Suppose the payoff of the market portfolio w˜ m has k possible values.

Denote these possible values by a1 < ··· < ak. For convenience, suppose ai − ai−1 is the same number  for each i. Suppose there is a risk-free asset with payoff equal to 1. Suppose there are k −1 call options on the market portfolio, with the exercise price of the ith option being ai. The payoff of the ith option is max(0,w˜ m − ai).

(a) Show for each i = 1,..., k −2 that a portfolio that is long one unit of option i and short one unit of option i +1 pays  if w˜ m ≥ ai+1 and 0 otherwise. (This portfolio of options is a bull spread.)

(b) Consider the following k portfolios. Show that the payoff of portfolio i is 1 when w˜ m = ai and 0 otherwise. (Thus, these are Arrow securities for the events on which w˜ m is constant.)

(i) i = 1: long one unit of the risk-free asset, short 1/ units of option 1, and long 1/ units of option 2. (This portfolio of options is a short bull spread.)

(ii) 1 < i < k − 1: long 1/ units of option i − 1, short 2/ units of option i, and long 1/ units of option i + 1. (These portfolios are butterfly spreads.)

(iii) i = k −1: long 1/ units of option k −2 and short 2/ units of option k −1.

(iv) i = k: long 1/ units of option k −1.

(c) Given any function f , define z˜ = f(w˜ m). Show that there is a portfolio of the risk-free asset and the call options with payoff equal to z˜.