Problem $3.$ Consider a market consisting of a safe asset $B$ with $B_t=1+t$

for all $t0$ and one risky asset $S$ with $S_0=1$ and

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

for each $t0,$ where $W$ is a standard Wiener process.

$($a$)$ What is the dynamics of $S$ under a risk$-$neutral probability

measure?

$($b$)$ Find a formula for the no$-$arbitrage price at time $0$ of a

European call option with maturity $5$ on the risky asset with

strike price $100.($Note that the payoff of this option at time $10$

equals $(S_5-100{)}^{+}.)$