Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 29L / e / Solution

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 $\alpha =1/12$.