MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 8
Background: Chapter 3 – Multi-Period Market Models.
Exercise 1 We consider the two-period market modelM = (B, S) with the
savings account Bt = (1+r)
t where the interest rate r = 0.1. The stock price
process S is represented under P by the following diagram
S2 = 10 ω1
S1 = 7
3
5
99rrrrrrrrrr
2
5
%%L
LL
LL
LL
LL
L
S2 = 6 ω2
S0 = 5
2
5
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3
5
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88
88
88
88
88
88
88
88
S2 = 4 ω3
S1 = 3
2
5
99rrrrrrrrrr
3
5
%%L
LL
LL
LL
LL
L
S2 = 2 ω4
(a) Find the risk-neutral probability measure Q for the modelM = (B, S).
(b) Find the replicating strategy for the digital call option with strike
K = 8 and maturity T = 2, that is, for the payoff X given by
X = h(S2) =
{
1, if S2 ≥ 8,
0, otherwise.
Find the arbitrage price process πt(X) for t = 0, 1, 2.
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(c) Compute the arbitrage price process for the Asian option with the
payoff at maturity T = 2 given by the following formula
Y =
(
1
3
(
S0 + S1 + S2
)
− 4
)+
.
Exercise 2 Consider the CRR model with T = 2 and S0 = 80, S
u
1 =
104, Sd1 = 88. Assume that the interest rate r = 0.2. Consider a European
contingent claim X maturing at T = 2 with the payoff given by the formula
X =
(
S2 − S1
)
1{S2−S1>20} =
{
S2 − S1, on the event {S2 − S1 > 20},
0, on the event {S2 − S1 ≤ 20}.
(a) Show explicitly that the contingent claim X is path-dependent.
(b) Find the risk-neutral probability measure P̃ for the modelM = (B, S)
and compute the arbitrage price of X using the risk-neutral valuation
formula
πt(X) = Bt EP̃
(
XB−1T | Ft
)
, t = 0, 1, 2.
(c) Find the replicating portfolio (φ0, φ1) for the claim X and check that
the equality Vt(φ) = πt(X) is satisfied for t = 0, 1, 2.
(d) Show that in any CRR model we have that EP̃(S2 − S1) = r(1 + r)S0.
Let Y =
(
S2 − S1
)
1{S2−S1≤20}. Find the price of Y at time 0 using
the additivity of arbitrage prices and the fact that X + Y = S2 − S1.
Confirm your result by computing
π0(Y ) = B0 EP̃
(
Y (B2)
−1).
(e) Find the unique probability measure P̂ on (Ω,F2) such that the process
B̂t := Bt/St, t = 0, 1, 2 is a martingale under P̂ with respect to the
filtration F = (Ft)t=0,1,2 and check that π0(Y ) = S0 EP̂
(
Y (S2)
−1
)
.
Exercise 3 (MATH3975) We consider a discrete-time stochastic process
X = (Xt, t = 0, 1, . . . ) defined on a finite probability space (Ω,F ,P) endowed
with a filtration F = (Ft)t≥0. It is assumed throughout that a process X is
adapted to the filtration F, that is, X is F-adapted.
(a) Assume that X has independent increments with respect to F, meaning
that for any t = 0, 1, . . . the increment Xt+1−Xt is independent of the
σ-field Ft. Show that the process Y , which is given by the following
expression
Yt := Xt − EP(Xt), t = 0, 1, . . . ,
is a martingale under P with respect to the filtration F.
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(b) Let A0 = 0 and for t = 0, 1, . . .
At+1 − At = EP(Xt+1 −Xt | Ft). (1)
(b1) Verify that the process Ỹ given by the equality Ỹt := Xt − At for
t = 0, 1, . . . is a martingale under P.
(b2) We assume that the process Ŷt := Xt − Ât for t = 0, 1, . . . is a
martingale under P where the process  satisfies: Â0 = 0 and
Ât+1 is Ft-measurable for every t = 0, 1, . . . (we then say that the
process  is F-predictable). Show that  = A where the process
A is given by formula (1) with A0 = 0.
Comment: In parts (b1)-(b2) we have shown that if a process X is
F-adapted, then there exists a unique F-predictable process A with
A0 = 0 such that the process Ỹ = X − A is a martingale under P.
(c) Assume that a process X = (Xt, t = 0, 1, . . . , T ) represents a gamble,
meaning here that if the game is played at time t then the (positive or
negative) reward at time t+1 per one unit of the bet equals Xt+1−Xt.
The random size of the bet is given by an arbitrary F-adapted process
H called a gambling strategy. The profits/losses after t rounds of the
game when a gambling strategy H is followed are given by the following
equality (by convention, G0 = 0)
Gt :=
t−1∑
u=0
Hu(Xu+1 −Xu).
Note that one does not pay any fee for the right to play the game X. By
definition, we then say that the game X is fair if there is no gambling
strategy H such that EP(Gt) 6= 0 for some t ≤ T .
(c1) Show that the game is fair if and only if X is a martingale under
P with respect to the filtration F.
(c2) Consider an arbitrary F-adapted process X. Argue that the cor-
responding game will become a fair game if the player is required
to pay at time t the fee At+1 − At per one unit of the bet where
A is the unique F-predictable process with A0 = 0 that satisfies
equality (1).
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