Solution: Why the Physical Drift Disappears

Exercise: Why the Physical Drift Disappears

Solution

The option payoff is replicated by dynamic trading in the stock and risk-free account. Once the Brownian risk is hedged locally, the remaining portfolio is riskless and must earn $r$. The expected stock return demanded by investors is therefore not needed to price the replicated payoff.

Equivalently, under the risk-neutral measure, the drift is changed from $\mu$ to $r$ while volatility remains $\sigma$.

Takeaways

• Replication removes exposure to the physical drift.
• Pricing uses $\mathbb{Q}$, not historical expectation under $\mathbb{P}$.
• Volatility remains because it controls the distribution of hedgeable uncertainty.