSt=μSt dt+σSt dWtPdS_t=\mu S_t\,dt+\sigma S_t\,dW_t^{\mathbb P} with constant σ>0\sigma>0 and risk-free rate rr. On a fixed horizon TT, define λ=(μ−r)/σ\lambda=(\mu-r)/\sigma and

ZT=exp⁡(−λWTP−12λ2T),dQdP∣FT=ZT.Z_T=\exp(-\lambda W_T^{\mathbb P}-\tfrac12\lambda^2T), \qquad \frac{d\mathbb Q}{d\mathbb P}\Big|_{\mathcal F_T}=Z_T.

The density is positive and integrates to one by the Gaussian moment-generating function. For this constant λ\lambda, Girsanov's theorem says WtQ=WtP+λtW_t^{\mathbb Q}=W_t^{\mathbb P}+\lambda t is a Brownian motion under Q\mathbb Q. Substitute dWtP=dWtQ−λ dtdW_t^{\mathbb P}=dW_t^{\mathbb Q}-\lambda\,dt to get

dSt=(μ−σλ)St dt+σSt dWtQ=rSt dt+σSt dWtQ.dS_t=(\mu-\sigma\lambda)S_t\,dt+\sigma S_t\,dW_t^{\mathbb Q} =rS_t\,dt+\sigma S_t\,dW_t^{\mathbb Q}.
The deterministic discount factor removes the remaining drift from e−rtSte^{-rt}S_t. The discounted process is a true martingale here because the resulting stochastic exponential has unit conditional expected growth. For a time-varying market price of risk, the stochastic exponential must still be a valid martingale. Novikov's condition is one sufficient criterion.

A reliable sign check is to start from the drift you want. You need the term added to μ\mu to be r−μr-\mu. Set dWP=dWQ−λ dtdW^{\mathbb P}=dW^{\mathbb Q}-\lambda\,dt so that the substitution contributes −σλ-\sigma\lambda. The exponential density with −λWTP-\lambda W_T^{\mathbb P} is the one that makes WP+λtW^{\mathbb P}+\lambda t Brownian under Q\mathbb Q. Its normalising term ensures a total probability of one. Changing the sign in the density while leaving the Brownian substitution alone would describe a different measure.

What changes for a one-year stock payoff

Take S0=100S_0=100, μ=0.10\mu=0.10, r=0.04r=0.04, σ=0.20\sigma=0.20, and T=1T=1. Then λ=0.30\lambda=0.30. A positive physical Brownian endpoint receives a smaller density weight than a negative endpoint of equal magnitude. The physical expectation of S1S_1 is 100e0.10≈110.52100e^{0.10}\approx110.52, whereas the risk-neutral expectation is 100e0.04≈104.08100e^{0.04}\approx104.08.

For a contract that pays S1−102S_1-102 at maturity, the time-zero value is

e−0.04EQ[S1−102]=100−102e−0.04≈2.00.e^{-0.04}\mathbb E_{\mathbb Q}[S_1-102] =100-102e^{-0.04}\approx2.00.
The same result follows by holding one share and financing 102102 due at maturity. This replication explains why the valuation uses rr and why the physical drift drops out.
The same calculation prices a call by replacing the linear payoff with (S1−K)+(S_1-K)^+. A simulation can draw directly from the lognormal distribution with drift rr under Q\mathbb Q.

References and next steps