MATHEMATICS 430 ASSIGNMENT 3
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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 answe
(b) Same question as above for Yn = M
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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
= 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.
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