Question #$1$Consider a single$-$period binomial model $($crudely$)$ representing the one$-$day evolution of the futures price of gold, for some given delivery date that is fixed throughout. The futures price of gold today is $$1,000.$ At the end of the single period, the futures price of gold can be either $$900$ or $$1,100.($The numbers are of course silly, but they serve to simplify the calculations.$)$ Suppose you go long one futures contract today and you close out the position tomorrow. What is the value of the futures today? What is your cash flow in the up and down scenarios? What does that imply for the risk$-$neutral probability p$*$ of an up move of the futures price?

Solution: Value of the future contract today $=1000$

Up cash flow $$100$

Down cash flow $-$$$100$

Risk$-$neutral probability p$*=0\mathrm{.}5$

Continuing in the context of the question above, use p$*$ and an assumed zero risk$-$free rate to price an at$-$the$-$money European call on the futures that expires at the end of the period. By definition, upon exercise, the call pays the difference between the futures price and the strike. Finally, explain how you can use the futures contract and risk$-$free borrowing or lending to replicate a position of two at$-$the$-$money European calls. $($You should use the p$*$ you computed in the last question, but you will not be penalized if you use the wrong p$*$ in this question, provided of course your p$*$ does not imply an arbitrage.$)$ Create table for the answer