A stock does not pay dividend. Annual stock volatility $\backslash$sigma is

$10\%.$ Annual compound interest rate is $8\%.$ Current stock price is $100$ $$.$ A

European call option has a strike of $100$ has a maturity of $2$ years. Throughout

this question, we assume the Black$-$Scholes model.

i$)$ Compute the price $($i$.$e$.$ premium, value$)$ of this option using the BlackScholes formula.

ii$)$ The price of the stock increases suddenly to $102$ $$,$ compute the new

price of this option using the Black$-$Scholes formula.

iii$)$ Recall that the $$ of a European call is e

$$q$($T $$t$)\backslash$Phi $($d$1),$ use the Deltaapproximation $($i$.$e$.$ first order Taylor$$s expansion$)$ to get an approximation of the new price of this European call. Please use St $=100$ to

calculate $.$

iv$)$ Recall that the $\backslash$Gamma of a European call is e$$q$($T $$t$)\backslash$phi $($d$1)$

St$\backslash$sigma

$$

T $$t

$,$ compute the Deltagamma$-$approximation of the new price of this European call. Please use

St $=100$ to calculate $\backslash$Gamma $.$

v$)$ Compare the two approximate prices with the theoretical Black$-$Scholes

price you find in question ii$).$ Which approximation is more accurate $?$

vi$)$ Assume that the current stock price is $102$ $$.$ Compute the price of a

cash$-$or$-$nothing call with a two$-$year time$-$to$-$maturity and a strike price

of $100,$ with payoff equal to $0$ if St$+2<100$ and equal to $1$ if St$+2>=100.$