Problem $2(25$p$)($Digital Option $-$ Probabilistic Approach$)$

Let $\{W_t$:$t0\}$ be a $P-$standard Wiener process on the probability space $(,F,P)$ and let the stock price $S_t$ follow a GBM with the following SDE

$d{S}_t=S_{t}dt+S_{t}d{W}_t$

where $$ is the drift parameter, $>0$ is the volatility parameter, and let $r>0$ denote the risk$-$free interest rate.

A digital $($or cash$-$or$-$nothing$)$ call option is a contract that pays $$1$ at expiry time $T$ if the spot price $S_T>K$ and nothing if $S_{T}K.$ In contrast, a digital $($or cash$-$or$-$nothing$)$ put pays $$1$ at expiry time $T$ if the spot price