MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 7
Background: Chapter 3 – Multi-Period Market Models.
Exercise 1 We consider the conditional expectation EP(X | G) where G is
generated by a finite partition (Ai)i∈I of the sample space Ω = {ω1, . . . , ωk}.
Specifically, let k = 5 and
A1 = {ω1, ω2}, A2 = {ω3}, A3 = {ω4, ω5}.
Let the probability measure P be given by
P(ω1) = P(ω2) = 0.1, P(ω3) = 0.3, P(ω4) = 0.2, P(ω5) = 0.3.
Consider the random variable X : Ω→ R given by X(ωi) = i for i = 1, . . . , 5.
(a) Find the probability distribution of the random variable X.
(b) Compute the conditional expectation EP(X | G).
(c) Find the probability distribution of the random variable Y := EP(X | G).
(d) Show that EP(X) = EP(EP(X| G)).
Exercise 2 Consider the two-period market model M = (B, S) with the
savings account B given by
B0 = 1, B1 = 1 + r, B2 = (1 + r)
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with r = 0.25 and the stock price S evolving according to the following
diagram
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S2 = 10 ω1
S1 = 7
1
5
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4
5
%%L
LL
LL
LL
LL
L
S2 = 6 ω2
S0 = 5
3
5
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2
5
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88
88
88
88
88
88
88
88
S2 = 4 ω3
S1 = 3
2
5
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3
5
%%L
LL
LL
LL
LL
L
S2 = 2 ω4
(a) Compute the probabilities of the states ω1, ω2, ω3, ω4.
(b) Compute the conditional expectation EP(S2 | F1):
(b1) using the formula
EP(S2| F1) =
m∑
i=1
1Ai
P(Ai)
∑
ω∈Ai
S2(ω)P(ω),
(b2) using directly the conditional probabilities.
(c) Compute EP(S2) directly and using the equality
EP(S2) = EP(EP(S2| F1)).
Exercise 3 (MATH3975) Consider a finite probability space (Ω,F ,P) and
an arbitrary σ-field G ⊂ F . Let X be any F -measurable random variable.
(a) Show that the conditional expectation EP(X | G) satisfies∑
ω∈G
X(ω)P(ω) =
∑
ω∈G
EP(X| G)(ω)P(ω), ∀G ∈ G.
Deduce from this equality that EP(X) = EP(EP(X| G)).
(b) Let η be a random variable such that η is G-measurable and∑
ω∈G
X(ω)P(ω) =
∑
ω∈G
η(ω)P(ω), ∀G ∈ G.
Show that η = EP(X| G).
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Exercise 4 (MATH3975) Let P and Q be two equivalent probability mea-
sures on a (finite) probability space (Ω,F). Let F be an arbitrary filtration.
For any fixed t = 0, 1, . . . , T , we denote by Lt the Radon-Nikodym density
of Q with respect to P when Q and P are restricted to the σ-field Ft.
(a) Show that EP(Ls | Ft) = Lt for every 0 ≤ t ≤ s ≤ T . You may use part
(b) in Exercise 3.
(b) Using the abstract Bayes formula, establish the following equality, for
an arbitrary Fs-measurable random variable Y and for every 0 ≤ t ≤ s
EQ(Y | Ft) = (Lt)−1 EP(Y Ls | Ft).
(c) Let M be a process such that Mt is Ft-measurable for every t. Show
that the following conditions are equivalent:
(i) EQ(Ms | Ft) = Mt for every 0 ≤ t ≤ s ≤ T ,
(ii) EP(LsMs | Ft) = LtMt for every 0 ≤ t ≤ s ≤ T .
If a process M is such that Mt is Ft-measurable for every t, then we say that
M is F-adapted. If for an F-adapted process M the equality EQ(Ms | Ft) = Mt
is satisfied for every 0 ≤ t ≤ s ≤ T , then we say that M is an F-martingale
under Q. Hence it was shown in part (c) that the following conditions are
equivalent for an F-adapted process M :
(i) the process M is an F-martingale under Q,
(ii) the process LM is an F-martingale under P.
Note also that it was shown in (a) that the Radon-Nikodym density process
L of Q with respect to P is an F-martingale under P.
Exercise 5 (MATH3975) Using the tower property of conditional expec-
tation, show that if M is an F-adapted process, then the following conditions
are equivalent:
(i) the process M is a martingale under P,
(ii) EP(Mt+1 | Ft) = Mt for every 0 ≤ t ≤ T − 1,
(iii) EP(MT | Ft) = Mt for every 0 ≤ t ≤ T .
Deduce that if X an FT -measurable random variable, then the process Mt :=
EP(X | Ft) is the unique martingale under P with the terminal value MT = X.
Exercise 6 (MATH3975) Consider the process S from Exercise 2.
(a) Show that S is not a martingale under P.
(b) Find the unique probability measure Q on (Ω,F2) such that S is a
martingale under Q.
(c) Find the Radon-Nikodym density process L of Q with respect to P and
show that L is a martingale under P.
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