Consider the following portfolio: Buy one call option contract for $100$ shares on a non$-$dividend paying stock with a strike price of $$52$ with maturity in $3-$months. Short sell $200$ shares of the same stock. Enter into a forward contract to buy $100$ shares in $6$ months. The current stock price is $$50.$ The risk$-$free interest rate is $10\%$ per annum and the stock price volatility is $25\%$ per annum.

$($a$)$ Calculate the value of the portfolio at $T=0.$

$($b$)$ Calculate the payoff and profits of the portfolio at $T=1,3$ months respectively as a function of the spot stock price. Draw the payoff diagrams for each of the times.

$($c$)$ Compute all the Greeks $($Delta$,$ Gamma, Theta, Vega, Rho$)$ for the portfolio at $T=0.$

Consider a non$-$dividend$-$paying stock with a current price of $15.$ The market prices for European call options on this stock with different strike prices and the same maturity of $3$ months are given below. The risk$-$free interest rate is $5\%$ per annum.

$\backslash$table$[[$Strike Price $(K),$Market Price of Call Option $(C)$