Consider a 2-period binomial tree model for a stock with spot price S0, volatility and risk-free interest rate r. Denote the time step of the tree .

i) Construct the tree of forward prices F(t,T) = S0er(Tt) for a forward contract on the stock, with maturity T = 2.

ii) Show that at each node in the tree (times t = 0, ) the forward prices satisfy the martingale condition

(1) F(t,T)=EQ[F(t+,T)]=pF(t+,T;up)+(1p)F(t+;down) er d with the usual tree probability p = ud .

This illustrates the martingale property of the forward price: the forward price at time t1 is equal to the expectation in the risk-neutral measure of the forward price at any later time t2 > t1 (2) F(t1;T) = EQ[F(t2;T)] This martingale property holds also for futures contracts.