Problem $1$ Let $S(0)=$$$100,r=8\%($compounded continuously$),$ and $=0.$ Consider a

single period binomial tree model with $u=1\mathrm{.}3,d=0\mathrm{.}8$ and $T=0\mathrm{.}5.$

$($a$)$ Find the price of a European call option with strike price $$90$ using the risk$-$neutral

pricing formula.

$($b$)$ Find the price of a European put option with strike price $$90$ using the method of

replication.

$($c$)$ Verify the validity of put$-$call parity.

$($b$)$ Suppose you observe a market price of a $90-$strike put is $$8.$ Demonstrate how arbitrage

profit can be obtained.