You are analyzing a stock trading at $\backslash (\backslash$$ $50\backslash )$ with a volatility of $\backslash (30\backslash \%\backslash ).$ The risk$-$free rate is $\backslash (4\backslash \%\backslash )$ per year, continuously compounded, and the stock pays no dividends. Consider European call and put options with a strike price of $\backslash (\backslash$$ $52\backslash )$ and $3$ months to expiration $(\backslash ($ T$=0\mathrm{.}25\backslash )$ years$).$ a$)(10$ points$)$ Construct a straddle strategy using the call and put options. Using the Black$-$Scholes$-$Merton model, calculate the cost of setting up the straddle. Calculate the payoffs at expiration and total profits for final stock prices of $\backslash (\backslash$$ $40\backslash ),\backslash (\backslash$$ $52\backslash ),$ and $\backslash (\backslash$$ $60\backslash ).$ b$)(8$ points$)$ Calculate the delta of the call and put options individually using the Black$-$ScholesMerton model. Then, compute the delta of the straddle portfolio. Explain why the straddle's delta is what it is$,$ referencing the properties of options. c$)(7$ points$)$ Suppose you want to make the straddle delta$-$neutral by trading the underlying stock. How many shares of the stock should you buy or sell, and why? d$)(5$ points$)$ Instead of a straddle, you consider a futures contract on the same stock with a delivery price of $\backslash (\backslash$$ $52\backslash ),$ maturing in $3$ months. Briefly explain how the futures price relates to the spot price and why a futures$-$based strategy might be less suitable than the straddle for betting on large price movements.