Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K (Stochastic Financial Models)

For $0\le s\le t$, the increment $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and is $N(0,t-s)$. HenceAlso $\mathbb{E}{M}_t=1$, so $M$ is the exponential Brownian martingale.

Setand define $Q$ on ${\mathcal{F}}_T$ byThe Cameron-Martin theorem for a linear drift says thatis a Brownian motion under $Q$. ThereforeThe discounted stock is an exponential Brownian martingale, so $e^{-rt}{S}_t$ is a $Q$-martingale. Thus $Q$ is a Risk-neutral measure for the Black-Scholes model.

Risk-neutral valuation givesSince $W_T^Q\sim N(0,T)$,Hence the Power payoff in the Black-Scholes model has price

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.