There are a lot of options more complicated than the call or put options we have covered in class. One example is the Lookback Option, which has a payoff given by the maximum or minimum value that the underlying asset reaches in a given period (from now to the maturity of the option).

Consider a European lookback option to sell a stock 2 years from now for the maximum price of the stock during the coming 2 years. This option allows one to time the market perfectly and get the most from selling the stock. Suppose that the current price of the stock is $100. It does not pay dividends, and each year it can go up by 25% or down by 20%. The annual risk free rate is 5%. The figure below shows the evolution of the stock price:

Find the value of this option.

(Hint: Notice that the payoffs of the option on the 4 nodes at =2 depend on how the nodes are reached, that is, they are "path-dependent". That's why we have to treat the two nodes in the middle differently, even though they both have stock price of $100 .)

t=0 t=1 t=2 156.25 125 100 100 100 80 64 t=0 t=1 t=2 156.25 125 100 100 100 80 64