You are a owner of a call option and have the option to buy a share at fixed 'striking price' p. You have up to N days to exercise it. If you sell it on day k, you can immediately make a profit Ik - p. If Destination do Figure 1: Shortest path problem you do not exercise it at the last day, you make a profit 0. Suppose the price of the share obeys the equation Tk+1 = 3 + w where u are i.i.d. random variables for which wxl Soo. The aim is to exercise the option optimally. (1) Let Ji (2) be the maximal expected profit when the share price is r and there are N - k days to go. Write down the Bellman Equation. (2)Show that Ji () J+1, and Ji(2) -r is non-increasing in r. (3)Find the optimal policy which can be characterised as follows: There exists a sequence {uk} such that an optimal policy is to exercise the option the first time that r uk, where r is the current price and there remaining N - k days to go to make decision. You are a owner of a call option and have the option to buy a share at fixed 'striking price' p. You have up to N days to exercise it. If you sell it on day k, you can immediately make a profit Ik - p. If Destination do Figure 1: Shortest path problem you do not exercise it at the last day, you make a profit 0. Suppose the price of the share obeys the equation Tk+1 = 3 + w where u are i.i.d. random variables for which wxl Soo. The aim is to exercise the option optimally. (1) Let Ji (2) be the maximal expected profit when the share price is r and there are N - k days to go. Write down the Bellman Equation. (2)Show that Ji () J+1, and Ji(2) -r is non-increasing in r. (3)Find the optimal policy which can be characterised as follows: There exists a sequence {uk} such that an optimal policy is to exercise the option the first time that r uk, where r is the current price and there remaining N - k days to go to make decision