# Describing the Minimum of a Brownian Motion

I'm working on characteristic functions of GBM and implementing them in option pricing in python. In deriving the cf for the minimum of a drifted BM I make this step:

First, note that by implementing the definition of a minimum, also using the symmetry property of Brownian Motion, we can rewrite $$\widetilde N_0^T$$ as follows $$$$\widetilde N_0^T = -\max_{0 \leq t \leq T}\big(-\widetilde W(t)\big) = -\max_{0 \leq t \leq T}\big(-W(t) -\alpha t\big) = -\max_{0 \leq t \leq T}\big(W(t) -\alpha t\big)$$$$

Is this expression for the minimum faulty? I have been sifting through the code implementation for errors elsewhere for hours now, so is this the problem? Thanks in advance!!

• The step $-\max(-W(t)-\alpha t) = -\max(W(t)-\alpha t)$ is not correct. This is true in distribution when $W(t)$ is a Brownian motion. If you are operating in a measure under which $W(t)$ is not a Brownian motion, or need to worry about the joint distribution of $\tilde N$ and any other random variable that is not independent of $W$, then this step may cause some problems. May 26 at 12:57