Girsanov and the risk-neutral measure

Changing the odds, not the prices

In the real world, with probabilities $P$, a stock follows $dS=\mu S\,dt+\sigma S\,dW$. The average growth $\mu$ is hard to know and every investor has a different view. Girsanov's theorem says you can re-weight the paths so that the drift becomes the risk-free rate $r$, while the volatility $\sigma$ stays the same. The new weights are the risk-neutral measure $Q$. You get there by shifting the random part: $\tilde W_t=W_t+\theta t$, where $\theta=(\mu-r)/\sigma$ is the market price of risk, the extra return you earn per unit of risk. Under $Q$ the process $\tilde W$ is a plain Brownian motion.

Why do this? An option can be hedged with the stock and cash, so its price cannot depend on how much you hope the stock will grow. Pricing under $Q$ removes $\mu$ from the problem. The price today is the average payoff under $Q$, discounted at $r$. Keep in mind that $Q$ is a calculation tool and not a forecast: nobody claims the stock really grows at $r$. It is just the set of odds that makes the arithmetic of hedging work.

• Market price of risk: $\theta=(\mu-r)/\sigma$, the extra return per unit of risk.
• Girsanov changes only the drift. The volatility $\sigma$ is the same under $P$ and $Q$.
• Price = discounted average payoff under $Q$. $Q$ is a tool for pricing, not a forecast.

Lab · what's graded

• The market price of risk15 pts
• The risk-neutral forward price15 pts
• The risk-neutral probability15 pts
• The value of the call15 pts
• What Girsanov changes15 pts
• Pricing with the risk-neutral method10 pts
• Why the real return drops out15 pts