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2003

Volume 45, Issue 4, pp. 625-845


Survey and Review

Randy LeVeque, Section Editor

SIAM Rev. 45, pp. 625-626 (2 pages)

Online Publication Date: August 04, 2006

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A story in Aristotle's Politics is often cited as the first published discussion of a financial derivative, an instrument whose value is derived from other assets or events. In Greece of the fifth century B.C., the philosopher Thales, believing six months in advance that the spring weather would be exceptionally favorable to olive growth, negotiated the right to rent oil presses for the next harvest at a bargain rate from press owners who wanted to hedge their bets against the possibility of a poor harvest. In today's parlance, Thales purchased a call option. When spring brought a lush olive harvest, Thales rented the machines to others at a much higher rate, making a tidy profit to support his philosophical inquiries.
Options and other financial derivatives have become extremely popular in recent years, for better or worse. Overinvestment in derivatives of questionable value has led to notorious financial disasters such as the bankruptcy of Orange County, California, and the collapse of Barings Bank. Warren Buffett famously described derivatives as "financial weapons of mass destruction" in the Berkshire Hathaway Annual Report of 2002. Nonetheless, derivatives are undoubtedly here to stay; in 2002, trade in derivatives worldwide was estimated at $100 trillion or more, equaling or exceeding the total value of "real" equities.
Readers of SIAM Review can take a special interest in derivatives because their analysis is an arena in which mathematics has had a huge impact on finance. In particular, the 1970s papers of Black and Scholes (1973) and Merton (1976) began a flood of research on options pricing that continues to this day.
This issue's Survey and Review paper focuses on the special case of a spread option, a derivative that is based on the difference in value of two assets. Suppose you have some money to invest and the choice between two stocks to buy. You suspect that one will do much better than the other over the next few years, but you don't know which one. An ideal situation might be if you could buy one of them and hold it for a certain length of time, and then have the option of trading it in for the other stock if it turns out that that one did better. Another way to achieve the same benefit would be if someone agreed to pay you the difference in value between the two stocks at some specified time if you discover with hindsight that you made the wrong decision. Of course no one will offer you such an option free of charge, and so the mathematical question is how to price this option. This question is even more challenging if the option has a strike price that must be paid in order to exercise it.
While spread options aren't publicly traded on arbitrary pairs of stocks, a variety of complex spread options are available in many markets, and are particularly popular in energy markets. Energy prices are extremely volatile compared to other commodities, a fact that most consumers are well aware of and that greatly affects producers, distributors, and speculators. Spread options are an important hedging and speculation tool, and are available in many different forms. For example, options may be based on the difference between the prices of two different forms of oil (the crack spread), between the prices of oil and electricity (the spark spread), or between the prices of a single commodity at two different times (calendar spreads) or locations (locational spreads).
In the following article by Carmona and Durrleman, section 2 is devoted to a detailed description of these and other spread options that are used in practice. The authors then summarize the classical Black--Scholes pricing paradigm for options based on a single commodity, and the multivariate extensions needed for the analysis of spread options. The underlying stochastic differential equations can be solved by various approaches, and several approximations are illustrated and compared. Particular difficulties associated with energy markets are discussed throughout and are a main focus of the latter sections.
Financial mathematics is as exciting growth area for applied mathematics, and we hope this paper will introduce readers to some challenging problems of current interest in this field.

Pricing and Hedging Spread Options

René Carmona and Valdo Durrleman

SIAM Rev. 45, pp. 627-685 (59 pages) | Cited 30 times

Online Publication Date: August 04, 2006

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We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.

Problems and Techniques

Joe Flaherty, Section Editor

SIAM Rev. 45, pp. 687-687 (1 page)

Online Publication Date: August 04, 2006

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This edition of Problems and Techniques features articles on orthogonal rational functions with applications in identification and control, and methods for cancer classification.
Our lead-off article, by Wahlberg, is motivated by applications in control theory and signal processing where finite impulse response (FIR) and infinite impulse response (IIR) models are often invoked. The former are closely related to orthogonal polynomials and have nice computational properties, while the latter often give more compact representations and improved accuracy. Our author develops a fractional transformation framework to map the original problem to a new domain where FIR descriptions have rapid convergence. When mapped back to the original domain, the FIR models generate an IIR description with an orthogonal rational basis and extends the theory available for FIR models to the IIR case.
Our second article, by Rifkin, Mukherjee, Tamayo, Ramaswamy, Yeang, Angelo, Reich, Poggio, Lander, Golub, and Mesirov, an interdisciplinary team centered at the Whitehead Institute/MIT Center for Genome Research, describes a molecular approach to the classification of cancers. This technology has important ramifications when developing correct diagnoses and appropriate cancer treatment strategies. The technique proposed by the authors uses DNA microarrays to assemble a large database of expressed genes in the tumor samples. Gene expression profiles are analyzed by combining multiple binary support vector machine (SVM) classifiers trained to predict class membership. The cover of this issue of SIAM Review depicts six different types of cancer of the fourteen studied in this article.

Orthogonal Rational Functions: A Transformation Analysis

Bo Wahlberg

SIAM Rev. 45, pp. 689-705 (17 pages) | Cited 2 times

Online Publication Date: August 04, 2006

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Finite impulse response (FIR) models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application. The connections to orthogonal polynomials are well known. However, infinite impulse response (IIR) models often provide much more compact descriptions and in many cases give improved performance. The objective of this paper is to present a simple framework for the derivation and analysis of orthogonal IIR transfer functions, which are directly related to orthogonal rational functions. Orthogonality simplifies approximation analysis and leads to improved numerical properties. The basic idea is to use a fractional transformation to map the problem to a new domain, where an FIR description is most appropriate. This FIR representation is then mapped back to the original domain to give an orthogonal IIR representation. It is then straightforward to extend many results for FIR models to IIR model structures with arbitrary stable poles; i.e., properties of orthogonal polynomials are easily generalized to orthogonal rational functions. Much of the theory to be presented is classical, e.g., Laguerre and Kautz functions, and we will make use of well-known results in orthogonal filter theory. However, our main contribution is to present a uniform and transparent theory which also covers more novel results that have mainly been presented in the signals, systems, and control literature in the last decade.

An Analytical Method for Multiclass Molecular Cancer Classification

Ryan Rifkin, Sayan Mukherjee, Pablo Tamayo, Sridhar Ramaswamy, Chen-Hsiang Yeang, Michael Angelo, Michael Reich, Tomaso Poggio, Eric S. Lander, Todd R. Golub, and Jill P. Mesirov

SIAM Rev. 45, pp. 706-723 (18 pages) | Cited 9 times

Online Publication Date: August 04, 2006

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Modern cancer treatment relies upon microscopic tissue examination to classify tumors according to anatomical site of origin. This approach is effective but subjective and variable even among experienced clinicians and pathologists. Recently, DNA microarray-generated gene expression data has been used to build molecular cancer classifiers. Previous work from our group and others demonstrated methods for solving pairwise classification problems using such global gene expression patterns. However, classification across multiple primary tumor classes poses new methodological and computational challenges. In this paper we describe a computational methodology for multiclass prediction that combines class-specific (one vs. all) binary support vector machines. We apply this methodology to the diagnosis of multiple common adult malignancies using DNA microarray data from a collection of 198 tumor samples, spanning 14 of the most common tumor types. Overall classification accuracy is 78%, far exceeding the expected accuracy for random classification. In a large subset of the samples (80%), the algorithm attains 90% accuracy. The methodology described in this paper both demonstrates that accurate gene expression-based multiclass cancer diagnosis is possible and highlights some of the analytic challenges inherent in applying such strategies to biomedical research.

SIGEST

SIAM Rev. 45, pp. 725-725 (1 page)

Online Publication Date: August 04, 2006

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The editors of SIAM Review are delighted to present a SIGEST paper about cryptography, a topic that appears frequently these days in mainstream news media. Stories about computer security, defenses against terrorism, privacy, and file swapping invariably refer to cryptographic systems, which thus represent an extraordinarily visible example of the practical application of mathematics and computer science. Encryption systems depend on a stream of new mathematical results and algorithms; it may take years of study by experts to prove that a proposed technique is secure (or not). An additional wrinkle is that humans tend to avoid any encryption procedure that they perceive to be too much trouble, which means that researchers in cryptography need to consider criteria like ease of use as well as more precise properties.
Cryptographers are blessed with, and continue to devise, engaging terminology and even some implied personalities. No discussion of cryptography is complete without Alice and Bob---so much more appealing than A and B---whose roles may, however, vary. For example, Alice and Bob are sometimes friends eager to communicate secrets without others learning them, and sometimes two people who need to communicate but do not know or trust each other; new variations on Alice, Bob, and their relationship are constantly emerging. Another fixture in cryptography is the presumably evil adversary, who may be passive (an eavesdropper limited to observing the information traffic sent between Alice and Bob) or active (someone who can see and modify the traffic). Wide use is made of "zero knowledge proof systems" (an apparently contradictory term), and this paper defines an evocatively named "garbage/not garbage" oracle that is useful in validity checks.
The standard requirement of semantic security for cryptosystems means that an adversary learns nothing about the original text (the "plaintext") from its encrypted form (the "ciphertext"). The stricter requirement of nonmalleability, needed to hide information from active adversaries, means, roughly speaking, that an adversary cannot use the ciphertext associated with an original plaintext to produce an encryption of a related plaintext. This issue's SIGEST paper, "Nonmalleable Cryptography," by D. Dolev, C. Dwork, and M. Naor, which first appeared in 2000 in volume 30 of the SIAM Journal on Computing, identified and addressed the concept of nonmalleability. The paper has been extremely influential, is widely cited, and has inspired a substantial level of related research.
For the paper's appearance in SIGEST, the authors have added an extended and completely new preface, intended for the general SIREV reader, that introduces much of the terminology, presents easily understood examples, and brings the reader up-to-date on related problems. Section 1.4, on deniable authentication, clarifies some of the many complications in defining new protocols and their desired (or undesired) security features. The original paper itself explores in detail the implications of nonmalleability, presenting nonmalleable schemes for three key problem classes and proving security against strong ciphertext attacks.
We are grateful indeed to the authors for their extra efforts to make this important paper accessible to nonexperts in cryptography.

Nonmalleable Cryptography

Danny Dolev, Cynthia Dwork, and Moni Naor

SIAM Rev. 45, pp. 727-784 (58 pages)

Online Publication Date: August 04, 2006

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The notion of nonmalleable cryptography, an extension of semantically secure cryptography, is defined. Informally, in the context of encryption the additional requirement is that given the ciphertext it is impossible to generate a different ciphertext so that the respective plaintexts are related. The same concept makes sense in the contexts of string commitment and zero-knowledge proofs of possession of knowledge. Nonmalleable schemes for each of these three problems are presented. The schemes do not assume a trusted center; a user need not know anything about the number or identity of other system users.
Our cryptosystem is the first proven to be secure against a strong type of chosen ciphertext attack proposed by Rackoff and Simon, in which the attacker knows the ciphertext she wishes to break and can query the decryption oracle on any ciphertext other than the target.

Education

Bobby Schnabel, Section Editor

SIAM Rev. 45, pp. 785-785 (1 page)

Online Publication Date: August 04, 2006

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This edition of the Education section is an excellent demonstration of the breadth of the section. The first paper is a general treatment of the subject of computational science and engineering that serves as an overview and introduction to this emerging field, while the second is a specific example from the field of finance that can be used in teaching and studying applied linear algebra.
The authors of the first paper, O. Yacar and R. Landau, characterize computational science and engineering (CSE) education as having moved recently from its "infancy" to its "early growth" stage. Their paper, "Elements of Computational Science and Engineering Education," serves as a guide to CSE educational programs at this still-early stage of their existence, concentrating on undergraduate offerings. (Thus it complements the SIAM working group study " Graduate Study in Computational Science and Engineering" that was printed in this section in volume 43 (2001), pp. 163--177.) The paper should be of keen interest to both students and faculty, from different perspectives. For students who are considering studying CSE, this paper provides a very useful perspective on what to expect from a CSE education. Student advisors will want to keep it handy for this purpose! Faculty and departments will find it's useful in designing CSE curricula. Its strengths include a comprehensive collection of the issues confronting CSE curricula, a comparison of approaches to undergraduate CSE programs, and suggestions of the emerging core components of an undergraduate CSE curriculum. Of particular interest to all readers is an analysis of computer science, CSE, and applied physics curricula that serves to highlight the balance between computing, mathematics, and applied science that appears to be a distinguishing feature of CSE curricula.
"Designing a 401(k): A Case Study," by P. Laumakis, provides a nice, nonstandard, and self-contained application of linear algebra that should be useful in introductory applied linear algebra courses or in the portion of introductory numerical computation courses that deal with the solution of linear equations. As the title states, the example is taken from finance, an increasingly important application area for numerical computation, but not one that is often used in examples in introductory courses. The setting for the example is making investment choices for a retirement plan---not the most pressing concern of most undergraduates---but even the financial issues that are covered have much more general applicability. The example utilizes knowledge of singularity of matrices, existence of solutions to linear equations, and Gaussian elimination and so can be used to reinforce all these notions nicely from a practical and real-world viewpoint. It certainly is ideal for a linear algebra or numerical computation course taught to business or finance students, but as the author's own experience shows, it is suitable for general undergraduate audiences as well.
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Elements of Computational Science and Engineering Education

Osman Yasar and Rubin H. Landau

SIAM Rev. 45, pp. 787-805 (19 pages) | Cited 5 times

Online Publication Date: August 04, 2006

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The multidisciplinary nature of computational science and engineering (CSE) and its relation to other disciplines is described. The stages through which CSE education is evolving, from initial recognition in the 1980s to present growth, are discussed. The challenges and benefits of different approaches to CSE education are discussed, as is the emergence of a set of core elements common to different approaches. The content of courses, curricula, and degrees offered in CSE are reviewed, and a survey is made of all undergraduate degree programs. The curricula of different programs are examined for the common "tool set" they define and analyzed for their relative weighting of computing, application, and mathematics. A trend toward a standard curriculum is noted.
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Designing a 401(k): A Case Study

Paul J. Laumakis

SIAM Rev. 45, pp. 806-813 (8 pages)

Online Publication Date: August 04, 2006

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Over the past decade or so, many private and publicly held companies have moved away from providing employees with company-funded pensions upon retirement to a retirement system that requires both employee and company contributions to fund an individual's retirement. These so-called 401(k) retirement plans usually require the employee to choose among different investment vehicles, including mutual funds, in order to allocate their retirement savings. This study presents the results of an analysis of the allocation of a lump-sum rollover of money into a typical 401(k) plan, subject to certain restrictions. The mathematics required to complete this analysis includes those topics that are typically covered in an undergraduate course in linear algebra. All of the following material was used in a sophomore-level engineering mathematics course taught at Rowan University.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 45, pp. 815-845 (31 pages)

Online Publication Date: August 04, 2006

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Getting a new book by Gerhard Wanner, Ernst Hairer, and coauthors is always a treat for folks interested in ordinary differential equations. Their fresh and scholarly takes on the field, especially regarding numerical algorithms, are deep and uniquely valuable. Their latest monograph on the important topic of geometric numerical integration, with Christian Lubich, is given a perceptive featured review by Prof. Robert McLachlan of Massey University that encourages "scientists with diverse backgrounds in dynamical systems, geometry, mechanics, and the numerous appications fields in physics and chemistry" to read it.
As usual, the other reviews in this issue span a wide variety of topics and levels of sophistication. Their authors have been unusually enthusiastic about the books reviewed, suggesting that we'll not be without books worth reading in the months ahead.
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