$6.$ A stock is currently trading at $55.$ You hold a portfolio of the following instruments:

$$ Long $200$ shares of stock.

$$ Long $200$ puts with a strike of $50$ and maturity of three months.

$$ Short $200$ calls with a strike of $60$ and maturity of three months.

You are given the following information:

Instrument Price Delta Gamma Vega Theta Rho

Call with K $=506\mathrm{.}3210\mathrm{.}8230\mathrm{.}0387\mathrm{.}1525\mathrm{.}5229\mathrm{.}730$

Put with K $=500\mathrm{.}7000\mathrm{.}1770\mathrm{.}0387\mathrm{.}1523\mathrm{.}0532\mathrm{.}615$

Call with K $=553\mathrm{.}0790\mathrm{.}5650\mathrm{.}05710\mathrm{.}8276\mathrm{.}8126\mathrm{.}993$

Put with K $=552\mathrm{.}3960\mathrm{.}4350\mathrm{.}05710\mathrm{.}8274\mathrm{.}0966\mathrm{.}586$

Call with K $=601\mathrm{.}2100\mathrm{.}2970\mathrm{.}0509\mathrm{.}5155\mathrm{.}5133\mathrm{.}779$

Put with K $=605\mathrm{.}4650\mathrm{.}7030\mathrm{.}0509\mathrm{.}5152\mathrm{.}55111\mathrm{.}035$

$($a$)$ What is the current value of your portfolio?

$($b$)$ What is the delta of your portfolio? the gamma? the vega? the theta? the rho?

$($c$)$ Suppose you want to make your portfolio gamma neutral. What is the cost of

achieving this using the $55-$strike call? What is the theta of your new position?

$($d$)$ What is the cost if you used the $55-$strike put? What is the theta of the new position?

$27.$ Using the same information as in Question $26,$ calculate the following quantities:

$($a$)$ The delta and gamma of a covered call portfolio with K $=55($i$.$e$.,$ a portfolio

where you are long the stock and short a call with a strike of $55).$

$($b$)$ The delta and gamma of a protective put portfolio with K $=50($long the stock

and long a put with a strike of $50).$

$($c$)$ The delta and gamma of a bull spread using calls with strikes of $55$ and $60($long a

$55-$strike call, short a $60-$strike call$).$

$($d$)$ The delta and gamma of a butterfly spread using calls with strikes of $50,55,$ and

$60($long a $50-$strike call, long a $60-$strike call, and short two $55-$strike calls$).$

$($e$)$ The delta and gamma of a collar with strikes $50$ and $60($long position in the stock,

long a $50-$strike put, short a $60-$strike call$)$