The use of sophisticated mathematical tools in modern finance is now commonplace. Researchers and practitioners routinely run simulations or solve differential equations to price securities, estimate risks, or determine hedging strategies. Some of the most important tools employed in these computations are optimization algorithms. Many computational finance problems ranging from asset allocation to risk management, from option pricing to model calibration, can be solved by optimization techniques. This book is devoted to explaining how to solve such problems efficiently and accurately using recent advances in optimization models, methods, and software.

Optimization is a mature branch of applied mathematics. Typical optimization problems have the objective of allocating limited resources to alternative activities in order to maximize the total benefit obtained from these activities. Through decades of intensive and innovative research, fast and reliable algorithms and software have become available for many classes of optimization problems. Consequently, optimization is now being used as an effective management and decision-support tool in many industries, including the financial industry.

This book discusses several classes of optimization problems encountered in financial models, including linear, quadratic, integer, dynamic, stochastic, conic, and robust programming. For each problem class, after introducing the relevant theory (optimality conditions, duality, etc.) and efficient solution methods, we discuss several problems of mathematical finance that can be modeled within this problem class. The reader is guided through the solution of asset/liability cash-flow matching using linear programming techniques, which are also used to explain asset pricing and arbitrage. Volatility estimation is discussed using nonlinear optimization models.

Optimization is a branch of applied mathematics that derives its importance both from the wide variety of its applications and from the availability of efficient algorithms. Mathematically, it refers to the minimization (or maximization) of a given objective function of several decision variables that satisfy functional constraints. A typical optimization model addresses the allocation of scarce resources among possible alternative uses in order to maximize an objective function such as total profit.

Decision variables, the objective function, and constraints are three essential elements of any optimization problem. Problems that lack constraints are called unconstrained optimization problems, while others are often referred to as constrained optimization problems. Problems with no objective functions are called feasibility problems. Some problems may have multiple objective functions. These problems are often addressed by reducing them to a single-objective optimization problem or a sequence of such problems.

If the decision variables in an optimization problem are restricted to integers, or to a discrete set of possibilities, we have an integer or discrete optimization problem. If there are no such restrictions on the variables, the problem is a continuous optimization problem. Of course, some problems may have a mixture of discrete and continuous variables. We continue with a list of problem classes that we will encounter in this book.

This chapter presents several applications of integer linear programming: combinatorial auctions, the lockbox problem, and index funds. We also present a model of integer quadratic programming: portfolio optimization with minimum transaction levels.

Combinatorial auctions

In many auctions, the value that a bidder has for acquiring a set of items may not be the sum of the values that he has for acquiring the individual items in the set. It may be more or it may be less. Examples are equity trading, electricity markets, pollution right auctions, and auctions for airport landing slots. To take this into account, combinatorial auctions allow the bidders to submit bids on combinations of items.

Specifically, let M = {1, 2, …,m} be the set of items that the auctioneer has to sell. A bid is a pair Bj = (Sj, pj) where Sj ⊆ M is a nonempty set of items and pj is the price offer for this set. Suppose that the auctioneer has received n bids B1, B2, …, Bn. How should the auctioneer determine the winners in order to maximize his revenue? This can be done by solving an integer program. Let xj be a 0,1 variable that takes the value 1 if bid Bj wins, and 0 if it looses.

In this chapter, we discuss Value-at-Risk, a widely used measure of risk in finance, and its relative, Conditional Value-at-Risk. We then present an optimization model that optimizes a portfolio when the risk measure is the Conditional Value-at-Risk instead of the variance of the portfolio as in the Markowitz model. This is achieved through stochastic programming. In this case, the variables are anticipative. The random events are modeled by a large but finite set of scenarios, leading to a linear programming equivalent of the original stochastic program.

Risk measures

Financial activities involve risk. Our stock or mutual fund holdings carry the risk of losing value due to market conditions. Even money invested in a bank carries a risk — that of the bank going bankrupt and never returning the money let alone some interest. While individuals generally just have to live with such risks, financial and other institutions can and very often must manage risk using sophisticated mathematical techniques. Managing risk requires a good understanding of quantitative risk measures that adequately reflect the vulnerabilities of a company.

Perhaps the best-known risk measure is Value-at-Risk (VaR) developed by financial engineers at J.P. Morgan. VaR is a measure related to percentiles of loss distributions and represents the predicted maximum loss with a specified probability level (e.g., 95%) over a certain period of time (e.g., one day). Consider, for example, a random variable X that represents loss from an investment portfolio over a fixed period of time.

Conic optimization problems are encountered in a wide array of fields including truss design, control and system theory, statistics, eigenvalue optimization, and antenna array weight design. Robust optimization formulations of many convex programming problems also lead to conic optimization problems, see, e.g., [8, 9]. Furthermore, conic optimization problems arise as relaxations of hard combinatorial optimization problems such as the max-cut problem. Finally, some of the most interesting applications of conic optimization are encountered in financial mathematics and we will address several examples in this chapter.

Tracking error and volatility constraints

In most quantitative asset management environments, portfolios are chosen with respect to a carefully selected benchmark. Typically, the benchmark is a market index, reflecting a particular market (e.g., domestic or foreign), or a segment of the market (e.g., large cap growth) the investor wants to invest in. Then, the portfolio manager's problem is to determine an index-tracking portfolio with certain desirable characteristics. An index-tracking portfolio intends to track the movements of the underlying index closely with the ultimate goal of adding value by beating the index. Since this goal requires departures from the underlying index, one needs to balance the expected excess return (i.e., expected return in excess of the benchmark return) with the variance of the excess returns.

The tracking error for a given portfolio with a given benchmark refers to the difference between the returns of the portfolio and the benchmark.

Introduction to robust optimization

In many optimization models the inputs to the problem are not known at the time the problem must be solved, are computed inaccurately, or are otherwise uncertain. Since the solutions obtained can be quite sensitive to these inputs, one serious concern is that we are solving the wrong problem, and that the solution we find is far from optimal for the correct problem.

Robust optimization refers to the modeling of optimization problems with data uncertainty to obtain a solution that is guaranteed to be “good” for all or most possible realizations of the uncertain parameters. Uncertainty in the parameters is described through uncertainty sets that contain many possible values that may be realized for the uncertain parameters. The size of the uncertainty set is determined by the level of desired robustness.

Robust optimization can be seen as a complementary alternative to sensitivity analysis and stochastic programming. Robust optimization models can be especially useful in the following situations:

• Some of the problem parameters are estimates and carry estimation risk.

• There are constraints with uncertain parameters that must be satisfied regardless of the values of these parameters.

• The objective function or the optimal solutions are particularly sensitive to perturbations.

• The decision-maker cannot afford to take low-probability but high-magnitude risks.

Recall from Chapter 1 that there are different definitions and interpretations of robustness; the resulting models and formulations differ accordingly. In particular, we can distinguish between constraint robustness and objective robustness.

Volatility is a term used to describe how much the security prices, market indices, interest rates, etc., move up and down around their mean. It is measured by the standard deviation of the random variable that represents the financial quantity we are interested in. Most investors prefer low volatility to high volatility and therefore expect to be rewarded with higher long-term returns for holding higher volatility securities.

Many financial computations require volatility estimates. Mean-variance optimization trades off the expected return and volatility of a portfolio of securities. The celebrated option valuation formulas of Black, Scholes, and Merton (BSM) involve the volatility of the underlying security. Risk management revolves around the volatility of the current positions. Therefore, accurate estimation of the volatilities of security returns, interest rates, exchange rates, and other financial quantities is crucial to many quantitative techniques in financial analysis and management.

Most volatility estimation techniques can be classified as either a historical or an implied method. One either uses historical time series to infer patterns and estimates the volatility using a statistical technique, or considers the known prices of related securities such as options that may reveal the market sentiment on the volatility of the security in question. GARCH models exemplify the first approach while the implied volatilities calculated from the BSM formulas are the best-known examples of the second approach. Both types of techniques can benefit from the use of optimization formulations to obtain more accurate volatility estimates with desirable characteristics such as smoothness.

Asset/liability management

The financial health of any company, and in particular those of financial institutions, is reflected in the balance sheets of the company. Proper management of the company requires attention to both sides of the balance sheet - assets and liabilities. Asset/liability management (ALM) offers sophisticated mathematical tools for an integrated management of assets and liabilities and is the focus of many studies in financial mathematics.

ALM recognizes that static, one-period investment planning models (such as mean-variance optimization) fail to incorporate the multi-period nature of the liabilities faced by the company. 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 liabilities and asset returns usually have random components, their optimal management requires tools of “optimization under uncertainty” and, most notably, stochastic programming approaches.

We recall the ALM setting we introduced in Section 1.3.4: let Lt be the liability of the company in year t for t = 1, …, T. The Lt's are random variables. Given these liabilities, which assets (and in which quantities) should the company hold each year to maximize its expected wealth in year T? The assets may be domestic stocks, foreign stocks, real estate, bonds, etc. Let Rit denote the return on asset i in year t. The Rit 's are random variables.

Derivative securities and the fundamental theorem of asset pricing

One of the most widely studied problems in financial mathematics is the pricing of derivative securities, also known as contingent claims. These are securities whose price depends on the value of another underlying security. Financial options are the most common examples of derivative securities. For example, a European call option gives the holder the right to purchase an underlying security for a prescribed amount (called the strike price) at a prescribed time in the future, known as the expiration or exercise date. The exercise date is also known as the maturity date of the derivative security. Recall the similar definitions of European put options as well as American call and put options from Section 1.3.2.

Options are used mainly for two purposes: speculation and hedging. By speculating on the direction of the future price movements of the underlying security, investors can take (bare) positions in options on this security. Since options are often much cheaper than their underlying security, this bet results in much larger earnings in relative terms if the price movements happen in the expected direction compared to what one might earn by taking a similar position in the underlying. Of course, if one guesses the direction of the price movements incorrectly, the losses are also much more severe.