This is a finmath problem. Problem $2$

Suppose that the bank account $B$ and a non$-$dividend$-$paying stock price $S$ have the following

dynamics under risk$-$neutral measure.

$d{B}_t=r{B}_{t}dt{B}_0=1$

$d{S}_t=r{S}_{t}dt+S_{t}d{W}_t{S}_0=216$

Let $T>0,$ and assume that exp$(-rT)=0\mathrm{.}96.$ Let $K=250$ and assume that a $T-$expiry $K-$strike

call on $S$ has time$-0$ price $34,$ and a $T-$expiry $K-$strike binary call on $S$ has time$-0$ price $0\mathrm{.}44.$

The expectation in $($d$)$ and probability in $($e$)$ are with respect to risk$-$neutral measure. Compute:

$($a$)$ The time$-0$ price of a $T-$expiry $K-$strike binary put on $S.$

$($b$)$ The time$-0$ price of a $T-$expiry $K-$strike put on $S.$

$($c$)$ The time$-0$ price of a $T-$expiry $K-$strike asset$-$or$-$nothing call on $S.$

$($d$)$ The time$-0$ expectation of $S_T.$

$($e$)P(S_T>K).$

$($f$)$ A portfolio holds $\{B,S\}$ in quantities that vary continuously in time. It is self$-$financing and

its time$-T$ value is $(K-S_T{)}^{+}.$ How many units of $S$ does it hold at time $0?$