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 the state of the art in optimization models, methods, and software.

Optimization is a mature branch of applied mathematics. Typical optimization problems have the goal 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 nonlinear 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 second edition includes a more detailed discussion of mean–variance optimization, multi-period models, and additional material to highlight the relevance to finance.

The book's structure has also been clarified for the second edition; it is now organized in four main parts, each comprising several chapters. Part I guides the reader through the solution of asset liability cash flow matching using linear programming techniques, which are also used to explain asset pricing and arbitrage. Part II is devoted to single-period models. It provides a thorough treatment of mean–variance portfolio optimization models, including derivations of the one-fund and two-fund theorems and their connection to the capital asset pricing model, a discussion of linear factor models that are used extensively in risk and portfolio management, and techniques to deal with the sensitivity of mean–variance models to parameter estimation.

Dynamic programming is an approach to model and solve multi-period decision problems. The fundamental principle of dynamic programming is the Bellman equation, a certain kind of optimality condition. As we detail in this chapter, the central idea of the Bellman equation is to break down a multi-stage problem into multiple two-stage problems. Under suitable conditions, the Bellman equation yields a recursion that helps in characterizing the solution and in computing it. Before embarking on a formal description, we illustrate the dynamic programming approach via some examples.

Some Examples

Example 13.1 (Matches puzzle) Suppose there are 30 matches on a table and I play the following game with a clever opponent: I begin by picking up 1, 2, or 3 matches. Then my opponent must pick 1, 2, or 3 matches. We continue alternating until the last match is picked up. The player who picks up the last match loses. How can I (the first player) be sure of winning?

Solution. If I can ensure that it will be my opponent's turn when 1 match remains, I certainly win. Let us work backwards one step: If I can ensure that it will be my opponent's turn when 5 matches remain, I will also win. The reason for this is that no matter what he does when there are 5 matches left, I can make sure that when he has his next turn, only 1 match will remain. Hence it is clear that I win if I can force my opponent to play when 5 matches remain. We can continue working backwards and conclude that I will ensure victory if I can force my opponent to play when 5, 9, 13, 17, 21, 25, or 29 matches remain. Since the game starts with 30 matches on the table, I can ensure victory by picking 1 match at the beginning, bringing the number down to 29.

Example 13.2(Knapsack problem) Given a set of items, each with a certain weight and value, select the collection of items with total maximum value such that their total weight does not exceed some fixed weight limit W.