Homework due Aug $12,202407$:$00$ CDT

Question $5$

$\frac{0\mathrm{.}0}{20\mathrm{.}0}$ points $($graded$)$

All assumptions of the Black$-$Scholes$-$Merton option pricing model hold. Stock $xYZ$ is priced at $S_{xYZ}=$$$40.$ It has volatility $=20\%$ per year. The annualized continuously$-$compounded risk$-$free interest rate is $r=3\mathrm{.}2\%.$

Compute the price of a European call option with strike price $K=$$$41,$ which matures in $6$ months.

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Compute the option Delta at time $t=0.$

Suppose that at time $t=0$ the stock price changes instantaneously from $S_{xYZ}=40$ to $S_{xYZ}=42.$ Compute the resulting change in the option price.

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Suppose that at time $t=0$ the stock price changes instantaneously from $S_{xYZ}=40$ to $S_{xYZ}=42.$ Compute the resulting change in the value of the replicating portfolio for this option.

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Suppose that at time $t=0$ the stock price changes instantaneously from $S_{xYZ}=40$ to $S_{xYZ}=38.$ Compute the resulting change in the option price.

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Suppose that at time $t=0$ the stock price changes instantaneously from $S_{xYZ}=40$ to $S_{xYZ}=38.$ Compute the resulting change in the value of the replicating portfolio for this option.

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