Question: The price of a security in each time period is its price in the previous time period multiplied either by u = 1.25 or by d = .8. The initial price of the security is 100. Consider the following "exotic" European call option that expires after five periods and has a strike price of 100. What makes this option exotic is that it becomes alive only if the price after two periods is strictly less than 100. That is, it becomes alive only if the price decreases in the first two periods. The final payoff of this option is payoff at time 5 = I(S(5) -100)+, where I = 1 if S(2) < 100 and I = 0 if S(2) = 100. Suppose the interest rate per period is r = .1.
(a) What is the no-arbitrage cost (at time 0) of this option?
(b) Is the cost of part (a) unique? Briefly explain.
(c) If each price change is equally likely to be an up or a down movement, what is the expected amount that an option holder receives at the time of expiration?