MATHEMATICS 430 ASSIGNMENT 3 Pleasse submit through Mycourses by March 21, 11:59 pm as a single pdf document. Only one submission is allow Instructions: Show all work and justify answers (even where...

Please help me with the following questions


MATHEMATICS 430 ASSIGNMENT 3 Pleasse submit through Mycourses by March 21, 11:59 pm as a single pdf document. Only one submission is allow Instructions: Show all work and justify answers (even where not explicitly requested). Marks may be deducted for lack of neatness (print if necessary). The assignment mark may be based on a randomly selected problem or problems instead of the whole assign- ment. Therefore be sure to solve each problem. In BLOCK CAPITALS, your LAST NAME , and your ID number, in the top right corner. 1. Let Mn be a martingale. (a) Let Xn = (Mn)+ = max(Mn, 0). Is Xn a martingale? a super-martingale? Justify your answer (b) Same question as above for Yn = M 3 n. 2. (Doob’s decomposition ) Let Xn be a super-martingale with respect to a filtration Pn. Define by induction M0 = X0 and Mn+1 = Mn + Xn+1 − E(Xn+1|Pn) for n = 1, 2, .... (a) show that Mn is a martingale. (b) Set An = Mn−Xn. Show that An is a nondecreasing and predictable process. ( Note; An is called the compensator process. The decomposition Xn = Mn − An where M0 = X0, Mn is a martingale and An is nondecreasing and predictable is called the Doob decomposition.) We will show below that the Doob decomposition is unique. (c) Show that a predictable martingale is a constant process ie if (Mn) is a pre- dictable martingale, then there exists a constant c such that Mn = C for every n. (d) deduce that the Doob’s decomposition of a super-martingale is unique. 3. (Chooser option)The setting in this problem is that of a N -step binomial model. Let m be an integer such that 1 ≤ m ≤ N − 1. A chooser option is an option which confers on its owner a right to receive either a(European) call or put at time m. The put or call expires at time N with strike K. The owner of the option may wait until time m to choose. Find the time-zero price of the chooser option. (hint: use put-call parity). 4. Consider a 5-steps binomial model where S0 = 100, u = 1.1, d = 0.9, r = 0. (a) Find the price process of an American option with payoff Min(30, (Sn − 88)+). (b) Find the compensator process ( refer to Problem 2 part c ). (c) Find an optimal stopping time. (d) Should the buyer naively decide to wait until T = 5 to exercise the option, find the profit accumulated by the seller. 5. Bermudan Option Setting: 4-Steps binomial option where S0 = 100, u = 1.15, d = 0.9, e r = 1, 05. Consider a Bermudan put option with strike K = 100 where the allowed exercise times are 0, 2 or 4. Find the price process of such option and an optimal exercise time. 3