Suppose that you have a stock in the 3-period binomial model with S0= 4, u= 2, d= 1/2, and r= 0. 1. You have a cash flow process that pays out 0.1·Sk on period k for each k, and which pays out an additional amount at the end of period 3 equal to the maximum value that the stock price has taken over periods 0 through 3. Work out the tree for the process V0,...,V3 for the value of the cash flow. Prove directly that the process (V0ν0,...,V3ν3) is a supermartingale and not a martingale. Do not work out the replicating portfolio.
Hint 1: For each k, the value Vk should be the amount needed to make payment Ck, plus the risk-neutral expectation of all future payments, discounted back to day k .
Hint 2: To simplify computations, you can try breaking up this security into two pieces: the piece that makes the payments each period and the piece that makes the lump-sum payment at the end. The value of the security will, in all states, be the sum of the values of these two securities.