An investor wants to manage a portfolio which consists of two assets, one is the riskless asset and the other is a certain stock. Let S⁰ be the price process for the riskless asset and let S¹ be the price process of the stock. Assume that there are N periods, and the interest rate per period is r. The evolution of the stock price follows a Binomial model: At the end of each period, it either goes up by a factor of (1+u ) with probability p or down by a factor of 1 + d with probability 1 - p. At the end of each period, he adjusts his portfolio so that his wealth is equally shared between the two assets.

a) Formulate his trading strategy as a predictable process (H_n )_n0, assuming that the initial wealth of the investor is V₀. (Hint: You can formulate H_n recursively; that is, start with defining H₀ and for general n, write each H_n as a function of H_n-1 and S_n-1 .)

b) Is this trading strategy self-financing?

c) What is the expected wealth of the investor at time N ?