A forward contract with delivery price $K$ obligates its holder to buy one share of the stock at expiration

time $T$ in exchange for payment $K.$ Let $f(t,x)$ be the value of forward contract with at earlier times

tin$[0,T]$ if the stock price at time $t$ is $S(t)=x.$

$($a$)$ Argue by arbitrage$-$free argument that the forward price must be

$f(t,x)=x-e^{-r(T-t)}K.$

$($b$)$ Since $f(t,x)$ satisfies the same BSM PDE as for $c(t,x)($Question $2($c$)),$ we must have the put$-$call

parity formula $($prove it$)$

$f(t,x)=c(t,x)-p(t,x),x0,0tT.$

Hint: Check terminal and boundary conditions.

$($c$)$ If $S(t)$ follows BSM model, use It$\text{'}$s lemma to derive the SDE for $f(t,S(t)).$ What is your

conclusion of the volatility for $f(t,S(t))?$