A stock currently costs $4 per share. In each time period, the value of the stock will either increase or decrease by 50%, and the risk-free interest rate is 10%. Let S0, S1, and S2 be the prices of the stock at times 0, 1, and 2, and suppose we are selling a European-style call option security expiring at time 2, with a strike price of sqrt(S1S2). That is, the value of the option at time 2 is (S2 - sqrt(S1S2))+. (This can be compared with the Asian option described in Exercise 1.8 in the text; the dierences are that here the strike price is also a random variable, and the average is geometric rather than arithmetic.)
(a) Find the risk-neutral price of this option at time 0.
(b) Describe the replication process for this option in the case that the rst movement of the stock is upwards (that is, assuming the rst coin-ip lands on Heads). That is, at times 0 and 1, nd the value of a portfolio which replicates this security, compute the number of shares of stock we buy or short-sell, and nd the amount of money we invest or borrow at the risk-free rate. Show that in each case, the portfolio that replicates the security.