BSM model Greeks.

It is Jun $13,2022$ and you consider trading options on a security priced at $$41$ with standard deviation on the continuously compounded returns of $10\%.$ The options are European with a strike price of $$42$ and expiration date Dec $7,2022.$ The continuously compounded risk$-$free rate associated with the expiration is $5\%.$ Assume that the security is a non$-$dividend paying stock and the Black$-$Scholes model is appropriate for pricing.

Notes:

The cumulative standard normal table is available on Moodle.

Use full accuracy in all the calculations except for finding $N(d_{1,2}).$ These values are supposed to come from the table, which has $2$ decimals accuracy. Thus, if you are using Excel you must round the arguments of $N(d_{1,2})$ to $2$ decimals.

Wherever needed, use $=3\mathrm{.}14$

a$.$ What is the value of the call option?

$

Round your answer to two decimal places.

b$.$ What is the value of the put option?

$

Round your answer to two decimal places.

c$.$ What is the number of call option contracts necessary to delta$-$neutral hedge $4,000$ shares of the stock $($with a sign!$)?$ contracts

Round your answer to the nearest integer.

d$.$ What is the put gamma for a stock price movement from $$41$ to $$40\mathrm{.}5?$

Round your answer to two decimal places.

e$.$ What is the value of the call scaled vega for a given shift in volatility from $10\%$ to $10\mathrm{.}8\%?$