[Math] Running maximum for Geometric Brownian Motion

brownian motionstochastic-processes

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[(\mu-\sigma^2/2)t + \sigma W_t] $

where $W_t$ is a Wiener process

Best Answer

  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".
Related Question