Problem $1[50$ marks$]$

Let $us$ consider a continuous time market, where the continuously com$-$

pounded interest rate $isr>0,$ the risky asset $S=(S_t{)}_{0}tT$ follows the Black$-$

Scholes model with drift $$ and volatility $,so$ that $S$ follows the dynamic

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

with Brownian motion $B.$

$a{S}_{t}as$ function $of(t,B_t).$

$bQ$ denote the $(equivalent)$ risk neutral probability, and $B^Q$

denote the corresponding Brownian motion under $Q.$ Write down the

dynamic $ofS$ under $Q,$ and then give the expression $of{S}_{t}as$ function $of$

$(t,B_t^Q).$

$c{1}_{\{S_T)}>K.$ Compute

the following expectation

$E^Q[e^{-rT}{1}_{\{S_T)}>K].$

Warning: Please note the sign difference with the midterm question.

$d|S_T-K|$

$(absolute$ value $of$ the difference$),$ for some constant $K>0.$ Compute the

following expectation

$E^Q[e^{-rT}|S_T-K|]$

Note: Please express the above expectation values $in$ terms $of$ the $pa-$

rameters $r,,,T$ and $K.$ You may use $$:Rlongrightarrow$[0,1]to$ denote the

cumulative distribution function $of$ the standard Gaussian distribution

$N(0,1).$