Question $4(30$ points$)$

Consider a binomial tree pricing model regarding a call option on a future. The current

future price is $$50.$ In the up state, the future price becomes $$70$; in the down state, it

becomes $$30.$ The strike price of the call option is $$60.$ The gross risk$-$free return $R$ is $1\mathrm{.}2.$

a$)$ Given that you long a future, what is the payoff in the up and down states? $(5$ points$)$

b$)$ What is the payoff of the call option in the up and down states? $(5$ points$)$

c$)$ Use the future and the risk$-$free bond to construct the synthetic call option on the future.

$(5$ points$)$

d$)$ Suppose it involves a transaction cost of $$2$ to long a unit of future, and a transaction

cost of $$4$ to short a unit of future. What are the upper bound and the lower bound of the

call premium so that there is no arbitrage? $(15$ points$)$