欧洲期权下的鲁棒投资组合优化：理论与应用
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本文研究了在欧洲期权环境下进行的鲁棒投资组合优化问题。标题"Robust Portfolio Optimization with European Options"聚焦于当投资者能够买卖股票以及这些股票对应的欧洲期权时，如何最大化在面临不确定性因素影响下的最坏情况回报。管理者的决策依据是对于驱动股票回报的因素，他们有一个预期范围，并且设定一个预算限制，用来控制这些因素相对于其名义值的规模偏差。
描述中的关键词"Robust optimization"表明作者采用了一种鲁棒性方法，这种方法考虑到了投资环境中的不确定性和潜在风险。"Portfoliomanagement"则强调了研究的核心在于如何通过合理的投资策略来应对这些不确定性，以确保投资组合的稳健性和稳定性。"Europeanoptions"则是研究焦点之一，因为它们在金融市场上具有重要地位，特别是对于对冲、杠杆和收益增强策略。
研究中，作者假设股票回报的不确定性被建模在一个多面体不确定集中，这种模型使得结果具有数学上的精确性和实用性。他们探讨了如何在考虑期权的存在时，优化投资组合配置，以便在面对可能的市场波动时，依然能够找到一个最优的解决方案。
理论部分，文章可能会深入分析如何构造和选择合适的不确定集，这可能涉及到线性规划或者凸优化技术，以便在满足预算限制的同时，寻找在所有可能的不确定情景下都能实现最大回报的策略。此外，可能还会讨论如何量化期权的“保险”特性对优化结果的影响，以及与只投资股票相比，投资组合中包含期权时的收益风险特征。
这篇研究提供了一个结合了鲁棒优化和金融衍生品（如欧洲期权）的决策框架，它不仅有助于投资者在金融市场波动中保护自己，还能理解期权如何改变投资组合的最优配置。这对于风险管理、金融机构和投资者来说，具有重要的实践指导意义。
Operations Research Perspectives 8 (2021) 100178
4
max
n
−
k=1
K
−
k
x
−
k
P
−
k
−
n
+
j=1
K
+
j
x
+
j
P
+
j
s.t. x +
n
+
j=1
S(0)
P
+
j
x
+
j
−
n
−
k=1
S(0)
P
−
K
x
−
k
= 0,
x +
n
+
j=1
x
+
j
+
n
−
k=1
x
−
k
= 1,
x, x
+
j
, x
−
k
≥ 0 ∀j, k.
(4)
This is a linear programming problem with two equality constraints and
a non-empty feasible set, so there exists an optimal solution with only
two non-zero components. Because we must have x +
n
+
j=1
S(0)
P
+
j
x
+
j
−
n
−
k=1
S(0)
P
−
k
x
−
k
= 0, the two non-zero components have to be one put and
either one call or the stock and a put.
Hence, we know the optimal portfolio will consist of either calls and
puts or stock shares and puts. When the optimal sub-portfolio consists of
one put (denoted put k for some k) and one call option (denoted call j for
some j) for a given stock, then the optimal worst-case return is:
Z
∗
= max
j,k
K
−
k
− K
+
j
P
+
j
+ P
−
k
, (5)
and the optimal allocations within the sub-portfolio are:
x
∗+
=
P
+
j
P
+
j
+ P
−
k
, x
∗−
=
P
−
k
P
+
j
+ P
−
k
.
where j and k are those that achieve the maximum in Eq. (5).
This is because, when the two equality constraints in Problem (4)
become (dropping the indices j and k):
x
+
P
+
=
x
−
P
−
,
x
+
+ x
−
= 1,
leading to Eq. (2.1). We nd the optimal j and k by reinjecting the al-
locations into the objective and picking the j and k that achieve the
highest objective value, yielding Eq. (5).
In a similar manner left to the reader, we can prove that, when the
optimal sub-portfolio for a given asset consists of one put (denoted put k
for some k) on the stock and that stock itself, then the optimal worst-case
return is:
Z
∗
= max
k
K
−
k
P
−
k
+ S(0)
, (6)
and the optimal sub-portfolio allocations in the stock and the put on that
stock are, with k the index that achieves the optimum in Eq. (6):
x
∗
=
S(0)
P
−
k
+ S(0)
, x
∗−
k
=
P
−
k
P
−
k
+ S(0)
Because Z
∗
in Eq. (6) is always positive, it is suboptimal to only invest
in the stock itself, for which the worst-case return is 0 in this framework.
It follows immediately by combining the results above that the
optimal worst-case return for the single-asset subproblem is given by:
Z
∗
= max
max
j,k
K
−
k
− K
+
j
P
+
j
+ P
−
k
, max
k
K
−
k
P
−
k
+ S(0)
. (7)
We now return to the original worst-case problem where we have
multiple underlying assets. Using Eq. (7) and introducing new decision
variables
α
i
as the fraction of the portfolio invested in asset i either by
buying shares of the stock itself or by buying call options on that stock or
by buying put options on that stock, we rewrite Problem (1) as:
max
n
i=1
max
max
j,k
K
−
ik
− K
+
ij
P
+
ij
+ P
−
ik
, max
k
K
−
ik
P
−
ik
+ S
i
(0)
α
i
s.t.
n
i=1
α
i
= 1,
α
i
≥ 0, ∀i.
This is a linear problem over a simplex, whose optimal solution is ach-
ieved at a corner point, yielding a non-diversied optimal portfolio and
Eq. (2) as the optimal objective.
Hence, at optimality, the manager who seeks to protect his portfolio
against the most adverse realization of the uncertainty invests in only
one asset, and for that asset invests in at most one call option and one put
option, or at most one put option and the stock itself. Although imple-
menting this approach will guarantee to the investor that his portfolio
return will not fall below the optimal value in Eq. (2), it is extremely
conservative because it assumes that the stock prices at the next time
period can take any possible non-negative value, so that the gross
returns can take any non-negative value as well. It is therefore natural to
ask how the optimal strategy will change if we consider a more realistic
description of uncertainty that uses our knowledge of the stock prices at
the present time period and limits the possible outcomes at the next time
period to a more realistic set of values. This is the purpose of the next
section.
2.2. Weak guarantee model
2.2.1. Formulation
Here, we consider a more realistic modeling of uncertainty inspired
by Bertsimas and Sim [5]. Specically, we assume that the stock returns
belong to a polyhedral set centered at the nominal values of the returns,
the size of which is parametrized by a budget of uncertainty, and model
correlation between stock returns using a factor model. The decision
variables are the fractions of the portfolio invested in each nancial
instrument. The investor seeks to maximize the worst-case portfolio
return over that polyhedral uncertainty set, subject to fractions summing
to one and being non-negative. The difference with the strong-guarantee
setting is that the previous uncertainty set consisted of all non-negative
stock returns. In the current setting, we assume that option prices also
are non-negative; however, this assumption is relaxed in Section 2.5.
We use the following notation, in addition to the one described in the
previous section.
Additional
parameters:
L the number of factors considered,
R
i
nominal value of stock return i for i = 1,…, n,
R
il
maximum deviation of stock return i from its nominal value
due to factor l
for i = 1, …, n and l = 1,…, L,
Γ budget of uncertainty (in [0, L]).
Additional decision
variables:
z
l
scaled deviation of factor l from its nominal value for
l = 1,…, L.
Let denote R the uncertainty set for the stock returns. We will dene
R in terms of the scaled deviations of the stock returns from their
nominal values, the sum of which cannot exceed the budget of uncer-
tainty. In mathematical terms, R is dened as:
R =
R
∃z,
L
l=1
z
l
≤ Γ,
z
l
≤ 1, ∀l, R
i
= R
i
+
L
l=1
R
il
z
l
∀i,
H. Ashra and A.C. Thiele
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