A stock does not pay dividend. Annual stock volatility

$\backslash$

sigma is

$1$

$0$

$\%$

$.$

$$Annual compound interest rate is

$8$

$\%$

$.$

$$Current stock price is

$1$

$0$

$0$

$$$

$.$

$$A

European call option has a strike of

$1$

$0$

$0$

$$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

$1$

$0$

$2$

$$$

$,$

$$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

$=$

$1$

$0$

$0$

$$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

$=$

$1$

$0$

$0$

$$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

$1$

$0$

$2$

$$$

$.$

$$Compute the price of a

cash

$-$

or

$-$

nothing call with a two

$-$

year time

$-$

to

$-$

maturity and a strike price

of

$1$

$0$

$0$

$,$

$$with payoff equal to

$0$

$$if St

$+$

$2$

$<$

$1$

$0$

$0$

$$and equal to

$1$

$$if St

$+$

$2$

$>$

$=$

$1$

$0$

$0$

$.$