Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 2 / f / Solution

The logarithm of the geometric average isBy stochastic Fubini theorem, ${\int }_0^T{W}_{u}du={\int }_0^T(T-v)d{W}_v$, a centered Gaussian random variable with variance $T^3/3$. ThereforeThe Geometric Asian option is consequently a call on this lognormal random variable. Completing the square as in part c givesThe averaging changes both the log-mean and log-variance; replacing $S_T$ by an ordinary stock price at a shorter maturity without these adjustments would give the wrong formula.