Drift Lives in the Measure: An Intuitive Look at Girsanov's Theorem
Why This Matters We want to price a derivative. Under the real world measure $\mathbb{P}$, we face two problems. First, we do not know the true drift $\mu$ of the underlying, and historical estimates are notoriously unreliable. Second, even if we knew $\mu$, taking the expected payoff under $\mathbb{P}$ would still not give the market price. Risky cash flows must be discounted more heavily than guaranteed ones because investors are risk averse. Pricing under $\mathbb{P}$ requires both the true probabilities of outcomes and a model for how the market prices risk. Both are fundamentally unobservable. So what can we do? ...