$(20$ points$)$ BS Model and Option Greeks

Consider a non$-$dividend$-$paying stock, with price $S_t,$ and a European call option on that stock, whose value can be modelled using the Black$-$Scholes model.

$($a$)(5$ points$)$ Write down the formula for the delta of this option under this model.

$($b$)(5$ points$)$ Suppose that the stock price at time $0$ is $S_0=40$ and the continuously compounded risk$-$free rate is $2\%$ per annum. The call option has strike price $45\mathrm{.}91,$ term to maturity $5$ years and a delta of $=0\mathrm{.}6179.$

Determine the implied volatility of the stock to the nearest $1\%.$

$($c$)(5$ points$)$ A second stock with price $R_t,$ is currently priced at $R_0=$$$30$ and has volatility ${}_R=\sqrt[\phantom{2}]{15\%}$ per annum.

An exotic option pays an amount $c$ at time $T$ if $T=1c=$$$50k_S=0\mathrm{.}8k_R=0\mathrm{.}6\frac{R_1}{R_0}$

Give a risk$-$neutral pricing formula for the value $of$ the option $at$ time $0if$ the two stocks are independent,defining any additional notation used.

$(d)(5$ points$)$ Assume now that the stock prices are independent. The option has term $T=1$ year. payoff $c=$$$50$ and strike prices $k_S=0\mathrm{.}8$ and $k_R=0\mathrm{.}6.$ Determine the value $of$ the option $at$ time $0.\frac{S_1}{S_0}$ and$\frac{R_1}{R_0}$

Give a risk$-$neutral pricing formula for the value $of$ the option $at$ time $0if$ the two stocks are independent,defining any additional notation used.

$(d)(5$ points$)$ Assume now that the stock prices are independent. The option has term $T=1$ year. payoff $c=$$$50$ and strike prices $k_S=0\mathrm{.}8$ and $k_R=0\mathrm{.}6.$ Determine the value $of$ the option $at$ time $0.$