A stock does not pay dividend. Annual stock volatility o 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 Black$-$

Scholes 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 A of a European call is e$-$q$($T$-$t$)($d$1),$ use the Delta$-$

approximation $($i$.$e$.$ first order Taylor's expansion$)$ to get an approxi$-$

mation of the new price of this European call. Please use St $=100$ to

calculate $4.$

iv$)$ Recall that the T of a European call is e $($T$-$t$)$o$($d$1)$

S$1$ovT$=$t $,$ compute the Delta$-$

gamma$-$approximation of the new price of this European call. Please use

St $=100$ to calculate T$.$

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$42<mtext\; id="EditorElement\_238302"></mtext><mn\; id="EditorElement\_238303">1</mn><mn\; id="EditorElement\_238304">0</mn><mn\; id="EditorElement\_238305">0</mn>$ and equal to $1$ if St$42>=100.$Question $1.$ A stock does not pay dividend. Annual stock volatility $$ 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 Black$-$

Scholes 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 $e^{-q(T-t)}(d_1),$ use the Delta$-$

approximation $($i$.$e$.$ first order Taylor's expansion$)$ to get an approxi$-$

mation of the new price of this European call. Please use $S_t=100$ to

calculate $.$

iv$)$ Recall that the $$ of a European call is $\frac{e^{-q(T-t)}(d1)}{S_t\sqrt[\phantom{2}]{T-t}},$ compute the Delta$-$

gamma$-$approximation of the new price of this European call. Please use

$S_t=100$ to calculate $.$

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

price you find in question $i$ i$).$ 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 $S_{t+2}<mn\; id="EditorElement\_235571">1</mn><mn\; id="EditorElement\_235572">0</mn><mn\; id="EditorElement\_235573">0</mn>$ and equal to $1$ if $S_{t+2}100.$