Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 29L (Stochastic Financial Models)

For times $0\le t_1<\cdots <t_m$, the vector $(W_{t_1},\dots ,W_{t_m})$ is a linear transformation of the independent Gaussian incrementsso Brownian motion is a Gaussian process. Its mean is zero, and for $s\le t$,

Fubini gives $\mathbb{E}{I}_t=0$. For $s\le t$,

For $s<t$,The increment is independent of the Brownian filtration at time $s$, and its exponential factor has mean one by the Gaussian moment-generating function. Hence the conditional expectation of the discounted time-$t$ price is the discounted time-$s$ price.

Risk-neutral valuation givesWith $v={\sigma }^{2}T$ and $m=K{e}^{-rT}/S_0$, this is exactly

The average log price isPart (b) gives $\mathrm{var}(I_T)=T^3/3$, so the Gaussian variance of the final term isRewriting the lognormal factor in the normalized form $e^{-v/2+\sqrt{v}Z}$ and discounting givesThus $\alpha =1/12$.