I need a help with this question. I was trying to figure out how to do it but it is about Lookback Option, an advanced topics.

There are a lot of options more complicated than the cail or put options we have covered in class. One example is the cookbook 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 ofthe 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 ofthe stock is 5100. 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 ofthe stock price: t=0 t=1 L12 mo 125/ 151.125 \mo \m /.... \m Zind the value of this option. (Hint: Notice that the payoffs of the option on the 4 nodes att = 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 $190.} 5 l 20.2233" l X