Exercise $4(13+7=20$ points$)$ Let K$,$ T $>0.$ Denote by CE $($t$,$ T$,$ K$),$ PE $($t$,$ T$,$ K$),$

CA$($t$,$ T$,$ K$)$ and PA$($t$,$ T$,$ K$)$ the prices at time t $$ T of a European Call Option, a Euro$-$

pean Put Option, an American Call Option, and an American Put Option respectively,

with maturity T and strike K$.$ Assume that the underlying stock pays no dividends.

$1.$ It was shown in the lecture that CE $($t$,$ T$,$ K$)<$ St for any t in $[0,$ T $],$ where St is the

value of the underlying asset at time t$.$ Using a no$-$arbitrage argument, show that

this extends to the American Call Option as well, i$.$e:

CA$($t$,$ T$,$ K$)<$ St

for any t in $[0,$ T $].$

$2.$ In the lecture we mentioned the Put$-$Call parity bounds:

St $$ KB$($t$,$ T $)$ CA$($t$,$ T$,$ K$)$ PA$($t$,$ T$,$ K$)$ St $$ K

for t in $[0,$ T $],$ where B$($t$,$ T $)$ is the value at time t of a bond that pays $1$ dollar at

time T $.$ Since the right inequality in the above expression was proven in the lecture,

you are now asked to prove the left one, i$.$e:

St $$ KB$($t$,$ T $)$ CA$($t$,$ T$,$ K$)$ PA$($t$,$ T$,$ K$)$

for t in $[0,$ T $].$ Hint: while you can argue by no$-$arbitrage, this is not needed!