Price a two-step American put by hand. Spot 100, strike 100, each step the stock goes up 20% or down 20%, and the gross interest per step is 1.05. Give the American value, the European value and the early exercise premium.

Price a two-step American put by hand. Spot 100, strike 100, each step the stock goes up 20% or down 20%, and the gross interest per step is 1.05. Give the American value, the European value and the early exercise premium.

Approach: Build the risk-neutral probability from the up and down factors, roll the tree back from expiry, and at each interior node compare continuation against immediate exercise.

7.99. On this binomial tree the risk-neutral probability is p = (1.05 - 0.8)/(1.2 - 0.8) = 0.625. Terminal nodes are 144, 96 and 64, giving put payoffs of 0, 4 and 36. At the up node the continuation value is (0.625*0 + 0.375*4)/1.05 = 1.43 and intrinsic is zero, so it holds. At the down node with spot 80 the continuation value is (0.625*4 + 0.375*36)/1.05 = 15.24 while intrinsic is 20.00, so backward induction exercises there and stamps 20.00. Rolling to the root gives (0.625*1.43 + 0.375*20.00)/1.05 = 7.99, against a European value of 6.29 when the down node keeps 15.24. The early exercise premium is 1.70, and it exists only because holding the put forgoes interest on the 100 of strike proceeds.

Follow-up: At what gross interest rate per step does the early exercise at the down node stop being optimal on this tree?

Key concepts: risk-neutral probability, backward induction, early exercise premium, binomial tree.