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 you do not exercise it at the last day, you make a profit 0. Suppose the price of the share obeys the equation 2+1 = 3 + wk, where wk are i.i.d. random variables for which Ewl so. 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() > Ji+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 > 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 you do not exercise it at the last day, you make a profit 0. Suppose the price of the share obeys the equation 2+1 = 3 + wk, where wk are i.i.d. random variables for which Ewl so. 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() > Ji+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 > Uk, where r is the current price and there remaining N-k days to go to make decision