27. In the context of Example 3.7, find a replicating portfolio for the put option with the payoff function g(s) = (100 s)+ = max(100 s, 0).

[Example 3.7 (Replication in a Single-Period Model) Consider a single-period model with r = 0.005, S(0) = 100, s1 = 101, and s2 = 99. The payoff Taf is trying to replicate is the European call option

g[S(1)]=[S(1)K ]+ =max[S(1)K ,0]

with K = 100. According to equation (3.44), Taf has to solve the system

0(1 + 0.005) + 1101 = 1 0(1 + 0.005) + 199 = 0

The solution is 0 = 49.2537, 1 = 0.5. So, Taf has to borrow 49.2537 from the bank (short-sell the bond or bank account) and buy half a share of the stock. We note that the

cost of the replicating strategy (the amount of necessary initial investment) is 0 B(0) + 1 S(0) = 0.746 As will be argued later, this is the fair price of the option in this model.]