这个作业是完成风险股票和储蓄资产的收益分析

MATH3075/3975
Financial Derivatives

练习1考虑一个单周期三州市场模型M =(B,S)
日期0和T =1。我们假设有两种资产:储蓄
帐户B的初始值为B0 = 1,而风险股票的初始值为
价格S0 =4。无风险简单利率r等于10%。假设
股票价格S1满足(S1(ω1),S1(ω2),S1(ω3))=(8、5、3)和真实世界
概率P满足P(ω1)= 0.3,P(ω2)= 0.3,P(ω3)= 0.4。
(a)直接证明模型M =(B,S)是无套利的,即没有
这种模型中存在套利机会。不要在这里使用FTAP
(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
对于模型M,空间Ω=(ω1,ω2,ω3)。
(e)查找所有期望值
情商
</ s> </ s> </ s>
(S1-4-)
1 + r
</ s> </ s> </ s>
其中Q超过所有风险中性概率测度的M类
对于型号M。
(f)(MATH3975)查找X的对冲价格,即最小
存在投资组合(x,φ)的初始end赋x
不平等
V1(x,φ)(ω)≥(S1(ω)− 4)+
每个ω∈Ω成立。
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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