$12\mathrm{.}14$ Consider a bull spread where you buy a $40-$strike call and sell a $45-$strike call.

Suppose $=0\mathrm{.}30,r=0\mathrm{.}08,=0,$ and $T=0\mathrm{.}5.$

a$.$ Suppose $S=$$$40.$ What are delta, gamma, vega, theta, and rho?

b$.$ Suppose $S=$$$45.$ What are delta, gamma, vega, theta, and rho?

c$.$ Are any of your answers to $($a$)$ and $($b$)$ different? If so$,$ why?Consider a bull spread where you buy a $40-$strike call and sell a $45-$strike call. Suppose S$=$$ $40,\backslash$sigma $=0\mathrm{.}30,$ r$=0\mathrm{.}08,\backslash$delta $=0,$ and T$=0\mathrm{.}5.$ Draw a graph with stock prices ranging from $ $20$ to $ $60$ depicting the profit on the bull spread after $1$ day, $3$ mopahs, and $6$ months$\mathrm{.}12\mathrm{.}15$ Consider a bull spread where you buy a $40-$strike put and sell a $45-$strike put. Suppose

$=0\mathrm{.}30,r=0\mathrm{.}08,=0,$ and $T=0\mathrm{.}5.$

a$.$ Suppose $S=$$$40.$ What are delta, gamma, vega, theta, and rho?

b$.$ Suppose $S=$$$45.$ What are delta, gamma, vega, theta, and rho?

c$.$ Are any of your answers to $($a$)$ and $($b$)$ different? If so$,$ why?

d$.$ Are any of your answers different in this problem from those in Problem

$12\mathrm{.}14?$ If so$,$ why?$13\mathrm{.}1$ Suppose you sell a $45-$strike call with $91$ days to expiration. What is delta? If the

option is on $100$ shares, what investment is required for a delta$-$hedged portfolio?

What is your overnight profit if the stock tomorrow is $$39?$ What if the stock price

is $$40\mathrm{.}50?13\mathrm{.}9$ Consider a $40-$strike call with $91$ days to expiration. Graph the results from the

following calculations.

a$.$ Compute the actual price with $90$ days to expiration at $$1$ intervals from $$30$

to $$50.$

b$.$ Compute the estimated price with $90$ days to expiration using a delta approx$-$

imation.

c$.$ Compute the estimated price with $90$ days to expiration using a delta$-$gamma

approximation.

d$.$ Compute the estimated price with $90$ days to expiration using a delta$-$gamma$-$

theta approximation.