Course: Stochastic Processes II - M375T Instructor: Gerard Brunick
Problem Set 4
We will go over this problem set on Monday 4/6.
We are working in the coin-flip setup with Mn,Pn
i=1
1
Hi−
1
Tifor all of the following
problems.
1. In this problem we look at what happens when you exponentiation a martingale.
(a) Set Xn=eMn. Is (Xn)n∈N0a martingale, submartingale, or supermartingale?
Hint: write Xn=e∆MnXn−1and use the convexity of the exponential function.
(b) Let Nn=eMn−nC for some constant C. Choose Cso that (Nn)n∈N0is a martingale.
Hint: write Nn=e∆Mne−CNn−1.
2. Let Nn=M2
n−nC for some constant C. Choose Cso that (Nn)n∈N0is a martingale.
Hint: write M2
n= (∆Mn+Mn−1)2and expand to get Nnwritten in the form Nn=
? + Nn−1.
3. Set Θ0= 0, Θn=Mn−1for n∈N, and In=Pn
i=1 Θn∆Mn.
(a) Write Inas a function of Mnand n. Hint: notice that Mn−1(∆Mn) = 1
2M2
n−
M2
n−1−(∆Mn)2, and the expression on the right is essentially a telescoping sum.
(b) Is (Θn)n∈N0adapted? Is it predictable? Is Ina martingale?
4. Let X= 2 P3
i=1
1
Hidenote twice the number of heads that occur in the first three flips
(so Xis a random variable not a process).
(a) Set Nn=E[X|Fn]. Compute the process (Nn)n∈N0explicitly and check that it is
a martingale. Hint: it may help to consider the cases n≤3 and n > 3 separately.
For the n≤3 case it may help to draw a tree of all possibilities.
(b) Find a process (Θn)n∈Nsuch that Nn=N0+Pn
i=1 Θi∆Mi. You may help to look
at the tree from the previous part and try to use backward induction.
(c) Suppose that in addition to the regular bets on heads and tails, you can also
buy a “magic double triple header ticket” for some cost $C, where Cis a non-
random constant. When you buy such a ticket, you get $2 for each heads that
appears in the next three flips. If C < 3, explain how you can take advantage
of this mispricing to make money with no risk. Such an opportunity is called an
arbitrage.
Hint: The “hedging strategy” that you computed in the previous part of this
problem gives you one way to produce a payoff equal to X, and the buying the
ticket gives you another. How can you play these two possibilities off against one
another?
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