A portfolio manager is interested in purchasing an instrument with a call option$-$like payoff but does not want to have to pay money up front. The manager learns from a banker that one can do this by entering into a break forward contract. The manager wants to learn if the banker is quoting a fair price. The stock price is $437\mathrm{.}55.$ The contract expires in $270$ days. The volatility is $18$ percent, and the continuously compounded risk$-$free rate is $3\mathrm{.}75$ percent. The exercise price will be set at the forward price of the stock.

a$.$ Determine the exercise price.

b$.$ The loan implicit in the break forward contract will have a face value of $40\mathrm{.}19.$ Determine if this is a fair amount by using your answer in part a and computing the value of K$.$

c$.$ Regardless of whether the break forward is found to be fairly priced, determine the value of the position if the stock price ends up at $465$ and at $425.$