MATH3075/3975
Financial Derivatives

（a）直接证明模型M =（B，S）是无套利的，即没有

（定理2.2.1），但请参考课程说明中的定义2.2.3。
（b）考虑到期日T = 1和行使价的看涨期权
K =4。检查此选项是否存在复制策略。
（c）明确找到所有可获得的或有债权的类别。
（d）找出所有measures度量Q =（q1，q2，q3）的M类

（e）查找所有期望值

</ s> </ s> </ s>
（S1-4-）
1 + r
</ s> </ s> </ s>

（f）（MATH3975）查找X的对冲价格，即最小

V1（x，φ）（ω）≥（S1（ω）− 4）+

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 Qe for the extended model Mf =
(B, S1
, S2
) in which S
1 = S and the risky asset S
2
is defined as the
claim Z traded at its initial price π0(Z) = −0.5, that is, S
2
0 = −0.5
and S
2
1 = Z. Is the market model Mf complete?
Exercise 3 (MATH3975) Let Ω = {ω1, ω2}. We consider a single-period
model M = (S
1
, S2
) with two risky assets with prices S
1 and S
2 given by
S
1
0 = s0 > 0, S2
0 = z0 > 0 and
S
1
1
(ωi) = si
, S2
1
(ωi) = zi
for i = 1, 2 where 0 < s1 < s2 and 0 < z1 < z2. There are two traded assets,
S
1 and S
2
, 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.

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