The following data are market prices on a given day:

The expirations are: $41$ days for JUL; $72$ days for AUG; $163$ days for OCT.

The respective simple annual risk$-$free rates for each expiration period are: $\mathrm{.}0503,$

$\mathrm{.}0535$ and $\mathrm{.}0571.$ The annual volatility of the returns on the underlying stock is $=\mathrm{.}21.$

Use DerivaGem and calculate the Greeks of the following strategies:

Q$1.$ Short a Butterfly with the $160,165$ and $170,$ OCT puts.

Q$2.$ Short a Bear spread with $100$ CBOE of the $155,170$ AUG calls.

Q$3.$ Short a Bull spread with $100$ CBOE of the $165,170$ OCT puts.

Q$4.$ Calculate the number of shares of the underlying stock that will Delta

neutralize the strategy in:

$4\mathrm{.}1$ Q$2.$

$4\mathrm{.}2$ Q$3.$

Q$5.$ A financial institution just sold $1,000$ CBOE of the $170,$ AUG calls. Explain how to

create a Delta$-$neutral position with these short calls and shares of the underlying

asset.

Q$6.$ You wish to create a Delta$-$neutral position with two different puts but without any

shares of the underlying assets. The market premiums of these puts are: $p_1$ and $p_2.$

The value of this portfolio is: $,V=(p_1)n(p_1,s)+(p_2)n(p_2,s).$

Derive the formula for the relationship between the two calls so as to create a Delta$-$neutral

position. Use the same method that I used on slide $31CH.17$ but with only the two calls:

no stock shares.

Q$7.$ A financial institution just sold $1,000$ CBOE of the $170,$ AUG puts. Use the formula

you derived in Q$6$ to explain what position will be taken in the OCT $165$ put to create a

Delta$-$neutral portfolio, using no shares of the underlying asset.

Q$8.$ Consider the following data:

Calculate the Delta, Gamma and Vega of the following portfolio:

Portfolio $=\{$short $6$ of calls $1$; short $15$ of puts $1$; long $15$ of calls $2$; short $20$ of calls $3\}$