MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 5
Background: Section 2.2 – Single-Period Market Models.
Exercise 1 Consider the market modelM = (B, S) introduced in Exercise
3 (Week 4). We thus have k = 3, r = 1
9
, S0 = 5 and the stock prices at time
1 are given by the following table
ω1 ω2 ω3
S1
60
9
40
9
30
9
Are there any values for K such that the call option (S1 − K)+ represents
an attainable contingent claim?
Exercise 2 Consider the stochastic volatility modelM = (B, S) introduced
in Example 2.2.3 from the course notes. Hence Ω = {ω1, ω2, ω3, ω4}, the
volatility v is the random variable on Ω given by
v(ω) =
{
h if ω = ω1, ω4,
l if ω = ω2, ω3,
where 0 < l < h < 1 and the stock price S1 satisfies: S0 > 0 and
S1(ω) =
{
(1 + v(ω))S0 if ω = ω1, ω2,
(1− v(ω))S0 if ω = ω3, ω4,
We assume, in addition, that 0 ≤ r < h. (a) Characterise the class of all attainable contingent claims in M and check whether the model M is complete. (b) Describe the class M of all risk-neutral probability measures for M. (c) (MATH3975) Describe the set of all arbitrage prices for the call option (S1 −K)+ where the strike K satisfies S0(1 + l) < K < S0(1 + h). (d) (MATH3975) Assume that r = 0. Check directly whether the call op- tion with strike K such that S0(1 + l) < K < S0(1 + h) is attainable and find the range of values of its arbitrage price. 1