Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K / d / Solution

Under the risk-neutral measure, writeLet $M_T={\max }_{t\le T}{X}_t$ and $m_T={\min }_{t\le T}{X}_t$. ThenThe time-reversed increment processhas the same finite-dimensional distributions as $X$, because $X$ has stationary independent increments. Moreover,Choosingtherefore makes $Y_2=S_0{e}^{X_T-m_T}$ equal in distribution to $Y_1$. Their discounted expectations, and hence their time-zero Black--Scholes prices, are equal. This is Brownian time reversal for fixed-strike lookback extrema.