Determine $.$

Q$4)$

You are considering the purchase of a $3-$month $41\mathrm{.}5-$strike American call option on a non$-$

dividend$-$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$).$

J$)$ 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$)$

Q$5)(10$ marks$)$

For a two$-$period binomial model for stock prices, you are given:

$($i$)$ Each period is $6$ months.

$($ii$)$ The current price for a non$-$dividend$-$paying stock is R$60\mathrm{.}00.$

$($iii$)u=1\mathrm{.}179,$ where $u$ is one plus the rate of capital gain on the stock per period if the

price goes up$.$

$($iv$)d=0\mathrm{.}890,$ where $d$ is one plus the rate of capital loss on the stock per period if the

price goes down.

$($v$)$ The continuously compounded risk$-$free interest rate is $4\%.$

a$)$ Prove that the formula of the risk neutral probability of the stock price going up is

given by:

$p^{**}=\frac{e^{(r-)h}-d}{u-d},is$ the dividend rate. $(03$ marks$)$

b$)$ Calculate the current price of a one$-$year American put option on the stock with ${?}^2$

strike price of $R70\mathrm{.}00.(07$ marks$)$

Q$6)(10$ marks$)$

The stochastic process $(R(t),t0)$ is given by:

$R(t)=R(0)e^{-t}+0\mathrm{.}05(1-e^{-t})+0\mathrm{.}1{\displaystyle \underset{0}{\overset{t}{}}}{e}^{s-t}\sqrt[\phantom{2}]{R(s)}dZ(s),$ where $\{R(t),t0\}is$ a standard

Brownian motion.

a$)$ Find $dR(t),t0,$ the Stochastic differential equation of $R(t).(07$ mark$)$

b$)$ Is $R(t)$ a martingale with respect to $Z(t)?.$ The proof is required $(03$ marks$)$