Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 29L (Stochastic Financial Models)

Under zero drift, reflection gives the maximum tail. Cameron-Martin changes from Brownian motion to drift $c$, weighting reflected paths by $e^{2cx}$, and yields $P({\max }_{t\le T}{X}_t\ge x)=P(X_T\ge x)+e^{2cx}P(-X_T\ge x)$.

Under the risk-neutral measure, $\log S_T=\log S_0+(r-{\sigma }^2/2)T+\sigma W_T^Q$. Therefore $\pi (\log S_T)=e^{-rT}[\log S_0+(r-{\sigma }^2/2)T]$.

The initial delta is the derivative with respect to $S_0$, namely $e^{-rT}/S_0$ shares.

Apply part (a) under the risk-neutral log-price drift $r-{\sigma }^2/2$, integrate the maximum and minimum tails using the supplied identities, and simplify the two exponential moments. Fubini then identifies the residual term as $({\sigma }^2/2){\int }_0^T{S}_{t}dt$. Discounted expectations on both sides give exactly the stated pricing identity.