9. (Equity swap o) Mr. A. Gaylord manages a pension fund and believes that his stock selection ability is excellent However, he is worried because the market could go down He considers entering an equity swap where each quarter i, up to quarter M, he pays counterparty B the previous quarter's total rate of return r; on the S&P 500 index times some notional principal and receives payments at a fixed rate on the same principal. The total rate of return includes dividends Specifically, += (S,+ d)/S-1. where S, and

d, are the values of the index at i and the dividends received from i-1 to i, respectively Derive the value of such a swap by the following steps: -

(a) Let V(Sd) denote the value at time of receiving S, +d, at time i Argue that V(Sd) = S and find Vi()

(b) Find Va(r)

(c) Find Vo(1)

(d) Find the value of the swap.