2. Consider a three-step binomial model. The stock prices S(n, j) and interest rates r(n, j) are shown in the two binomial pricing trees below. S(0,0) = $30 S(1, 1) = $33.00 S(1,0) r(0,0) = 5% S(2, 2) = $34.50 = S(2, 1) = $33.00 $31.00 S(2,0) = $30.00 r(1, 1) = 3% r(1,0) = 4% S(3, 3) = $36.00 S(3, 2) = $35.00 S(3, 1) = $32.00 S(3,0) = $29.50 r(2, 2) = 4% r(2, 1) = 5% 2 r(2,0) = 6% (a) Verify that there is no arbitrage opportunity at any time. (b) Calculate the forward price and future price for forward and future contracts maturing at time N = 3 and begun at node (n, j), with n = 0, 1, 2, 3 and 0 jn. (c) Although interest rates are stochastic, you should find that G(2,j) = F(2, j) for all j = 0, 1, 2. Why is this? (d) Make the interest rate deterministic with r(0) = 0.05, r(1) = 0.03 and r(2) = 0.04 and show that the forward and future prices are now equal.

2. Consider a three-step binomial model. The stock prices S(n,j) and interest rates r(n,j) are shown in the two binotial pricing trees below. (a) Verify that there is no arbitrage opportunity at any time. (b) Calculate the forward price and future price for forward and future contracts maturing at time N=3 and begun at node (n,j), with n=0,1,2,3 and 0jn. (c) Although interest rates are stochastic, you should find that G(2,j)=F(2,j) for all j=0,1,2. Why is this? (d) Make the interest rate deterministic with r(0)=0.05,r(1)= 0.03 and r(2)=0.04 and show that the forward and future prices are now equal