Problem 2: We consider a binomial model for the interest rates with r₀ := 16% and

r_n+1 = r_nn+1, n = 0, 1, ..., N 1,

where the iid random variables _n take values u := 2 and d := 0.5 with the respective single period risk-neutral probabilities p := 0.2 and q := 0.8.

We consider a futures contract on the interest rate with maturity N := 4 and the delivery price

F_N := 1 r_N ,

where r_N is the single period interest rate determined at maturity. Recall that it costs nothing to enter both long or short positions in futures, and that if we buy (long position) futures at time m, then the payments take place at times m + 1 k N and are given by

F_k F_k1, k = m + 1, ..., N.

Here F_k is the futures price at time k.

We consider the European call option which expires at M := 1 where it pays the cash amount

V_M := max{F_M K, 0}.

Here F_M is the futures price at time M and the strike is K := 0.92. Com- pute the arbitrage-free price V₀ and the number of futures contracts ₀ in

the replicating strategy for this option at time n = 0.