# Continuous option pricing: Brownian Bridge

I have a question on the proof of the formula of Sup(S) between 2 simulation points.

Do you know how the prove the following formula? Thanks

• I'd have to look at it further, but why not fix $t=0$ and $t+\Delta t = 1$ to start. Then should the solution to the SDE not look like $$S_{t} = S_{0}^{1-t}S_{1}^{t}e^{\sigma B_{t}}$$ where $B_{t}$ is a standard Brownian bridge with $B_{1}=0$? Where does the problem come from? I think context would help. – Christopher K Dec 3 '20 at 23:03