金融中的优化方法：《Optimization Methods in Finance》解析

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"《Optimization Methods in Finance》是Gerard Cornuejols和Reha Tütüncü合著的一本关于金融领域优化方法的书籍，它深入探讨了在金融决策中广泛应用的各种优化技术。该书源自他们在卡内基梅隆大学硕士课程的教学内容，旨在解决从资产配置到风险管理、期权定价到模型校准等一系列计算金融问题。" 正文: 《Optimization Methods in Finance》一书详细阐述了金融领域中不同类型的优化问题及其解决方案。首先，书中介绍了线性、二次、整数、动态、随机、锥形和鲁棒规划等优化问题类别，这些类别覆盖了金融建模的广泛领域。每种问题类别的讨论都始于相关理论基础，如最优性条件和对偶性，接着探讨如何运用高效的求解方法。对于马克维茨的均值-方差优化模型这一经典模型，书中不仅做了详尽解析，还引入了一些针对各种金融问题的新颖优化模型。这些新模型可能包括但不限于改进的风险管理策略、更复杂的资产配置模型以及适应市场不确定性变化的鲁棒优化模型。书中还涵盖了金融模型校准的问题，这是现代金融市场中不可或缺的部分。模型校准涉及找到最佳参数设置，以使模型预测与实际市场行为最吻合。优化技术在这一过程中起到了关键作用，能够高效地搜索参数空间，确保模型的准确性和实用性。此外，书中可能还探讨了动态优化和随机优化问题，这些方法在处理时间序列数据和不确定性时特别有用。例如，动态编程可以应用于投资组合再平衡问题，而随机优化则可以用于处理金融市场中的随机因素，如价格波动和未来事件的不确定性。最后，书中可能还包括了关于如何在实际金融环境中应用这些优化技术的案例研究和实例，帮助读者将理论知识转化为实践技能。通过对这些复杂问题的分析，读者能够更好地理解和解决实际金融决策中的优化挑战。《Optimization Methods in Finance》是一本全面介绍金融优化方法的权威教材，它结合了理论深度和实践应用，对于金融专业人员和研究人员来说，是理解并掌握金融优化技术不可或缺的参考资料。通过阅读此书，读者将能够利用现代优化技术提升其在金融领域的决策效率和精度。

16 CHAPTER 1. INTRODUCTION

The constraint robust formulation of the resulting problem is equivalent to

(1.14).

Constraint robustness and objective robustness are concepts that arise

in conservative decision making and are not always appropriate for opti-

mization problems with data uncertainty.

1.3 Financial Mathematics

Modern ﬁnance has become increasingly technical, requiring the use of so-

phisticated mathematical tools in both research and practice. Many ﬁnd the

roots of this trend in the portfolio selection models and methods described

by Markowitz in the 1950’s and the option pricing formulas developed by

Black, Scholes, and Merton in the late 1960’s. For the enormous eﬀect these

works produced on modern ﬁnancial practice, Markowitz was awarded the

Nobel prize in Economics in 1990, while Scholes and Merton won the Nobel

prize in Economics in 1997.

Below, we introduce topics in ﬁnance that are especially suited for mathe-

matical analysis and involve sophisticated tools from mathematical sciences.

1.3.1 Portfolio Selection and Asset Allocation

The theory of optimal selection of portfolios was developed by Harry Markowitz

in the 1950’s. His work formalized the diversiﬁcation principle in portfolio

selection and, as mentioned above, earned him the 1990 Nobel prize for

Economics. Here we give a brief description of the model and relate it to

QPs.

Consider an investor who has a certain amount of money to be invested

in a number of diﬀerent securities (stocks, bonds, etc.) with random re-

turns. For each security i = 1, . . . , n, estimates of its expected return µ

and variance σ

are given. Furthermore, for any two securities i and j, their

correlation coeﬃcient ρ

is also assumed to be known. If we represent the

proportion of the total funds invested in security i by x

, one can compute the

expected return and the variance of the resulting portfolio x = (x

, . . . , x

)

as follows:

E[x] = x

+ . . . + x

= µ

and

V ar[x] =

i,j

= x

where ρ

≡ 1, Q

= ρ

, and µ = (µ

, . . . , µ

The portfolio vector x must satisfy

= 1 and there may or may

not be additional feasibility constraints. A feasible portfolio x is called

eﬃcient if it has the maximal expected return among all portfolios with the

same variance, or alternatively, if it has the minimum variance among all

portfolios that have at least a certain expected return. The collection of

eﬃcient portfolios form the eﬃcient frontier of the portfolio universe.

1.3. FINANCIAL MATHEMATICS 17

Markowitz’ portfolio optimization problem, also called the mean-variance

optimization (MVO) problem, can be formulated in three diﬀerent but equiv-

alent ways. One formulation results in the problem of ﬁnding a minimum

variance portfolio of the securities 1 to n that yields at least a target value

R of expected return. Mathematically, this formulation produces a convex

quadratic programming problem:

min

x = 1

x ≥ R

x ≥ 0,

(1.15)

where e is an n-dimensional vector all of which components are equal to

1. The ﬁrst constraint indicates that the proportions x

should sum to 1.

The second constraint indicates that the expected return is no less than the

target value and, as we discussed above, the objective function corresponds

to the total variance of the portfolio. Nonnegativity constraints on x

are

introduced to rule out short sales (selling a security that you do not have).

Note that the matrix Q is positive semideﬁnite since x

Qx, the variance of

the portfolio, must be nonnegative for every portfolio (feasible or not) x.

As an alternative to problem (1.15), we may choose to maximize the

expected return of a portfolio while limiting the variance of its return. Or,

we can maximize a risk-adjusted expected return which is deﬁned as the

expected return minus a multiple of the variance. These two formulations

are essentially equivalent to (1.15) as we will see in Chapter 8.

The model (1.15) is rather versatile. For example, if short sales are per-

mitted on some or all of the securities, then this can be incorporated into

the model simply by removing the nonnegativity constraint on the corre-

sponding variables. If regulations or investor preferences limit the amount

of investment in a subset of the securities, the model can be augmented with

a linear constraint to reﬂect such a limit. In principle, any linear constraint

can be added to the model without making it signiﬁcantly harder to solve.

Asset allocation problems have the same mathematical structure as port-

folio selection problems. In these problems the objective is not to choose

a portfolio of stocks (or other securities) but to determine the optimal in-

vestment among a set of asset classes. Examples of asset classes are large

capitalization stocks, small capitalization stocks, foreign stocks, government

bonds, corporate bonds, etc. There are many mutual funds focusing on

speciﬁc asset classes and one can therefore conveniently invest in these as-

set classes by purchasing the relevant mutual funds. After estimating the

expected returns, variances, and covariances for diﬀerent asset classes, one

can formulate a QP identical to (1.15) and obtain eﬃcient portfolios of these

asset classes.

A diﬀerent strategy for portfolio selection is to try to mirror the move-

ments of a broad market population using a signiﬁcantly smaller number of

securities. Such a portfolio is called an index fund. No eﬀort is made to

identify mispriced securities. The assumption is that the market is eﬃcient

and therefore no superior risk-adjusted returns can be achieved by stock

18 CHAPTER 1. INTRODUCTION

picking strategies since the stock prices reﬂect all the information available

in the marketplace. Whereas actively managed funds incur transaction costs

which reduce their overall performance, index funds are not actively traded

and incur low management fees. They are typical of a passive management

strategy. How do investment companies construct index funds? There are

numerous ways of doing this. One way is to solve a clustering problem where

similar stocks have one representative in the index fund. This naturally leads

to an integer programming formulation.

1.3.2 Pricing and Hedging of Options

We ﬁrst start with a description of some of the well-known ﬁnancial options.

A European call option is a contract with the following conditions:

• At a prescribed time in the future, known as the expiration date, the

holder of the option has the right, but not the obligation to

• purchase a prescribed asset, known as the underlying, for a

• prescribed amount, known as the strike price or exercise price.

A European put option is similar, except that it confers the right to sell

the underlying asset (instead of buying it for a call option). An American

option is like a European option, but it can be exercised anytime before the

expiration date.

Since the payoﬀ from an option depends on the value of the underlying

security, its price is also related to the current value and expected behavior

of this underlying security. To ﬁnd the fair value of an option, we need

to solve a pricing problem. When there is a good model for the stochastic

behavior of the underlying security, the option pricing problem can be solved

using sophisticated mathematical techniques.

Option pricing problems are often solved using the following strategy. We

try to determine a portfolio of assets with known prices which, if updated

properly through time, will produce the same payoﬀ as the option. Since the

portfolio and the option will have the same eventual payoﬀs, we conclude

that they must have the same value today (otherwise, there is arbitrage)

and we can therefore obtain the price of the option. A portfolio of other

assets that produces the same payoﬀ as a given ﬁnancial instrument is called

a replicating portfolio (or a hedge) for that instrument. Finding the right

portfolio, of course, is not always easy and leads to a replication (or hedging)

problem.

Let us consider a simple example to illustrate these ideas. Let us assume

that one share of stock XYZ is currently valued at $40. The price of XYZ

a month from today is random. Assume that its value will either double or

halve with equal probabilities.

=$40



*

80=S

(u)

20=S

(d)

1.3. FINANCIAL MATHEMATICS 19

Today, we purchase a European call option to buy one share of XYZ stock

for $50 a month from today. What is the fair price of this option?

Let us assume that we can borrow or lend money with no interest be-

tween today and next month, and that we can buy or sell any amount of the

XYZ stock without any commissions, etc. These are part of the “frictionless

market” assumptions we will address later. Further assume that XYZ will

not pay any dividends within the next month.

To solve the option pricing problem, we consider the following hedging

problem: Can we form a portfolio of the underlying stock (bought or sold)

and cash (borrowed or lent) today, such that the payoﬀ from the portfolio at

the expiration date of the option will match the payoﬀ of the option? Note

that the option payoﬀ will be $30 if the price of the stock goes up and $0

if it goes down. Assume this portfolio has ∆ shares of XYZ and $B cash.

This portfolio would be worth 40∆+B today. Next month, payoﬀs for this

portfolio will be:

=40∆+B



*

80∆+B=P

(u)

20∆+B=P

(d)

Let us choose ∆ and B such that

80∆ + B = 30

20∆ + B = 0,

so that the portfolio replicates the payoﬀ of the option at the expiration

date. This gives ∆ =

and B = −10, which is the hedge we were looking

for. This portfolio is worth P

= 40∆ + B =$10 today, therefore, the fair

price of the option must also be $10.

1.3.3 Risk Management

Risk is inherent in most economic activities. This is especially true of ﬁ-

nancial activities where results of decisions made today may have many

possible diﬀerent outcomes depending on future events. Since companies

cannot usually insure themselves completely against risk, they have to man-

age it. This is a hard task even with the support of advanced mathematical

techniques. Poor risk management led to several spectacular failures in the

ﬁnancial industry during the 1990’s (e.g., Barings Bank, Long Term Capital

Management, Orange County).

A coherent approach to risk management requires quantitative risk mea-

sures that adequately reﬂect the vulnerabilities of a company. Examples of

risk measures include portfolio variance as in the Markowitz MVO model,

the Value-at-Risk (VaR) and the expected shortfall (also known as condi-

tional Value-at-Risk, or CVaR)). Furthermore, risk control techniques need

to be developed and implemented to adapt to rapid changes in the values

of these risk measures. Government regulators already mandate that ﬁ-

nancial institutions control their holdings in certain ways and place margin

requirements for “risky” positions.

20 CHAPTER 1. INTRODUCTION

Optimization problems encountered in ﬁnancial risk management often

take the following form. Optimize a performance measure (such as expected

investment return) subject to the usual operating constraints and the con-

straint that a particular risk measure for the company’s ﬁnancial holdings

does not exceed a prescribed amount. Mathematically, we may have the

following problem:

max

RM[x] ≤ γ

x = 1

x ≥ 0.

(1.16)

As in the Markowitz MVO model, x

represent the proportion of the total

funds invested in security. The objective is the expected portfolio return and

µ is the expected return vector for the diﬀerent securities. RM[x] denotes

the value of a particular risk measure for portfolio x and γ is the prescribed

upper limit on this measure. Since RM[x] is generally a nonlinear function

of x, (1.16) is a nonlinear programming problem. Alternatively, we can

minimize the risk measure while constraining the expected return of the

portfolio to achieve or exceed a given target value R. This would produce a

problem very similar to (1.15).

1.3.4 Asset/Liability Management

How should a ﬁnancial institution manage its assets and liabilities? A static

mean-variance optimizing model, such as the one we discussed for asset al-

location, fails to incorporate the multiple liabilities faced by ﬁnancial insti-

tutions. Furthermore, it penalizes returns both above and below the mean.

A multi-period model that emphasizes the need to meet liabilities in each

period for a ﬁnite (or possibly inﬁnite) horizon is often required. Since li-

abilities and asset returns usually have random components, their optimal

management requires tools of “Optimization under Uncertainty” and most

notably, stochastic programming approaches.

Let L

be the liability of the company in period t for t = 1, . . . , T . Here,

we assume that the liabilities L

are random with known distributions. A

typical problem to solve in asset/liability management is to determine which

assets (and in what quantities) the company should hold in each period

to maximize its expected wealth at the end of period T. We can further

assume that the asset classes the company can choose from have random

returns (again, with known distributions) denoted by R

for asset class i in

period t. Since the company can make the holding decisions for each period

after observing the asset returns and liabilities in the previous periods, the

resulting problem can be cast as a stochastic program with recourse:

max

i,T

]

(1 + R

i,t−1

−

i,t

= L

, t = 1, . . . , T

i,t

≥ 0 ∀i, t.

(1.17)

The objective function represents the expected total wealth at the end of

the last period. The constraints indicate that the surplus left after liability

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