MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 6
Background: Section 2.2 – Single-Period Market Models.
Exercise 1 Consider a single-period three-state market model M = (B, S)
with the dates 0 and T = 1. We assume that there are two assets: the savings
account B with the initial value B0 = 1 and a risky stock with the initial
price S0 = 4. The risk-free simple interest rate r equals 10%. Assume that the
stock price S1 satisfies (S1(ω1), S1(ω2), S1(ω3)) = (8, 5, 3) and the real-world
probability P satisfies P(ω1) = 0.3, P(ω2) = 0.3, P(ω3) = 0.4.
(a) Show directly that the model M = (B, S) is arbitrage free, that is, no
arbitrage opportunities exist in this model. Do not use here the FTAP
(Theorem 2.2.1), but refer instead to Definition 2.2.3 in Course Notes.
(b) Consider the call option with the expiry date T = 1 and strike price
K = 4. Examine the existence of a replicating strategy for this option.
(c) Find explicitly the class of all attainable contingent claims.
(d) Find the class M of all martingale measures Q = (q1, q2, q3) on the
space Ω = (ω1, ω2, ω3) for the model M.
(e) Find all expected values
EQ
(
(S1 − 4)+
1 + r
)
where Q ranges over the class M of all risk-neutral probability measures
for the model M.
(f) (MATH3975) Find the superhedging price for X, that is, the minimal
initial endowment x for which there exists a portfolio (x, φ) such that
the inequality
V1(x, φ)(ω) ≥ (S1(ω)− 4)+
holds for every ω ∈ Ω.
1
Exercise 2 Consider a single-period market model M = (B, S) on the
sample space Ω = {ω1, ω2, ω3}. Assume that the savings account equals
B0 = 1, B1 = 1.1 and the stock price equals S0 = 5 and
S1 = (S1(ω1), S1(ω2), S1(ω3)) = (7.7, 5.5, 4.4).
The real-world probability P is such that P(ωi) > 0 for i = 1, 2, 3.
(a) Find the class M of all martingale measures for the model M. Is this
market model complete?
(b) Show that the claim X = (X(ω1), X(ω2), X(ω3)) = (5.5, 3.3, 2.2) is
attainable and compute its arbitrage price π0(X) using the replicating
strategy for X.
(c) Consider the contingent claim Y = (3, 1, 0). Show that the expected
value
EQ
(
Y
B1
)
does not depend on the choice of a martingale measure Q ∈M. Is this
claim attainable?
(d) Consider the contingent claim Z = (4.4, 0,−3.3). Find the range of
arbitrage prices
π0(Z) = EQ
(
Z
B1
)
where Q ∈M. Is this claim attainable?
(e) Find the unique martingale measure Q̃ for the extended model M̃ =
(B, S1, S2) in which S1 = S and the risky asset S2 is defined as the
claim Z traded at its initial price π0(Z) = −0.5, that is, S20 = −0.5
and S21 = Z. Is the market model M̃ complete?
Exercise 3 (MATH3975) Let Ω = {ω1, ω2}. We consider a single-period
model M = (S1, S2) with two risky assets with prices S1 and S2 given by
S10 = s0 > 0, S
2
0 = z0 > 0 and
S11(ωi) = si, S
2
1(ωi) = zi
for i = 1, 2 where 0 < s1 < s2 and 0 < z1 < z2. There are two traded assets, S1 and S2, so the wealth of a strategy φ equals Vt(φ) = φ 1 tS 1 t + φ 2 tS 2 t for t = 0, 1. It should be stressed that the existence of the savings account B is not postulated. Hence the process B should not be used at all in your solution. 2 (a) Under which assumptions on the (relative) values of s0, s1, s2, z0, z1 and z2 the modelM = (S1, S2) is arbitrage-free? To answer this question in terms of some inequalities satisfied by s0, s1, s2, z0, z1 and z2, examine the relative wealth V̂ (φ) = V (φ) S2 . (b) Assume that s0, s1, s2, z0, z1 and z2 are such that the model M = (S1, S2) is arbitrage-free. Check whether the model M = (S1, S2) is complete. (c) Assume that s0, s1, s2, z0, z1 and z2 are such that the model M = (S1, S2) is arbitrage-free. Find the price and the replicating strategy for the contingent claim X = (S11 − S21)+ with maturity date T = 1. Was it necessary to assume here that the model is complete? (d) Find the price of the contingent claim with the payoff Y = (S21 − S11)+ using part (c) and a suitable version of the put-call parity relationship. 3