Running maximum for Geometric Brownian Motion

Question

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ evolves as

$$ X_t = X_0 \exp\left[\left(\mu-\frac{\sigma^2}{2}\right)t + \sigma W_t\right] $$

where $W_t$ is a Wiener process?

Answer 1

1. First of all, it will be easier to find the distribution of the running maximum of $\log X$ which in your notation will be $\log M$.
2. Assume for a moment that $\log X$ has zero drift. Reflection principle then tells you that for any $a > \log X_0$ $$ P(\log M_t > a) = 2 P(\log X_t > a ).\tag{1} \label{1}$$ Similarly for $a<\log X_0$, with reversed inequalities in \eqref{1}. Now since $\log X_t$ is Gaussian, one can evaluate the rhs of \eqref{1}. This gives you CDF and after differentiation a PDF of $\log M_t$. This should be enough for your purposes.
3. To deal with the non-zero drift, change measure, apply Girsanov to get zero drift, apply reflection principle as in 2. and change measure back. Alternatively search for "maximum of Brownian motion with drift".