- (40) Consider a special type of lookback option whose payoff at maturity is given by the difference between the maximum and minimum stock prices attained over the life of the option.

LT = max St − min St t∈[0,T ] t∈[0,T ]

Consider pricing this option using the binomial model. Let the current price of stock in Hindsight Inc. be S0 = 216 and consider an option maturing in n = T = 3 periods, so that ∆t = 1 per period. Suppose for the sake of simplicity that the risk free rate is r = 0, and that each period the stock price either doubles u = 2, or fallsbyhalfd=1 =1.

u2

(a) (5) Calculate the risk neutral probability of an uptick p.

- (b) (5) An important property of lookback options not shared by vanilla calls and puts is that they are path dependent. That is, their payoffs depend not only on the final stock price but on how the price moved over the life of the option. Show that this option is path dependent, that is, find two price paths that end at the same price.
- (c) (15) Draw the stock price tree. Note that there are 2n = 8 possible price paths, in contrast to the n+1 = 4 different final prices.
- (d) (15) Using backward induction, calculate the initial price of this option.

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