Q$2)$

You are considering the purchase of a $3-$month $41\mathrm{.}5-$strike American call option on a nondividend$-$paying stock $G(t).$ You are given: $($i$)$ The Black$-$Scholes framework holds. $($ii$)$ The stock is currently selling for $40.($iii$)$ The stock's volatility is $30\%.($iv$)P(N(0,1)z_0+\sqrt[\phantom{2}]{T})=0\mathrm{.}5,$ where $z_0$ is such that $P(G(T)K=41\mathrm{.}5)=1.$

a$)$ Prove that an American call option on a non$-$dividend$-$paying stock has the same price as a European call option on a nondividend$-$paying stock. $(02$ marks$)$

b$)$ Find the risk free interest rate $r.(03$ marks$)$

c$)$ Solve the diffusion process of the stock under the risk$-$neutral probability, where the risk free interest rate is the one calculated in a$).$ You need to define precisely the new Brownian motion under this risk$-$neutral probability. $(07$ marks$)$

d$)$ Derive the formula to calculate the value $C$ of the American call option on the nondividend$-$paying stock. $(08$ marks$).$

e$)$ Calculate the value $C$ of the American call option. $(03$ marks$).$

f$)$ Prove and explain the Call $-$ Put parity relation $(02$ marks$)$

g$)$ Do you think that the value of an American Put option on a nondividend$-$paying stock should be different of the value of an European Put option on a nondividend$-$paying stock? Explain. $(05$ marks$)$

Please do question $2$