Find probability that payoff function is in $[10,20]$

Question

In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by:

$$f=(\max_{t\in[0,T]} S_t -110)^{+}$$

where $S_t$ satisfies

$$dS_t=15dW_t$$ $$S_0=95$$

Find probability $P(f\in[10,20])$. Use cumulative density fucntion of standard normal distribution to express it.

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Obviously

$$S_t=95+15W_t$$

$$f=(15 \max_{t\in[0,T]} W_t -15)^{+}$$

But i don't have any idea how can I move further. Any help please?

Answer 1

You need to use the density function of the "running maximum" of the Brownian motion. This density function is well known, see for example here.