249 references, last updated Tue Apr 8 11:29:05 2008

Jorge Aarao. A harmonic note. Math Horizons, 14:14, September 2006. harmonic series.

Høakan Andersson and Tom Britton. Stochastic Epidemic and Their Statistical Analysis. Springer, 2000.

David Aldous and Persi Diaconis. Shuffling card and stopping times. American Mathematical Monthly, pages 333–348, May 1986. stopping times.

Jesper Andreason and Martin Dahlgren. At the flick of a switch. Energy Risk, pages 71–75, February 2006. energy risk.

S. Asmussen and P. W. Glynn. Stochastic Simulation, volume 57 of Stochastic Modelling and Applied Probability. Springer Verlag, 2007.

Marco Allavenada and Peter Laurence. Quantitative Modeling of Derivative Securities. Chapman and Hall, 2000. HG 6024 A3A93 2000.

Marco Avellaneda and Peter Laurence. Quantitative Modeling of Derivative Securities: From Theory to Practice. Chapman and Hall, 2000. HG 6024 A3A93 2000.

Linda J. S. Allen. An Introduction to Stochastic Processes with Applications to biology. Pearson Prentice-Hall, 2003.
Chapter 8 covers Diffusion Processes and Stochastic Differential Equations with Section 8.3, random Walks and Brownian Motion, Section 8.4 diffusion Processes, Section 8.5 Kolmogorov Differential Equations, Section 8.6 Wiener Processes, Section 8.7 Ito Stochastic Integral, Section 8.8 Ito Stochastic Differential Equation, Section 8.9, Numerical methods for Solving SDEs. These sections are followed by biological applications.

R. Almgren. Financial derivatives and partial differential equations. The American Mathematical Monthly, 109:1–12, 2002.
One of my favorite articles on the subject, very nice introduction, but is explicitly PDE based, with only a hint of the risk-neutral martingale approach.

David Applebaum. Lévy processes – from probability to finance and quantum groups. Notices of the American Mathematical Society, 51(11):1336–1347, December 2004.
This survey article is an introduction to a class of stochastic processes called Lévy processes. Their basic structure was understood during the ``heroic age'' of probability in the 1930s and 1940s and much of this was due to Paul Lévy himself, the Russian mathematician A. N. Khintchine and to K. Ito in Japan. During the past ten years, there has been a revival of interest in these processes, due to new theoretical developments and also a wealth of novel applications, particularly to option pricing in mathematical finance.

David Applebaum. Lèvy processes – from probability to finance and quantum groups. Notices of the AMS, 51(11):1336–1347, December 2004. Lévy processes.

Kerry Back. A Course in Derivative Securities. Springer, 2005.
From the advertising blurb: ``This book aims at a middle ground between the introductory books on derivative securities, and those that provide advanced mathematical treatments. It is written for mathematically capable students who have not necessarily had prior exposure to probability theory, stochastic calculus, or computer programming. It provides derivations of pricing and hedging formulas (using the probabilistic change of numeraire technique) for standard options, exchange options, options on forwards and futures, quanto options, exotic options, caps, floors, and swaptions, as well as VBA code implementing the formulas. It also contains in introduction to Monte Carlo, binomial models, and finite-difference methods.

Emilio Barucci. Financial Markets Theory. Springer Finance. Springer-Verlag, 2003.
From the advertising blurb: ``Financial Markets Theory is the only textbook to address the economics foundations of financial markets from a mathematically rigorous standpoint and to offer a self-contained critical discussion, based on empirical results. It is an advanced book, well-suited for a first graduate course in financial mathematics. It is self-contained and introduces topics in a setting accessible to economists and practitioners equipped with a basic mathematical background.'' This book sounds like a foundational book in economics and the economics behind financial markets and therefore is in a different direction than most of the course as I am teaching it.

Bartholomew and Biggs. Nonlinear Optimization with Financial Applications. Springer, 2005.

D. Beaglehole and A. Chenbanier. A two-factor mean-reverting model. Risk, pages 65–69, July 2002.
Discusses the pricing of flex options on Nymex WTI using a two-factor MR model for the spot with spot-dependent volatility. Flex options have some swing-like characteristics, they permit the exercise of up to a specified number of options on futures.

Jamil Baz and George Chako. Financial Derivatives: Pricing, Applications and Mathematics. Cambridge University Press, 2004.

F. E. Benth. Option Theory with Stochastic Analysis. Universitext. Springer Verlag, 2004.
From the advertising blurb: ``The objective of this textbook is to provide a very basic and accessible introduction to option pricing, invoking only a minimum of stochastic analysis. Although short, it covers the theory essential to the statistical modeling of stocks, pricing of derivatives (general contingent claims) with martingale methods and computational finance including both finite-difference and Monte Carlo methods. The reader is led to an understanding of the assumptions inherent in the Black and Scholes theory, of the main idea behind deriving prices and hedges, and of the use of numerical methods to compute prices for exotic contracts.'' I have not seen this text. This sounds like a perhaps satisfactory textbook for a course such as I am teaching here, yet it also sounds like like a lot of material to pack into approximately 172 pages.

Serge Bernstein. Démonstration de théorèeme de weierstrass fondée sur le calcul de probabilités. url http://www.math.technion.ac.il/hat/fpapers/PO3.PDF, 1913. history.

P. Bernstein. Capital Ideas: The Improbable Origins of Modern Wall Street. Free Press, 1992. popular history.

Peter L. Bernstein. Against the Gods: The Remarkable Story of Risk. John Wiley and Sons, 1996. HD61 B4666 1996.

Peter L. Bernstein. Capital Ideas Evolving. John Wiley, 2007. popular history.

R. Bhar and S. Hamori. Hidden Markov Models: Applications to Financial Economics. Kluwer Academic, 2004.

Bhar. Option Theory with Stochastic Analysis. Springer, 2004.
Need more information, Have only the barest details.

Bhar. Empirical Techniques in Finance. Springer Verlag, 2005.
Need more information, Have only the barest details.

Tomas Björk. Arbitrage Theory in Continuous Time. Oxford Finance Series. Oxford University Press, second edition edition, 2004.
From the publisher's description: ``This accessible introduction to the mathematical underpinnings of finance concentrates on the probabilistic theory of continuous arbitrage pricing of financial derivatives including stochastic optimal control theory and Merton's fund separation theory.''

N. H. Bingham and Rudiger Kiesel. Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer Verlag, 1998. HG4515.2 B56.

Nicholas Bingham and Rüdiger Kiesel. Risk-Neutral Valuation, second edition. Springer Finance. Springer Verlag, second edition edition, 2004.
Since its introduction in the early 80's the risk-neutral valuation principle has proved to be an important tool in the pricing and hedging of financial derivatives. This second edition - completely up-to-date with new exercises - provides a comprehensive and self-contained treatment of the probabilistic theory behind the risk-neutral valuation principle and its application to pricing and hedging of financial derivatives. On the probabilistic side, both discrete and continuous time stochastic process are treated with special emphasis on martingale theory, stochastic integration, and change-of-measure techniques.'' I have not see this text, but it sounds like a comprehensive and standard introduction to the basic modern martingale theory that I skirt around.

Fischer Black. The pricing of commodity contracts. Journal of Financial Economics, 3:167–179, January/February 1976.
Describes the ``Black 76'' model which is used to price options on energy forward and futures contracts.

Damiano Brigo and Fabio Mercurio. Interest Rate Models, Theory and Practice. Springer-Verlag, 2001. (HB539 B785).

D. Brigo and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance. Springer, 2nd edition edition, 2005.
The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and the of ran reduction have been added and LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. This fast growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of convertible bonds and inflation-linked derivatives. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique is analogous to interest-rate modeling, Credit Derivatives, – mostly Credit Default Swaps (CDS) and CDS Options, – are discussed, building on the basic short rate models and market models introduced earlier for the default free market.

Truman Botts. Probability theory and the lebesgue integral. Mathematics Magazine, 42:105–111, 1969. History.

Nicolas Bouleau. Financial Markets and Martingales. Springer-Verlag, 2003.
From the advertising blurb: ``Is it really possible to make money on the financial markets? This is just one of the questions posed in this practical and thought-provoking book, winner in the original French version, of the ``Best financial economic book'' prize 1999 from the Institut de Haute Finance and the ``Prix FNAC-Arthur Anderson de meilluer livre -denterprise 2000''. Starting from games of chance, from which probability theory was born, Nicolas Bouleau explains how the finance markets operate and demonstrates how the application of mathematics has turned finance into a high-tech business and well as a formidable and efficient tool. Concise and accessible, with no previous knowledge of finance or mathematics required, the aim of this book is simply to articulate the main ideas and put them into perspective, leading readers to a fresh understanding of this complex area.'' I have not seen this book, but it looks interesting and useful as an introduction to the area.

M. Baxter and A. Rennie. Financial Calculus: An introduction to derivative pricing. Cambridge University Press, 1996. HG 6024 A2W554.

T. R. Bielecki and M. Rutkowski. Credit Risk: Modeling, Valuation and Hedging. Finance. Springer-Verlag, 2001.
From a review in Mathematical Reviews used as the book blurb: ``A fairly complete overview f the most important recent developments of credit risk modeling from the viewpoint of mathematical finance. It provides an excellent treatment of mathematical aspects of credit risk and will be useful as a reference for technical details to traders and analysts dealing with credit-risky assets. It is a worthwhile addition to the literature and will serve as highly recommended reading for students and researchers in the subject area for some years to come.''

Paolo Brandimarte. Numerical Methods in Finance: A MATLAB-based Introduction. Wiley Series in Probability and Statistics. J. W. Wiley and Sons, 2002.
From the book's back cover blurb: ``Numerical Methods in Finance bridges the gap between financial theory and computational practice while helping students and practitioners exploit MATLAB for financial applications. [The book] covers the basics of finance and numerical analysis and provides background material that suits the needs of students from both financial engineering and economics perspectives. Classical numerical analysis methods; optimization, including such less familiar topics such as stochastic and integer programming; simulation, including low discrepancy sequences; and partial differential equations are covered in detail. Extensive illustrative examples of the application of the all these methodologies are provided.'' The chapters are: begin enumerate item Financial problems and numerical methods item Basics of numerical analysis item Classification of optimization problems item Numerical methods for unconstrained optimization item Methods for constrained optimization item Principles of Monte Carlo Simulation item Finite different methods for partial differential equations item Optimization methods for portfolio management item Option valuation by Monte Carlo simulation item Option valuation by finite difference methods item Appendix: Introduction to MATLAB programming end enumerate

Paolo Brandimarte. Numerical Methods in Finance: A MATLAB-based Introduction, chapter Optimization models for portfolio management. J. W. Wiley and Sons, 2002.

Leo Breiman. Probability. Addison Wesley, 1968.

Leo Breiman. Probability. SIAM, 1992.

Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81:637–659, 1973.
The Nobel prize winning classic paper. The one that started it all by laying the foundation for modern valuation of derivatives.

Andrei N. Borodin and Paavo Salminen. Handbook of Brownian Motion – Facts and Formulae. Birkhäuser, 1996.
From the back cover: ``The purpose of the book is to give an easy reference to a large number of facts and formulae associated to Brownian motion. The book consists of two parts. The first one– theory part – is devoted to properties of linear diffusions in general and Brownian motion in particular. Results are given mainly without proofs. The second one– formula part– is a table of distributions of functionals of Brownian motion and related processes. The collection contains more than 1500 numbered formula. The book is of value as basic reference material to researchers, graduate students, and people doing applied work with Brownian motion, It can also be be used as a source of explicit examples when teaching stochastic processes.''

R. Buff. Uncertain Volatility Models – Theory and Application. Springer Verlag, 2002.

Sid Browne and Ward Whitt. Portfolio choice and the Bayesian Kelly criterion. Adv. in Appl. Probab., 28(4):1145–1176, 1996. Kelly criterion.

Z. Brezniak and T. Zastawniak. Basic Stochastic Processes. Springer Undergraduate Mathematics Series. Springer, 1st edition edition, 1999. 3-540-76175-4.

Andrew J. G. Cairns. Interest Rate Models. Princeton University Press, 2004.
From the advertising blurb: `` dots interest rate markets and bond markets, being much richer in structure than equity-derivative models, are particularly fascinating and complex. This book introduces the tools required for the arbitrage-free modeling of the dynamics of these markets. Andrew Carins covers not only seminal works but also modern developments.''

James Case. Can science outperform the shamans in global financial markets? SIAM News, pages 9–10, October 2005.
A book review of The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward by B. Mandelbrot and R. Hudson. From the review: Mandelbrot and Hudson have produced an eminently readable – if somewhat quirky – book about risk, reward and modern theories of finance.

James Case. Two theories of relativity, all but identical in substance. SIAM News, page 7 ff, September 2005. popular history.

A. B. Clarke and R. L. Disney. Probability and Random Processes, Second Edition. John Wiley and Sons, 1985.

[ u Cer03]
Ale u s u Cern y. Mathematical Techniques in Finance. Princeton University Press, 2003.
From the advertising blurb: `` dots offers a carefully crafted blend of numerical applications and theoretical grounding in economics, finance, and mathematics. In the best engineering tradition, Ale u s u Cern y mixes tools from calculus, linear algebra, probability theory, numerical mathematics, and programming to analyze in an accessible way the most intriguing problems in financial economics.

H. Crauel and M. Gundlach. Stochastic Dynamics. Springer, 1999.

Alexandre J. Chorin and Ole H. Hald. Stochastic Tools in Mathematics and Science. Surveys and Tutorials in the Applied Mathematical Sciences. Springer-Verlag, 2006.
Form the publisher's blurb: Stochastic Tools in Mathematics and Science is an introductory book on probability based modeling. It covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics covered include conditional expectations, stochastic processes, Brownian motion and its relation to partial differential equations, Langevin equations, the Liouville and Fokker-Planck equations, as well as Markov chain Monte Carlo algorithms, renormalization and dimensional reduction, and basic equilibrium and non-equilibrium statistical mechanics. The applications include data assimilation, prediction from partial data, spectral analysis and turbulence. A noteworthy feature of the book is the systematic analysis of memory effects. The presentation is mathematically attractive and should form a useful bridge between theoretical treatments familiar to mathematical specialists and the more practical questions raised by specific applications.

S. J. Chapman. The Kelly criterion for spread bets. IMA J. Appl. Math., 72(1):43–51, 2007. Kelly criterion.

Kai-Lai Chung. Green, Brown, and Probability and Brownian Motion on the Line. World Scientific, 2002.

P. Cizek, W. Härdle, and R. Weron. Statistical Tools for Finance and Insurance. Springer, 2004.
From the publisher's description: ``Statistical Tools for Finance and Insurance presents ready to use solutions, theoretical developments and method construction for many practical problems in quantitative finance and insurance The book provides the tools, instruments and online algorithms for recent techniques in quantitative finance and modern treatments in insurance calculations. Te book covers topics such as heavy tailed distributions, implied trinomial trees, pricing of CAT bonds, simulation of risk processes, and ruin probability approximation. This book presents modern tools for quantitative analysis in finance and insurance. It provides a smooth introduction into advanced techniques applicable to a wide range of practical problems.''

Sasha Cygnanowski, Peter Kloeder, and Jerzy Ombach. From Elementary Probability to Stochastic Differential Equations with Maple. Universitext. Springer Verlag, 2003.
From the back description of the book: ``The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduates and graduate students, not necessarily in mathematics, providing an overview and intuitive background for more advanced studies as well as some practical skills in the use of Maple in the context of probability and its applications. Although the book contains definitions and theorems, it differs from conventional mathematics books in its use of Maple worksheets instead of formal proofs to enable the reader to gain an intuitive understanding of the ideas under consideration.`` Chapter 7 covers Stochastic Calculus, Stochastic Integrals, Stochastic Differential Equations, Stochastic Chain Rule and Ito Formula. The book also covers Stratanovich Stochastic Calculus.

S. Cetinkaya and M. Parlar. Optimal nonmyopic gambling strategy for the generalized Kelly criterion. Naval Res. Logist., 44(7):639–654, 1997. Kelly criterion.

L. Clewlow and C. Strickland. Energy Derivatives: Pricing and Risk Management. Lacima Publications, 2000.

R. Cont and P. Tankov. Financial Modeling with Jump Processes. Chapman and Hall/CRC, 2004.

R. A. Carmona and M. R. Tehranchi. Interest Rate Models: an Infinite Dimensional Stochastic Process Perspective. Springer Finance. Springer Verlag, 2006.
This book studies the mathematical issues that arise in modeling the interest rate term structure. These issues are approached by casting the interest rate models as stochastic evolution equations in infinite dimensional spaces. The book is comprised of three parts. Part I is a crash course on interest rates, including a statistical analysis of the data and an introductions to the some popular interest rate models. Part II is a self-contained introduction to infinite dimensional stochastic analysis, including SDE in Hilbert spaces and Malliavin calculus. Part III presents some recent results in interest rate theory. including finite dimensional realizations of HJM models, generalized bond portfolios, and the ergodicity of HJM models.

K. L. Chung and J. B. Walsh. Markov Processes, Brownian Motion, and Time Symmetry, volume 249 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 2005.
``The authors' aim is to present some of the best features of Markov processes, an n particular of Brownian motion with a minimum of prerequisites and technicalities. The volume is very useful for people who wish to learn Markov processes.'' adapted from a review by H. J. Englebert on MathSciNet

M. Carpinski and T. Zasawniak. Mathematics for Finance. Undergraduate Mathematics. Springer Verlag, 2003.
From a review in Zentrallblatt Math serving as the publisher's blurb: ``This text is an excellent introduction to mathematical finance. Armed with knowledge of basic calculus and probability can use this book to learn about derivatives, interest rates, and their term structure and portfolio management. The test serves as an easily understood introduction to the economic concepts but also manages to cover the topics in a mathematically rigorous manner. The book contains many worked examples and exercises and would make a useful textbook for a first course in financial mathematics.'' See also the review cited in [protter04monthly]

Davis. The Math of Money. Springer Verlag, 2001.
Need more information, Have only the barest details.

Mark Davis and Alison Etheridge, editors. Louis Bachelier's Theory of Speculation. Princeton University Press, 2006.
March 29, 1900 is considered by many to be the day mathematical finance was born. On that day a French doctoral student, Louis Bachelier, successfully defended his thesis emph Theórie de la Spéculation at the Sorbonne. This book provides a new translation with commentary and background of Bachelier's work.

G. DePrato. An Introduction to Infinite-Dimensional Analysis. Universitext. Springer, 2006.

Emanuel Derman. My Life as a Quant: Reflections on Physics and Finance. John Wiley and Sons, 2004.
Autobiography of a theoretical physicist turned serious programmer, turned Wall Street quantitative finance wizard. FROM THE PUBLISHER My Life as a Quant is Derman's account of his search for answers as he undergoes his transformation from ambitious young scientist to managing director. His book is simultaneously wide-ranging and personal. He tells the story of his passage between two worlds; he recounts his adventures with physicists, quants, options traders, and other highfliers on Wall Street; he analyzes the incompatible personas of traders and quants; and he meditates on the dissimilar natures of knowledge in physics and finance. Throughout his tale, he reflects on the appropriate way to apply the refined methods of physics to the hurly-burly world of markets. SYNOPSIS Wall Street is no longer the old-fashioned business it once was. In recent years, investment banks and hedge funds have increasingly turned to quantitative trading strategies and derivative securities for their profits, and have raided academia for PhDs to model these volatile products and manage their risk. Nowadays, the fortunes of firms and the stability of markets often rest on mathematical models. ``Quants'' the scientifically trained practitioners of quantitative finance who build these models have become key players on the Wall Street stage. And no Wall Street quant is better known than Emanuel Derman. One of the first high-energy particle physicists to migrate to Wall Street, he spent seventeen years in the business, eventually becoming managing director and head of the renowned Quantitative Strategies group at Goldman, Sachs & Co. There he coauthored some of today's most widely used and influential financial models. Physics and quantitative finance look deceptively similar. But, writes Derman, ``When you do physics you're playing against God; in finance, you're playing against God's creatures.'' How can one justify using the precise methods of physics in the frenzied world of financial markets? Is it reasonable to treat the economy and its markets as a complex machine? Or is quantitative finance merely flawed thinking masquerading as science, a brave whistling in the dark? My Life as a Quant is Derman's entertaining and candid account of his search for answers as he undergoes his transformation from ambitious young scientist to managing director. His book is simultaneously wide-ranging and personal. He tells the story of his passage between two worlds; he recounts his adventures with physicists, quants, options traders, and other highfliers on Wall Street; he analyzes the incompatible personas of traders and quants; and he meditates on the dissimilar natures of knowledge in physics and finance. Throughout his tale, he reflects on the appropriate way to apply the refined methods of physics to the hurly-burly world of markets. My Life as a Quant is a unique first-person story and a perceptive and revealing exploration of the quantitative side of Wall Street.

D. J. Daley and J. Gani. Epidemic Modelling: An Introduction. Cambridge University Press, 1999.

Persi Diaconis, Susan Holmes, and Richard Montgomery. Dynamical bias in the coin toss. url http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf, December 2004. coin toss, statistics, bias.

Sean Dineen. Probability Theory in Finance: A mathematical Gide to the Black-Scholes Formula, volume 70 of Graduate Studies in Mathematics. American Mathematical Society, 2005.

R.-A Dana and M. Jeanblanc-Picque. Financial Markets in Continuous Time. Springer Verlag, 2003.

Richard de Neufville. Real options: Dealing with uncertainty. real options.

J. L. Doob. What is a martingale? American Mathematical Monthly, 78(5):451–463, May 1971. History, martingale.
A historical and expository article by the author of the area. The article is not necessarily easy reading. The emphasis is theoretical rather than practical, a few non-trivial non-elementary examples.

J. L. Doob. The development of rigor in mathematical probability. American Mathematical Monthly, 103(7):586–595, August-September 1996. History.

M. d. r. Grossinho, A. N. Shirayev, and P. E. Oliveira. Stochastic Finance. Springer Verlag, 2006.
Since the pioneering work of Black, Scholes, and Merton in the field of financial mathematics, research has led to the rapid development of a substantial body of knowledge, with plenty of applications to the common function of the world's financial institutions. The high-tech character of modern business has increased the need for advanced methods, which rely to a large extent on mathematical techniques. It has become essential for the financial analyst to posses a high degree of proficiency in these mathematical techniques. The essays in Stochastic Calculus describe many of these techniques.

Freddy Delbaen and Walter Schachermayer. What is a dots free lunch. Notices of the American Mathematical Society, 51(5), 2004.
A very readable article that summarizes the current state of mathematical theory behind option pricing. The majority of the article is an explanation of the no arbitrage principle, followed by the derivation of the risk-neutral martingale measure for a single period binomial model for option pricing. The last few paragraphs of the article relate that example to the modern theory of equivalent martingale measures, stochastic integration and modern extensions of arbitrage, called a ``free lunch''.

F. Delbaen and W. Schachermayer. The Mathematics of Arbitrage. Springer Finance. Springer, 2005.
From the publisher's description: ``This book aims at a rigorous mathematical treatment of the theory of pricing and hedging of derivative securities by the principle of 'no arbitrage'. The first part presents a relatively elementary introduction, restricting itself to the case of finite probability spaces. The second part consists of an updated edition of seven original research papers by the authors, which analyzes the topic in the general framework of semi-martingale theory.

R. Elliott and P. E. Kopp. Mathematics of Financial Markets. Springer Verlag, 1999.

R. J. Elliott and P. E. Kopp. Mathematics of Financial Markets. Springer Finance. Springer, 2nd edition edition, 2005.
From the publisher's description: ``This book presents the mathematics that underpins pricing models for derivative securities, such as options,futures and swaps. The idealized continuous-time models built upon the famous Black-Scholes theory require sophisticated mathematical tools drawn from modern stochastic calculus. However, many of the underlying ideas can be explained more simply within a discrete time framework. This is developed extensively in this substantially revised second edition to motivate the technically more demanding continuous-time theory, which includes a detailed analysis of the Black-Scholes model and its generalizations, American put options, term structure models and consumption-investment problems. The mathematics of martingales and stochastic calculus is developed where it is needed. The new edition adds substantial material from current areas of active research, notably, a new chapter on coherent risk measures, with applications to hedging, a complete proof of the first fundamental theorem of asset pricing for general discrete market models, the arbitrage interval for incomplete discrete-time markets, characterization of characterization of complete discrete-time markets, using extended models of risk and return, and sensitivity analysis for the Black-Scholes models. The treatment remains careful and detailed rather than comprehensive with a clear focus on options. The test should prove useful to graduates with a sound mathematical background, ideally a knowledge of elementary concepts from measure-theoretic probability, who wish to understand the mathematical models on which the bewildering multitude of current financial instruments used in derivative markets and credit institutions is based. The first edition has been used successfully in a wide range of Master's programs in mathematical finance and this new edition should prove even more popular in this expanding market. It should be equally useful to risk managers and practitioners looking to master the mathematical tools needed for modern pricing and hedging techniques.''

Alison Etheridge. A Course in Financial Calculus. Cambridge University Press, 2002.

A. Eydeland and K. Wolnyiec. Energy and Power Risk Management - New Developments in Modeling Pricing and Hedging. Wiley Finance, 2003.

William Feller. An Introduction to Probability Theory and Its Applications, Volume I, Third Edition, volume I. John Wiley and Sons, third edition edition, 1973. QA 273 F3712.

M. R. Fengler. Semiparametric Modeling of Implied Volatility. Springer Finance. Springer Verlag, 2005.
The implied volatility surface is a key financial variable for the pricing and risk management of plain vanilla and exotic options alike. Consequently, statistical models for the implied volatility surface are of immediate importance in practice, they may appear as estimates of the current surface or as fully specified dynamic models describing its propagation throughout space and time. This book fills a gap in the financial literature by bringing together both recent advances in the theory of implied volatility and refined semi-parametric estimation strategies and dimension reduction methods for functional surface: the first part of the book is devoted to smile-consistent pricing approaches. The theory of implied and local volatility is presented concisely and the vital smile-consistent modeling approaches such as implied trees, mixture diffusion, or stochastic implied volatility models are discussed in detail. The second part of the book familiarizes the read with estimation techniques that are natural candidates to meet the challenges in implied volatility modeling, such as the rich functional structure of observed implied volatility surfaces and the necessity for dimension-reduction, non- and semi-parametric smoothing techniques. The book introduces Nadaraya-Watson, local polynomial and least squares kernel smoothing and dimension reduction methods such as common principle components, functional principle components models and dynamic semi-parametric factor models. Throughout, most methods are illustrated with empirical investigations, simulations and pictures.

E. Robert Fernholz. Stochastic Portfolio Theory. Applications of Mathematics. Springer-Verlag, 2002. UNL HG4529.5.F47 2002.
From the advertising blurb: ``This book is an introduction to stochastic portfolio theory for investment professionals and for students of mathematical finance. Each chapter includes a number of problems of varying levels of difficulty and a brief summary of the principal results of the chapter with proof.'' I have not seen this book.

J. Franke, J. Härdle, and C. M. Hafner. Statistics of Financial Markets. Universitext. Springer-Verlag, 2004.
From the publisher's blurb: ``Statistics of Financial Markets presents in a vivid yet concise style the necessary statistical and mathematical background for financial engineers and introduces the main ideas in mathematical finance and financial statistics. Topics covered include option valuation, financial time-series analysis, value-at-risk, copulas, statistics of the extremes, The underlying structure of the book, (basic tools in mathematical finance, financial time series analysis, and applications to given problems of financial markets) allows it to be used as a basis for lectures, seminars and even crash courses on the topic.''

B. Fristedt, N. Jain, and N. Krylov. Filtering and Prediction: A Primer, volume 38 of Student Mathematical Library. American Mathematical Society, 2007.

Jin Feng and Thomas G. Kurtz. Large Deviations for Stochastic Processes, volume 131 of Mathematical Surveys and Monographs. American Mathematical Society, 2006.
This book is devoted to the results on large deviations for a class of stochastic process. Part I gives necessary and sufficient conditions for tightness that are analogous to conditions for tightness in the theory of weak convergence. Part 2 focuses on Markov processes in metric spaces. For a sequence of such processes, convergence of Fleming's logarithmically transformed nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence semigroups in weak convergence. Part 3 discusses methods for verifying the comparison principle for viscosity solutions and applies the theory to obtain a variety of new and known results on large deviations for Markov processes.

David Freedman. Brownian Motion and Diffusions. Holden-Day, 1971. QA274.75F74.

Frenkel. Risk Management. Springer, 2005.

Joseph Glaz, Martin Kulldorff, Vladimir Pozdnyakov, and J. Michael Steele. Gambling teams and waiting times for patterns in two-state markov chains. Gambling, teams, waiting times, patterns, success runs, failure runs, Markov chains, martingales, stopping times, generating functions.

M. Gundlach and F. Lehrbass. CreditRisk+ in the Banking Industry. Springer Finance. Springer-Verlag, 2004.
From the publisher's blurb: ``CreditRisk+ is an important and widely-implemented default-mode model of portfolio credit risk, based on a methodology borrowed from actuarial mathematics. The book gives an account of the status quo as well as of new and recent development of the credit risk model CreditRisk+ which is widely used in the banking industry. It gives an introduction to the model itself, and to its ability to describe, manage and price credit risk.

P. Glasserman. Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability. Springer Verlag, 2003.

P. Glasserman. Monte Carlo Methods in Financial Engineering. Applications of Mathematics. Springer-Verlag, 2003.

Michael B. Gordy. A comparative anatomy of credit risk models. credit risk, December 1998.

Mircea Grigoriu. Stochastic Calculus: Applications in Science and Engineering. Birkhauser, 2003.
From the advertising blurb: ``The applications range from climate dynamics over materials sciences and mechanics to pattern formation, physics and finance.''

Charles M. Grinstead and J. Laurie Snell. Introduction to Probability. American Mathematical Society, 1997.

Geoffrey Grimmett and David Stirzaker. One Thousand Exercises in Probability. Oxford University Press, 2001.
From the advertising blurb: ``This is an updated and greatly expanded version of an already well-established and popular exercise manual. It provides a wide-ranging selection of illuminating, informative and entertaining problems, together with their solution. Topics include modeling and many applications of probability theory as well as theoretical aspects. There are questions at all ability levels, the majority being of elementary or intermediate standard.

Geoffrey Grimmett and David Stirzaker. Probability and Random Processes. Oxford University Press, 2001.
From the advertising blurb: ``The third edition of this well-established and popular textbook provides a wide-ranging and entertaining introduction to probability and random processes and many of their practical applications The emphasis is on modelling and understanding rather than abstraction, but beginners will encounter aspects of more advanced work. The book is largely self-contained. Many important random processes are developed in the text and through informative real-life examples.

Max Gunzburger. Numerical methods for stochastic PDES. SIAM News, 40(5), June 2007. Stochastic PDEs, survey.

H. Geman and O. Vasicek. Plugging into electricity. Risk, pages 93–97, August 2001.
Develops a new formalism for pricing electricity derivatives( and generally non-storable commodities, for example bandwidth) from the futures or forward curve.

Geoffrey Grimmet and Dominic Welsh. Probability: An Introduction. Oxford University Press, 1986.

Floyd B. Hanson. Applied Stochastic Processes and Control for Jump Diffusions: Modeling, Analysis and Computation. Number 13 in Advances in Design and Control. SIAM, 2007.

Espen Gaarder Haug. The Complete Guide to Option Pricing Formulas. McGraw-Hill, 1998. LC Number: HG 6024 A3H38.
From the introduction: ``This collection of options pricing formulas s not intended as a textbook in options pricing theory but rather as a reference book for those who are already familiar with basic finance theory. To better illustrate the use and implementation of options pricing formulas, included are examples of programming codes for several of them. The book differs from other texts on options pricing in the the accompanying text is cut to the bone. The collection of options pricing formulas is an ideal supplement for quant traders, financial engineers, students taking courses in options pricing theory, or anyone else working with financial options.'' To me, this is an easy to use and easy to read reference book, with good examples, and some useful pseudo-code.

Ronald Huisman and Cyriel de Jong. Option pricing for power pricing with spikes. Energy Risk, February 2003. energy risk.

David Heath. Introduction to models for the evolution of term structure of interest rates. In David C. Heath and Glen Swindle, editors, Introduction to Mathematical Finance, volume 57 of Proceedings of Symposia in Applied Mathematics, pages 59–64. American Mathematical Society, American Mathematical Society, 1999. interest rate rate models.

Fern Hunt and et al. An optimization approach to multiple sequence alignment. Appl. Math. Lett., 16(5):785–790, 2003. optimization, genomics, bioinformatics.

T. Hida. Brownian Motion. Springer-Verlag, 1980.

Desmond J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3):525–546, 2001.

Desmond J. Higham. An Introduction to Financial Option Valuation. Cambridge University Press, 2004. stochastic differential equations.
From the advertising blurb: The book includes many figures and examples, and well as computations based on real stock market data, Each chapter comes with an accompanying stand-alone MATLAB code to illustrate a key idea.

David Heath, Robert Jarrow, and Andrew Morton. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60(1):77–105, January 1992. interest rate models.

P. J. Hunt and J. E. Kennedy. Financial Derivatives in Theory and Practice. John Wiley and Sons, 2000. HG 6024 A3H86 2000.

Fern Y. Hunt, Anthony J. Kearsley, and Agnes O'Gallagher. A linear programming based algorithm for multiple sequence alignment. Proceedings of the Computational Systems Bioinformatics, 2003. optimization, genomics, bioinformatics.

W. Haerdel, T. Kleinow, , and G. Stahl. Applied Quantitative Finance. Springer Verlag, 2002.

S. D. Howison, F. P. Kelly, and P. Wilmott, editors. Mathematical Models in Finance. The Royal Society, Chapman and Hall, 1995. HF 332, M384 1995.

David C. Heath and Glen Swindle, editors. Introduction to Mathematical Finance, volume 57 of Proceedings of Symposia in Applied Mathematics, Providence, RI, 1997. American Mathematical Society, American Mathematical Society. QA 1 P72 v.57.

John C. Hull. Options, Futures, and other Derivative Securities. Prentice-Hall, second edition edition, 1993. economics, finance, HG 6024 A3H85.

J. C. Hull and A. White. Efficient procedures for valuing european and american path-dependent options. Journal of Derivatives, pages 21–31, Fall 1993.
Introduces a method to extend lattice-based contingent claim valuation techniques to problems with path-dependent payoffs and American-style exercise rights.

J. C. Hull and A. White. Numerical procedures for implementing term-structure models I: Single factor models. Journal of Derivatives, pages 7–16, Fall 1994.
Introduces a simple and computationally efficient trinomial lattice for constructing term-structure models based on a spot rate.

K. Ito and H. McKean. Diffusion Processes and their Sample Paths. Number XVI in Classics in Mathematics. Springer, 1974.

Kiyosi Ito. Essentials of Stochastic Processes, volume 231 of Translations of Mathematical Monographs. American Mathematical Society, 2006.
The famous Japanese mathematician Kiyosi Ito has made important contributions through his wok on stochastic integrals and other areas of probability theory including additive processes, stationary processes and Markov processes. This English translation of an Ito textbook includes updates to his discussion of the most important classes of stochastic processes. This classical treatment of the subject outlines Ito's approach to the theory of stochastic processes. It is the only English translation of an Ito textbook that includes an introduction to stationary processes. A chapter is devoted to the one-dimensional diffusion theory. significant as a basic prototype of the study of Markov processes.

Jacod. Probability Esentials. Springer, 2004.

Robert Jarrow. In honor of the nobel laureates robert C. merton and myron S. scholes: A partial differential equation that changed the world. Journal of Economic Perspectives, 13(4):229–248, Fall 1999. popular history.

N. F. Johnson, P. Jeffries, and Pak Ming Hui. Financial Market Complexity. Oxford Finance Series. Oxford University Press, 2003.
From the publisher's description: ``This book takes a fresh look at understanding how financial markets behave. Using recent ideas from the highly topical science of complexity and complex systems, the book provides the basis for a unified description of how today's markets really work.''

Norman L. Johnson and Samuel Kotz. Continuous Univariate Distributions-1. John WIley and Sons, 1970.
An old favorite of mine, because it categorizes and summarizes exotic special distributions. Many of the distributions arise naturally in applications, which is what led me to this book originally, and I liked it. Many of the distributions rely on special functions. Note that this is a companion book to several other similar books by Johnson and Kotz. The chapters are Continuous distributions, Normal distributions, Lognormal, Inverse Gaussian (Wald) Distributions, Cauchy Distributions, Gamma Distributions (including Chi-Square), Exponential Distributions, Pareto Distributions, Weibull Distributions and Extreme Value Distributions. QA 273.6 J6 1970x

J Jannssen and R. Manca. Applied Semi-Markov Processes. Springer Verlag, 2006.
This book aims to give the reader the tools necessary to apply semi-Markov processes in real life problems. The book is self-contained and starting from a low level of probability concepts gradually brings the reader to a deep knowledge of semi-Markov processes. It presents homogeneous and non-homogeneous processes as well as Markov and semi-Markov rewards processes. The concepts are fundamental for many applications, but they are not as thoroughly presented in other books on the subject as they are here.

M. Joshi. C++ Design Patterns and Derivatives Pricing. Cambridge University Press, 2004.
Contents: 1. A simple Monte Carlo model, 2. Encapsulation, 3. Inheritance and virtual functions, 4. Bridging with virtual constructors, 5. Strategies, decorations and statistics, 6. A random numbers class, 7. An exotics engine and the template pattern, 8. Trees, 9. Solvers, templates and implied vols, 10. The factory, 11. Design patterns revisited, Appendix A: Black-Scholes Formulas, Appendix B. Distribution functions, Appendix C. A simple array class, Appendix D. The code.

M. Joshi. The Concepts and Practice of Mathematical Finance. Cambridge University Press, 2004.
Contents: 1: Risk, 2. Pricing methodologies and arbitrage, 3. Trees and option pricing, 4. Practicalities, 5. The Ito calculus, 6. Risk neutrality and martingale measures, 7. The practical pricing of a European option, 8. Continuous barrier options, 9. Multi-look exotic options, 11. Multiple sources of risk, 12. Options with early exercise features, 13. Interest rate derivatives, 14. The pricing of exotic interest rate derivatives, 15. Incomplete markets and jump-diffusion processes, 16. Stochastic volatility, 17, Variance gamma models, 18. Smile dynamics and the pricing of exotic options, Appendix A. Financial and mathematical jargon, Appendix B. Computer Projects, Appendix C. Elements of Probability theory, Appendix D. Hints and Answers to questions.

M. Joshi. The Concepts and Practice of Mathematical Finance. Cambridge University Press, 2004.

R. Jarrow and P. Protter. A short history of stochastic integration and mathematical finance the early years, 1880-1970. In IMS Lecture Notes, volume 45 of IMS Lecture Notes, pages 75–91. IMS, 2004. popular history.

Eric Jondeau, Ser-Huang Poon, and Michael Rockinger. Financial Modeling under Non-Gaussian Distributions. Springer Finance. Springer, 2007.
Non-Gaussian distributions are the key theme of this book which addresses the causes and consequences of non-normality and time dependency in both asset returns and option prices. The aim is to bridge the gap between theoretical developments and the practical implementations of what many users and researchers perceive as sophisticated models. The emphasis throughout is is on practice. There are abundant empirical illustrations of the models and techniques described, may of which could be equally applied to other financial time series, such as exchange and interest rates.

P. Jaillaet, E. I. Ronn, and S. Tompaidis. Valuation of commodity-based swing options. Management Science, 2003.
In depth paper about valuation of gas swing in a commercial setting. Presents a single-factor mean-reverting spot price model that the authors attempt to calibrate to a set of futures and options prices in the US natural gas market. As well as a number of technical issues, it emphasizes some real-world technical and legal issues.

Jessica James and Nick Webber. Interest Rate Modeling. John Wiley and Sons, 2001.

Ioannis Karatzas. Lectures on the Mathematics of Finance, volume 8 of CRM Monograph Series. American Mathematical Society, 1997.

J. B. Keller. The probability of heads. American Mathematical Monthly, 93(3):91–197, March 1986.
The renowned applied mathematician J. B. Keller investigated coin flips in this article. He assumed a circular coin with negligible thickness flipped from a given height y-0 = a >0, and considered its motion both in the vertical direction under the influence of gravity, and its rotational motion imparted by the flip until it lands on the surface y = 0. The initial conditions imparted to the coin flip are the initial upward velocity and the initial rotational velocity. Under some additional simplifying assumptions Keller shows that fraction of flips which land heads approaches 1/2 if the initial vertical and rotational velocities are high enough. Actually, Keller shows more, that for high initial velocities there are very narrow alternate bands of initial conditions which inevitably and deterministically lead to heads and tails. Now we see that the randomness comes from the choice of initial conditions. Because of the narrowness of the bands of initial conditions, very slight variations of initial upward velocity and rotational velocity lead to different outcomes. A very readable article.

Valery Kholodnyi. Valuation and hedging of european contingent claims on power with spikes: a non-markovian approach. Journal of Engineering Mathematics, 49:233–252, 2004.

Valery A. Kholodnyi. Valuation and hedging of power-sensitive contingent claims for power with spikes: A non-markovian approach. url http://www.riskx.net/VKholodnyiQuantitative.ppt, February 2004. energy risk, slides, accessed March 2008.

Davar Khoshnevisan. Probability, volume 80 of Graduate Studies in Mathematics. American Mathematical Society, 2007.
From the publisher's description: ``This text departs from others designed for graduate level work by presenting a cohesive graduate course in measure-theoretic probability hat has a one-semester student in mind. Early chapter's of the book focus on a proof of the DeMoivre-Laplace central limit theorem and on basic results in analysis. Later chapters discuss Brownian motion, and stochastic integration, including the Ito integral, an important ingredient in the applications of probability to mathematical finance. The author has included numerous exercises of varying difficulty in the text. Each chapter also concludes with historical notes. While this book has ample material to cover a year-long course at a more leisurely pace, it is ideal for a one-semester course in that it focuses on the themes most central to probability.

Ralf Korn and Elke Korn. Option Pricing and Portfolio Optimization, volume 31 of Graduate Studies in Mathematics. American Mathematical Society, 2001.

Fima C. Klebener. Introduction to Stochastic Calculus with Applications. World Scientific, 2005.
From the publisher's advertisement: The second edition contains a new chapter on bonds, interest rates and their options. New materials include worked out examples in all chapters, best estimators, more results on change of time, change of measure, random measures, new results on exotic options, FX options, stochastic and implied volatility, models of the age-dependent branching process and the stochastic Lotka-Volterra model in biology, non-linear filtering in engineering. from a review: It provides a good introduction to stochastic analysis, leaving out several of the more technical proofs. The variety of examples and exercises suggests to use the book for self-study.

W. S. Kendall, F. Liang, and J-S Wang. Markov Chain Monte Carlo: Innovations and Applications. World Scientific, 2005.
Markov Chain Monte Carlo originated in statistical physics, but spilled over into various application areas, leading to a corresponding variety of techniques and methods. The variety stimulates new ideas and developments from many places. This book presents five expository essays by leaders in the field, drawing from perspectives in physics, statistics, and genetics and shows how different aspects of MCMC come to the fore in different contexts.

Robert W. Kolb. Financial Derivatives. Institute of Finance, 1993.

Clifford Konold. Confessions of a coin flipper and would-be instructor. The American Statistician, 49(2):203–209, May 1995. Penney-Ante.
Computer Simulations, Intuitions, Wait Time, Penney-Ante

Emmanuel Kowalski. Bernstein polynomials and brownian motion. American Mathematical Monthly, 113:865–886, December 2006.

I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Springer Verlag, second edition edition, 1997.

Ioannis Karatzas and Steven Shreve. Methods of Mathematical Finance, volume 39 of Applications of Mathematics. Springer Verlag, 1998. (HF 5691 K3382.

L. Koralov and Ya. G. Sinai. Probability Theory, Random Processes, Random Fields. Universitext. Springer Verlag, 2nd edition edition, 2006.
This book consist of two parts. The first, on classical probability theory, contains a detailed analysis of Markov chains, limit theorems, and their relation to Renormalization Group Theory. It also describes several applications of probability theory which are not discussed in most other textbooks. The second part includes the theory stationary random processes, martingales, stochastic integrals, and stochastic differential equations. On section is devoted to the theory of Gibbs random fields. The book contains essential material for an undergraduate or a graduate course in probability and the theory of random processes. It may also serve as a reference for scientists who use modern probability theory in their research.

S. Karlin and H. Taylor. A First Course in Stochastic Processes, Second Edition. Academic Press, 1975.

S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.
Volume 2 of the classic two-volume set from Karlin and Taylor. This volume would be suitable for a beginning graduate-level class on stochastic processes and applied probability. Another excellent model of mathematical good taste, excellent exposition and the right level of mathematical sophistication. Does not require measure theoretic ideas.

H. Kuo. Introduction to Stochastic Integration. Universitext. Springer Verlag, 2006.
The theory of stochastic integration, also called the Ito calculus, has a large spectrum of applications in virtually every scientific area involving random functions, but it can be a very difficult subject for people without much mathematical background. The Ito calculus was originally motivated by the construction of Markov processes from infinitesimal generators. Previously, the construction of such process required several steps, whereas, Ito constructed these diffusions directly in a single step as the solutions of stochastic integral equations associated with the infinitesimal generators. Moreover, the the properties of theses diffusion processes can be derived from the stochastic integral equations and the Ito formula. This introductory textbook on stochastic integration provides a concise introduction to the Ito calculus and covers th following topics: Construction of Brownian motion, Stochastic integrals for Brownian motion and martingales, the Ito formula, Multiple Ito-Wiener integrals. Stochastic differential equations, applications to finance, filtering theory and electric circuits. The reader should have a background in advanced calculus and elementary probability theory as well as a basic knowledge of measure theory and Hilbert spaces. Each chapter ends with a variety of exercises.

H. Kuo. Introduction to Stochastic Integration. Springer, 2006.

Y. K. Kwok. Mathematical Models of Financial Derivatives. Springer Verlag, 1998. HG6024 A3K85.

Y. K. Kwok. Mathematical Models of Financial Derivatives. Springer Finance. Springer, 2nd edition edition, 2005.
From the publisher's description: ``Mathematical Models of financial Derivatives is a textbook on the theory behind modeling derivatives and their risk management, focusing on the valuation techniques that are common to most derivative securities. A wide range of financial derivatives commonly traded in the equity and fixed income markets are analyzed, emphasizing aspects of pricing, hedging, and practical usage. The readers are guided through the texts on new advances in analytic techniques and numerical methods for solving various types of derivative pricing models. In the second edition, more emphasis has been placed on the discussion of Ito's calculus and Girsanov's Theorem and in particular on the concepts f risk neutral measure and equivalent martingale pricing approach. A new chapter on credit risk models and pricing of credit derivatives has been added. Most recent research results and concepts are made accessible to the readers through extensive, well-thought out exercises at the end of each chapter.''

M. Lefebvre. Applied Probability and Statistics. Springer Verlag, 2006.
This textbook is designed for a one-term course on probability and statistics for applied mathematics, engineering and computer science students. The exposition, enriched by many examples form real life, unfolds systematically from an introductory chapter to such topics as random variables and vectors, and stochastic processes as well as formulas quantities sampling distributions in the appendices. There are interesting historical notes, along with numerous exercises and solutions and useful bibliography and index.

Emmanuel Lesigne. Heads or Tails: An Introduction to Limit Theorems in Probability, volume 28 of Student Mathematical Library. American Mathematical Society, 2005.

Shou-Yen Robert Li. A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Annals of Probability, 8(6):1171–1176, December 1980. martingale, Penney-Ante.
From the abstract: ``We apply the concept of stopping times of martingales to problems in classical probability regarding the occurrence of sequence patterns in repeated experiments. For every finite collection of sequences of possible outcomes, we compute the expected waiting time until one of them is observed in a run of experiments. Also we compute the probability for each sequence to be the first to occur. The main result, with a transparent proof is a generalization of some well-known facts on Bernoulli processes including formulas of Feller and the `leading number' algorithm of Conway.''

R. J. Larsen and M. L. Marx. An Introduction to Probability and its Applications. Prentice Hall, 1985.

A. W. Lo and A. C. MacKinlay. A Non-Random Walk Down Wall Street. Princeton University Press, 1999. HG 4915 L6 1999.

Roger Lowenstein. When Genius Failed: The Rise and Fall of Long-Term Capital Management. Random House, 2000. HG 4930 L69 2000.

David G. Luenberger. Investment Science. Oxford University Press, New York, Oxford, 1998.
Major sections include:Deterministic Cash Flow Streams, Single-Period Random Cash Flows, Derivative Securities, and General Cash Flow Streams.As you may guess from the title and the sections, the emphasis is on investments and finance. The science part tends to be mostly operations research methods, for example, dynamic programming applied to backtracking values through binomial trees. HG4515.2.L84 1998

Yuh-Dauh Lyuu. Financial Engineering and Computation: Principles Mathematics and Algorithms. Cambridge University Press, Cambridge, 2002.
HG176.7 L97

Burton Malkiel. A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing. W. W. Norton, 8th edition, 2003.

Benoit Mandelbrot. Fractals and Scaling in Finance. Springer, 2005.
From the advertising blurb: '' This is the first book in the Selecta, the collected works of Benoit Mandelbrot. This volume incorporates his original contributions to finance and is a major contribution to the understanding of how speculative prices vary in time. The chapters consist of much new material prepared for this volume, as well as reprints of his classic papers. Much of this work helps to lay a foundation for evaluating risks in trading strategies. From the contents: Intro; Major Themes; New Methods in statistical economics; Historical Background; States of randomness; self-similarity and self-affinity; rank sized plots; proportional growth and other explanations of scaling; a case against the log distribution; Personal incomes and firm sizes; Random flight on Wall Street; nonlinear forecasts.''

W. Margrabe. The value of an option to exchange one asset for another. Journal of Finance, 33:177–186, March 1978.
Looks at spread options, to buy one asset and sell another simultaneously.

Henry P. McKean. Stochastic Integration. AMS Chelsea Publishing, 1969.
`` dots this volume originally published in 1969, back into print. This well-written book has been used for many years to learn about stochastic integrals. The author starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Ito lemma. The rest of the book is devoted to various topics of stochastic integral equations and stochastic integral equations on manifolds. This classic book is ideal for supplementary reading or independent study. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications.

Meester. A Natural Introduction to Probability Theory. Birkhauser, 2003.
I have looked at this book very briefly, and it looks interesting. It seems to approach probability theory in the way that I do for this course,, that is, intuitively, leading from elementary results towards measure-theoretic axiomatic probability theory. A book worth considering for the course.

Shirin Mehraban. A linear programming based algorithm for multiple sequence alignment by using markov decision processes. undergraduate thesis, University of Maryland, May 2004. optimization, genomics, bioinformatics.

Robert M. Merton. Theory of rational option pricing. Bell Journal of Economics and Management Science, 4:141–183, 1973.
The other Nobel prize winning classic.

Robert C. Merton. Continuous-Time Finance. Blackwell, 1990.
Learn from the master.

Robert C. Merton. Influence of mathematical models in finance on practice: past, present and future. In S. D. Howison, F. P. Kelly, and P. Wilmott, editors, Mathematical Models in Finance. Chapman and Hall, 1995. popular history.

Attilio Meucci. Risk and Asset Allocation. Springer, 2005.
This encyclopedic, detailed exposition spans all the steps of one-period allocation from the foundations to the most advanced developments. Multivariate estimation methods are analyzed in depth, including non-parametric, maximum-likelihood under non-normal hypotheses, shrinkage, robust, and very general Bayesian techniques. Evaluation methods such as stochastic dominance, expected utility, value at risk and coherent measures are thoroughly discussed in a unified setting and applied in a variety of contexts, including prospect theory, total return and benchmark allocation. Portfolio optimization is presented with emphasis on estimation risk, which is tackled by means of Bayesian, resampling and robust optimization techniques. All the statistical and mathematical tools, such as copulas, location-dispersion ellipsoids, matrix-variate distributions, cone programming, are introduced from the basics. Comprehension is supported by a large number of figures and examples, as well as real trading and asset management case studies. At symmys.com the reader will find freely downloadable complementary materials: the Exercise Book; a set of thoroughly documented MATLAB applications; and the Technical Appendices with all the proofs. More materials and complete reviews can also be found at symmys.com.

O. Moeschlin and E. Grycko. Experimental Stochastics in Physics. Springer Verlag, 2006.
Note this is a CD with a booklet. The electronic monograph texit Experimental Stochastics in Physics deals with random experiments on the computer from the field of molecular dynamics. Based on stochastic dynamic models of movement on the computer, macro quantities of interest, such as momentum distribution, temperature, pressure, and angular velocity of fluids are estimated statistically. The standard techniques of stochastic simulation can be found in Experimental Stochastics, (2nd Ed. Springer 2003) This approach yields new and unexpected insights into physical behavior which cannot be established by laboratory physics. The method can also be applied to non-real contexts, where certain principles of real physics are dispensed with. Although supported theoretically, the working method allows the reader to gain experimental skills on a CD-ROM.

Benoit B. Mandelbrot and Richard L. Hudson. The (mis)Behavior of Markets: A Fractal View of Risk, Ruin and Reward. Basic Books, 2004.
From a review in Mathematics Magazine Vol. 78, No. 2, April 2005: ``We celebrate this year the 100th anniversary of Einstein's year of marvelous papers on the size of molecules, light as composed of particles, special relativity theory and the existence of atoms by measuring Brownian motion of particles in solution. Just 5 years earlier, Louis Bachelier had presented his thesis that financial markets can be described by the laws of Brownian motion. Bachelier's ideas sank from sight, but were resurrected half a century later in what has become financial orthodoxy, with its key assumption that price changes in financial markets are normally distributed and independent of past changes. This orthodoxy culminated in the Nobel-prize winning Black-Scholes method of valuing options. But Benoit Mandelbrot, father of fractal geometry, strongly disputes those assumptions, and asserts that price changes are correlated (with a long memory) and follow a power law of scaling (longer tails). The results he says are that ``trouble runs in streaks'', markets are riskier than portrayed, and financial ``bubbles'' are inevitable. He gives data to support his claims and offers a ``multifractal'' model instead. His new book is eminently readable and engaging, the older book reprints the original papers on which his theories of scaling in finance are based.'' From a book review in SIAM News by James Case October, 2005, page 7: ``Mandelbrot and Hudson have produced an eminently readable – if somewhat quirky – book about risk, reward and modern theories of finance. dots Mandelbrot observes that the modern theory of finance is largely the work of five men: Louis Bachelier, Harry Markowitz, William Sharpe, and the team of Fisher Black and Myron Scholes. He devotes an entire chapter to Bachelier, another to the other four, and a third to his case against the theory he attributes mainly to the five. Of them all, he clearly finds Bachelier the most sympathetic.

Thomas Mikosch. Elementary Stochastic Calculus, with Finance in View. World Scientific, 1998.
Contents: Basic Concepts from Probability Theory, Stochastic Processes, Brownian Motion, Conditional Expectation, Martingales, The Riemann and Riemann-Stieltjes Integrals, The Ito Integral, The Stratonovich and Other Integrals, Deterministic Differential Equations, Ito Stochastic Differential Equations, The General Linear Differential Equation, Numerical Solution, The Black-Scholes Option Pricing Formula, A Useful Technique, Change of Measure, Modes of Convergence, Inequalities, Non-Differentiability and Unbounded Variation of Brownian Sample Paths, Proofs of Existence of the General Ito Stochastic Integral, The Radon-Nikodym Theorem, Proof of the Existence and Uniqueness of the Conditional Expectation.

Thomas Mikosch. Non-Life Insurance Mathematics. Universitext. Springer Verlag, 2004.
From the advertising blurb: ``This book offers a mathematical introduction to non-life insurance and at the same time, to a multitude of applied stochastic processes. It gives detailed discussions of the fundamental models for claim sizes, claim arrivals, total claim amount, and their probabilistic properties. Throughout the book the language of stochastic processes is used fro describing the dynamics of an insurance portfolio in claim size space and time. What makes the book special are more than 100 figures illustrating and visualizing the theory. Every section ends with extensive exercises. They are an integral part of this course, since they support the access to the theory. The book can serve as either a text for an undergraduate/graduate course on non-life insurance mathematics or applied stochastic processes.'' I have not seen this text, but it sounds interesting as a possible source for examples in stochastic processes.

R. Koch Median and S. Merino. Mathematical Finance and Probability: Discrete Introduction. Birkhauser, 2002.

Marek Musielaa and Marek Rutkowski. Martingale Methods in Financial Modeling. Springer Verlag, 1997. (HG6024 A3M87.

Marek Musiela and Marek Rutkowski. Martingale Methods in Financial Modelling, volume 36 of Applications of Mathematics. Springer-Verlag, second edition edition, 2005.
From the advertising blurb: ``This long-awaited new edition of an outstandingly successful well-established book, concentrating on the most pertinent and widely accepted modeling approaches, provides the reader with a text focused on practical rather than theoretical aspects of financial modeling. In the second edition, some sections of Part I are omitted for better readability and a brand new chapter is devoted to volatility risk. As a consequence, hedging of plain-vanilla options and valuation of exotic options is no longer limited to the Black-Scholes framework with constant volatility. The theme of stochastic volatility also reappears systematically in the second part of the book, which has been revised fundamentally, presenting much more detailed analyses of the various interest-rate models available: the author's perspective throughout is that the choice of a model should be based on the reality of ow a particular sector of the financial market functions, never neglecting to examine liquid primary and derivative assets and identifying the sources trading risk associated.''

Daniel J. Maki and Maynard Thompson. Mathematical Model and Applications. Prentice Hall, 1973.

P Malliavin and A. Thalmaier. Stochastic Calculus of Variation in Mathematical Finance. Springer Finance. Springer, 2005.
From the publisher's description: ``Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous path stochastic processes. The calculus provides formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for mathematical finance starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formula provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. Weak convergence of numerical integration of SDE is interpreted as a functional belonging to a Sobolev space of negative order. Insider information is expressed as an infinite dimensional drift. The last chapter gives an introduction to same objects in the context of jump processes where incomplete markets appear.''

R. Mansuy and M. Yor. Aspects of Brownian Motion. Universitext. Springer Verlag, 2006.
Stochastic Calculus and excursion theory are very efficient tools to obtain either exact or asymptotic results about Brownian motion and related processes. The emphasis of the book is on special classes of Brownian functionals like: Gaussian subspaces of the Gaussian space of Brownian Motion,, Brownian quadratic functionals, Brownian local times, Exponential functionals of Brownian motion with drift, Winding number of one or several points or a straight line or a curve, Time spent by Brownian motion below a multiple of its one-sided supremum. Besides it obvious audience of students and lecturers the book also address the interests of researchers from core probability theory out to applied fields such as polymer physics and mathematical finance.

Leonard C. MacLean and William T. Ziemba. Growth versus security tradeoffs in dynamic investment analysis. Ann. Oper. Res., 85:193–225, 1999. Stochastic programming. State of the art, 1998 (Vancouver, BC) Kelly criterion.

S. N. Neftci. An Introduction to the Mathematics of Financial Derivatives. Academic Press, 2000. HG 6024 A3N44 2000.

Roger B. Nelsen. How close are pairwise and mutual independence? From the Introduction: ``The important distinction between pairwise independence and mutual independence of n ge 3 is difficult to grasp.'' This paper compares the distributions of pairwise independent continuous random variables to the distribution of mutually independent random variables., January 2008.

Siu-Ah Ng. Hypermodels in Mathematical Finance: Modeling via Infinitesimal Analysis. World Scientific, 2003.

R. S. Nickerson. Penney-ante: Counterintuitive probabilities in coin-tossing. UMAP Journal, 28(4), Winter 2007. Penney Ante, referee.

Lars Tyge Nielsen. Pricing and Hedging of Derivative Securities. Oxford University Press, 1999.
The theory pricing and hedging of derivative securities is mathematically sophisticated. This book is an introduction to the use of advanced probability theory in financial economics, presenting the mathematics in a precise and rigorous manner. dots enables the reader to approach the literature with confidence, apply the methods to new problems or to do original research in the field.

D. Nualart. The Malliavin Calculus and Related Topics. Probability and Its Applications. Springer Verlag, 2nd edition edition, 2006.
The Malliavin calculus (or stochastic calculus of variation) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to Hörmander's ``sum of squares'' theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on Wiener space and then uses this to establish the regularity of probability laws and to prove Hörmander's theorem. The regularity of the law of stochastic partial differential equations driven by space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin calculus with the anticipating stochastic calculus, and the Markov property of solutions of stochastic differential equations with boundary conditions. The second edition of this monograph includes recent applications of the Malliavin calculus in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.

Andrei Okounov. Limit shapes, real and imaginary. The notes from Fields medalist Andrei Okounkov's lectures at the Joint Mathematics Meetings in January 2007. The first lecture is about the 2-dimensional Ising crystal and its zero-temperature limit which is the random-walk. The second lecture is devoted to zero-temperature interfaces in the 3-dimensional Ising model, also known as stepped surfaces. The third lecture discusses an application of limit shape ideas to gauge theory., January 2008.

Bernt Øksendal. Stochastic Differential Equations. Universitext. Springer Verlag, fourth edition edition, 1995.
From the back description of the book: ``This book is recommended to analysts (in particular those working in differential equations and deterministic dynamical systems and control) who want to learn quickly what stochastic differential equations are all about The 4th edition contains martingale representation theory, variational inequalities related to optimal stopping, stochastic control with terminal conditions and more.

Bernt Øksendal. Stochastic Differential Equations, volume XXVII of Universitext. Springer Verlag, 6th edition edition, 2003.
This book gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text in order to motivate and illustrate the theory and show its importance for many applications in economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case (which nevertheless are often sufficiently general for many purposes) in order to be able to reach quickly the parts of the theory which is most important for the applications. For the 6th edition, the author has added many exercises and, for the first time, solutions to many of the exercises are provided.

Bernt Øksendal and Agnes Sulem. Applied Stochastic Control of Jump Diffusions. Springer-Verlag, 2005.
From the advertising blurb: ``The main purpose of the book is to give a rigorous, yet mostly non-technical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. The type of control problems covered include classical stochastic control, optimal stopping, impulse control, and singular control. The text emphasizes applications. Main results are illustrated by examples. Exercises are given with complete solutions.'' Sounds like a book which takes applications in a different direction, but might be useful to study for purposes of understanding jump diffusions.

Emmanuel Parzen. Stochastic Processes. SIAM, 1999.

Eckhard Platen and David Heath. A Benchmark Approach to Quantitative Finance. Springer Undergraduate Mathematics. Springer, 2006.
The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk-neutral pricing theory. It permits a unified treatment of portfolio optimization, derivative pricing, integrated risk management and insurance risk modeling. The existence of an equivalent risk-neutral pricing measure is not required. Instead it leads to pricing formula with respect to the real-world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation of realistic parsimonious market models. The first part of the book describes the necessary tools from probability theory of stochastic differential equations with jumps. The second part is devoted to financial modeling by the benchmark approach. Various quantitative methods for the real-world pricing and hedging of derivatives are explained. The general framework is used to provide an understanding of the nature of stochastic volatility.

Vladimir Pozdnyakov and Martin Kulldorff. Waiting times for patterns and a method of gambling teams. American Mathematical Monthly, 113:134–143, February 2006. penney-ante game.

M. Poovey. Can numbers ensure honesty? unrealistic expectations and the U. S. accounting scandal. Notices of the American Mathematical Society, pages 27–35, January 2003. popular history.

William Poundstone. Fortune's Formula. Hill and Wang, 2005. popular history, Kelly criterion.
The book is subtitles: The untold story of the betting system that beat the casinos and Wall Street. This is a popular history of some of the people and ideas at the intersection of information theory, economics, finance and investing and especially about the Kelly criterion. The central individuals profiled in the book are Claude Shannon, the creator of information theory and Edward Thorp, the mathematician and investor who developed card-counting in blackjack and then who helped create modern computer-driven trading. The book is divided into seven sections, and a summary of the sections gives a sense of the book's contents. The first section, entitled Entropy, is about Claude Shannon and the creation of information theory. The second section, entitled blackjack is about Ed Thorp and the first use of card-counting in blackjack. The third section is Arbitrage and gives a survey of Paul Samuelson and the random-walk and efficient market hypotheses about the market. The fourth section is the St. Petersburg Wager and is about Daniel Bernoulli, the St. Petersburg paradox, and utility theory. The fifth section is titled RICO and is about Rudy Giuliani and the prosecution of the insider trading scandals of the 1980s. The sixth section is about the spectacular failure of LTCM and several other lesser known spectacular failures of hedge funds in the 1990s. The last section is titled Signal and Noise and is like an epilogue, returning to the application of the Kelly criterion to gambling, especially horse-racing. The book is written in short, simple declarative sentences and the chapters are short. The story is episodic almost to the point of being choppy. This gives a sense of reading a sequence of Reader's Digest articles. As with all popular histories, the mathematics is frustratingly sketchy leaving a sense of incompleteness and confusion. Still, I would recommend the book to someone who wanted a layman's introduction to the people behind the concepts of the efficient market, arbitrage, card-counting, and information theory. HV6710.P68 2995, 795.01

J.-L. Prigent. Weak Convergence of Financial Markets. Springer Verlag, 2002.

Philip Protter. Stochastic Integration and Differential Equations, Second Edition. Applications of Mathematics, Vol. 21. Springer Verlag, 2004.
From the advertising blurb: In spite of the apparent simplicity of approach, none of these books [other texts on stochastic integration] has used the functional analytic method of presenting semimartingales and stochastic integration. dots addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises.

Phillip Protter. Review of Mathematics for Finance: An Introduction to Financial Engineering by marek capinski and tomasz zastawniak. The American Mathematical Monthly, 111(10):923–927, December 2004.
This review argues (persuasively) that mathematical finance is still an inherently advanced graduate level subject in measure-based probability and stochastic process theory. Although ``a PDE-based pedagogic approach'' could be used, the reviewer ``does not favor such an approach.since [he] thinks it gives a biased view of the subject, one favored primarily by mathematicians who know and like PDEs but are rather innocent of the more significant results coming from a probabilistic approach. This leads to the misleading perception that the subject has a lot to do with PDEs which – while important – are only a tangential element of the theory and its applications. Nevertheless, a lot of mathematicians know and like PDEs, and perhaps for such a reason, such an approach is popular.''

James G. Propp and David B. Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms, 9(1&2):223–252, 1996.

James Propp and David Wilson. Coupling from the past: a user's guide. In D. Aldous and J. Propp, editors, Microsurveys in Discrete Probability, volume 41 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 181–192. American Mathematical Society, 1998.

Lawrence R. Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(3):257–286, February 1989. hidden Markov model, speech recognition.

Sidney Redner. A guide to first-passage processes. Cambridge University Press, 2001.
An interesting looking book, a pedagogical ``attempt to give a unified presentation of first-passage processes and illustrate some of their beautiful and fundamental consequences.'' Chem QA 274.2 R44

Marc Renault. Lost (and found) in translation: André's Actual method and its application to the generalized ballot problem. American Mathematical Monthly, 115(4):358–363, April 2008. ballot problem, reflection principle.

L. R. Rabiner and B. H. Juang. An introduction to hidden markov models. IEEE ASSP Magazine, pages 4–16, January 1986. hidden markov models.

Steven Roman. Introduction to the Mathematics of Finance. Undergraduate Texts in Mathematics. Springer Verlag, 2004.
The mathematics of finance has become a hot topic ever since the discovery of the Black-Scholes option pricing formula in 1973. Unfortunately, there are very few undergraduate texts in the area. This book is specifically written for advanced undergraduate or beginning graduate students in mathematics, finance, or economics. The mathematics is not watered-down but is appropriate for the intended audience. With the exception of an optional chapter on the Capital Asset Pricing Model, the book concentrates on discrete derivative pricing models, culminating in a careful and complete derivation of the Black-Scholes model as a limiting case of the Cox-Ross-Rubinstein discrete model. All necessary probability theory is developed on a ``need-to-know'' basis. The final Chapter is devoted to American options.

Sheldon Ross. A First Course in Probability. Macmillan, 1976.

Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2000.
From the publisher's advertisement: This textbook is an introduction to probability theory using measure theory. It is designed for graduate students in a variety of fields, including mathematics, statistics, economics, management, finance, computer science, and engineering who require a working knowledge of probability theory that is mathematically precise, but without excessive technicalities. This text provides complete proofs of all the essential introductory results. Nevertheless, the treatment is focused, and accessible, with he measure theory and mathematical details presented in terms of the intuitive probabilistic concepts, rather than separate imposing subjects. Contents: The Need for Measure theory; Probability Triples; Further Probabilistic Foundations, Expected Values, Inequalities and the Laws of Large Numbers. Distributions of Random Variables, Stochastic Processes and Gambling Games, Discrete Markov Chains, Some Further Probability Results, Weak Convergence, Characteristic Functions, Decomposition of Probability Laws, Conditional Probability and Expectation, Martingales, Introduction to Other Stochastic Processes.

Sheldon Ross. An Elementary Introduction to Mathematical Finance. Cambridge University Press, second edition edition, 2003.

Sheldon M. Ross. Introduction to Probability Models. Elsevier, 8th edition edition, 2003.
Note that this is now the 8th edition! A classic, I use a much earlier edition for many examples and ideas of probability theory and stochastic processes.

Sheldon M. Ross. Introduction to Probability Models. Academic Press, 9th edition edition, 2006.
Note that this is now the 9th edition! A classic, I use a much earlier edition for many examples and ideas of probability theory and stochastic processes. From the publisher's advertisement: ``With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. A new section on (3.7) on Compound Random Variables, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions. A new section (4.11) on hidden Markov chains, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequences of states. Simplified approach for analysing non-homogeneous Poisson processes.''

M. Rao and R. J. Swift. Probability Theory with Applications, volume 582 of Mathematics and Its Applications. Springer Verlag, 2006.
This book is a revised and expanded edition of a successful graduate and reference text. The material in the book is designed for a standard graduate course in probability theory, including some important applications. This new edition contains a detailed treatment of the core area of probability and both structural and limit results are presented in full detail. Key features of the book include: indicating the need for abstract theory even in applications and showing the inadequacy of existing results for certain apparently simple real-world applications. attempting to deal with existence problems for various classes of random families that figure in the main results of the subject; and providing a treatment of conditional expectations and of conditional probabilities that is more complete than in other existing textbooks. Since this a textbook, all proofs are given in complete detail (even at the risk of repetition) and some key results are given multiple proofs when each argument has something to contribute.

Louis M. Rotando and Edward O. Thorp. The Kelly criterion and the stock market. Amer. Math. Monthly, 99(10):922–931, 1992. Kelly criterion.

Mark Rubinstein. Implied binomial trees. The Journal of Finance, LXIX(3):771–817, July 1994.
From the abstract: ``This article develops a new method for inferring risk-neutral probabilities from the simultaneously observed prices of European options.'' More important for my purposes is the following analysis in Section I. ``In early research on 30 of [ S & P 500 ] component equities dots and quotes on their options covering a 2 year period during 1976-1978, I found that the Black-Scholes formula seemed to provide reasonably accurate values.''

Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer-Verlag, third edition edition, 1999.

Ed Sandifer. How euler did it: St. Petersburg paradox. url http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2045%20St%20Petersburg%20paradox.pdf, July 2007. St. Petersburg Paradox.

E. O. Schwartz. The stochastic behaviour of commodity prices: Implications for valuation and hedging. Journal of Finance, 52:923–953, July 1997.
Important step in the general energy price modeling literature.

W. Schoutens. Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003.

Bernd Schmid. Credit Risk Pricing Models, second Edition. Springer Finance. Springer Verlag, 2004.
From the advertising blurb: ``This book gives a deep insight into the latest basic and advanced credit risk modeling techniques covering not only the standard structural, reduced form, and hybrid approaches but also showing how these models can be applied to practice.'' I have not seen this book. From the title, it appears to be a specialized text on a specialized subject.

Ben Stein and Phil DeMuth. Yes, You can Time the Market. J. Wiley and Sons, 2003.
Yes, it really is that Ben Stein. However, I think that Ben Stein will win your money if you buy the book, you may not get any money. From the book jacket blurb: ``In this smart, simple book, Stein and DeMuth show you how to use the tools of technical analysis to determine the relative 'cheapness or 'expensiveness' of the market as a whole at any given moment. They demonstrate that basic criteria like dividend yield and price to earnings ratio are like clocks that can tell you when it's a good time to jump into an index fund, the stock market, or when you'd be better off putting your money in bonds, real estate, or cash.'' Actually, this book serves as an indicator of the correctness of our model of the market as an efficient market. If the market is efficient, has good, even perfect information flow, and the market is 'cheap', then everyone will know it, the demand for stocks will be high and the 'cheapness' will evaporate. If information flow is imperfect, then perhaps on short intervals, timing the market may be possible. Our mathematical assumptions are not universally accepted!

R. Seydel. Tools for Computational Finance. Springer Verlag, 2002.

Rüdiger Seydel. Tools for Computational Finance, second edition. Universitext. Springer Verlag, 2004.
From two reviews of the book quoted in he advertising blurb: ``Remarkably, Seydel addresses students of both mathematics and business, presumes only minimal background in either subject, yet ventures deep into he subject in little more than 200 pages.'' Choice and ``In an increasingly crowded filed of financial engineering titles, Seydel's Tools for Computational Finance stands out as filling an unmet need. It is an intermediate level text with an extremely practical focus. dots this is the kind of book you can read quickly gaining a broad understanding of practical techniques of financial engineering. On the other hand, you can go thorough it slowly, working through all the examples and exercises in order to gain an in-depth practical knowledge you can use on the job.'' Glyn Holton, Contingency Analysis.

Rüdiger Seydel. Tools for Computational Finance, third edition. Universitext. Springer Verlag, 2006.
From two reviews of the book quoted in he advertising blurb: ``Remarkably, Seydel addresses students of both mathematics and business, presumes only minimal background in either subject, yet ventures deep into he subject in little more than 200 pages.'' Choice and ``In an increasingly crowded filed of financial engineering titles, Seydel's Tools for Computational Finance stands out as filling an unmet need. It is an intermediate level text with an extremely practical focus. dots this is the kind of book you can read quickly gaining a broad understanding of practical techniques of financial engineering. On the other hand, you can go through it slowly, working through all the examples and exercises in order to gain an in-depth practical knowledge you can use on the job.'' Glyn Holton, Contingency Analysis.

Niel Shephard, editor. Stochastic Volatility. Oxford University Press, 2003.
From the advertising blurb: Determining whether there are patterns in the size and frequency of [movements in the price of securities], or in their cause and effect, is critical to devising strategies for the investment at the micro level and monetary stability at the macro level. Shephard has brought together a set of the classic and central papers that have contributed to our understanding of financial volatility. They cover stocks, bonds, and currencies and range from 1973 up to 2001.

Ichiro Shigekawa. Stochastic Analysis, volume 224 of Translations of Mathematical Monographs. American Mathematical Society, 2004.

Steven R. Shreve. Quantitative methods for portfolio management. In David C. Heath and Glen Swindle, editors, Introduction to Mathematical Finance, volume 57 of Proceedings of Symposia in Applied Mathematics, pages 1–24. American Mathematical Society, American Mathematical Society, 1999.

Steven E. Shreve. Stochastic Calculus For Finance, volume Volume I of Springer Finance. Springer Verlag, 2004.
From the advertising blurb: ``This book evolved from the first ten years of the Carnegie Mellon professional Master's program in Computational Finance. The contents have been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs. But more importantly, intuitive explanations, developed and refined through classroom experience with this material, are provided throughout the book. Volume I introduces the fundamental concepts in a discrete-time setting and Volume II builds on this foundation to develop stochastic calculus, martingales, risk-neutral pricing, exotic options, and terms structure models, all in continuous time. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties.

Steven E. Shreve. Stochastic Calculus For Finance, volume Volume II of Springer Finance. Springer Verlag, 2004.
From the advertising brochure: ``This book evolved from he first ten years of the Carnegie Mellon professional Master's program in Computational Finance. The contents of the book have been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both both precise statements of results, plausibility arguments and even some proofs. But more importantly, intuitive explanations developed and refined through classroom experience with this material, are provided throughout the book. Volume I introduces the fundamental concepts in a discrete-time setting and Volume II builds on this foundation to develop stochastic calculus, martingales, risk-neutral pricing, exotic options and term-structure models all in continuous time. The book contains a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties.'' I have not seen this book, but the author is one of the masters of the field, has written other important tests in the area. I would certainly consider this book as a possible text for this course. From a review in SIAM 2005 ``Steven Shreve's comprehensive two-volume Stochastic Calculus for Finance may well be the last word, at least for a while, in the flood of Master's level books, dots , a detailed and authoritative reference for ``quants''. The books are derived from lecture notes that have been available on the web for years and that have developed a huge cult following among students, instructors, and practitioners. The key ideas presented in these works involve the mathematical theory of securities pricing based on the ideas of classical finance dots the beauty of the mathematics is partly in the fact that it is self-contained and allows us to explore the logical consequences of our hypotheses. The material of this volume of Shreve's text is a wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions dots In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach. It is accessible to a broad audience and has been developed after years of teaching the subject. It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance.''

Kenneth J. Singleton. Empirical Dynamic Asset Pricing. Princeton University Press, 2006.
This book focuses on the interplay between model specification, data collections, and econometric testing of dynamic asset pricing models.

A. V. Skorokhod. Studies in the Theory of Random Processes. Dover, 1982.

Thomas S. Y. So and Sang Bin Lee. The Oxford Guide to Financial Modeling. Oxford University Press, 2003.
From the advertising blurb: This book discusses the theory and applications of more than 122 financial models currently in use and includes the financial models of stock and bond options, exotic options, investment grade and high yield bonds, convertible bonds, mortgage-backed securities, liabilities and financial institution's business models and corporate models.

J. Laurie Snell. Gambling, probability and martingales. Mathematical Intelligencer, 4(3):118–124, 1982. martingale.
A nice little article, expository to the point of being conversational. Has an excellent explanation of the rules of craps and the relationship to craps (not) being a martingale. Good historical survey of the concept of martingale and two standard elementary examples.

Frank Spitzer. Principles of Random Walk. Undergraduate Texts in Mathematics. Springer Verlag, second edition, 2001.

J. Michael Steele. Stochastic Calculus and Financial Applications. Springer-Verlag, 2001. QA 274.2 S 74.

David Stirzaker. Stochastic Processes and Models. Oxford University Press, 2005.
This book provides a concise and lucid introduction to simple stochastic processes and models. Including numerous exercises, problems and solutions, it covers the key concepts and tools in particular: random walks, renewals, Markov chains, martingales, the Wiener process model for Brownian motion, and diffusion processes, concluding with a brief account of the stochastic integral, and stochastic differential equations, as they arise in option-pricing. Ideal For A Undergraduate Second Course In Probability.

R. Stojanovic. Computational Financial Mathematics using MATHEMATICA. Springer Verlag, 2002.

Daniel Stroock. An Introduction to Markov Processes, volume 230 of Graduate Texts in Mathematics. Springer Verlag, 2005.
From the advertising blurb: ``This book provides a rigorous but elementary introduction to the theory of Markov processes on a countable state space. It covers Doeblin's theory, general ergodic properties, continuous time processes, and reversible processes, with the use of their associated Dirichlet forms.'' I don't think the book would be elementary enough for the purposes of understanding Markov processes for the mathematical finance course, but it would be a reference book.

A. A. Svesnikov. Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions. Dover, 1968.

Nasim Nicholas Taleb. Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life. Texere, 2001. HG 4521 T35.

Stephen J. Taylor. Asset Price Dynamics, Volatility and Prediction. Princeton University Press, 2005.
This book shows how current and recent market prices convey information about the probability distributions that govern future prices. Moving beyond purely theoretical models, Stephen Taylor applies methods supported by empirical research of equity and foreign exchange markets to show how daily and more frequent asset prices and the prices of options contracts can be used to construct and assess predictions about future prices, their volatility and their probability distributions.

A. C. Thompson. Valuation of path-dependent contingent claims with multiple exercise decisions over time: The case of take-or-pay. Journal of Financial and Quantitative Analysis, 2:271–293, June 1995.
Possibly the first paper to seriously address the issue of gas swing contracts. It uses a Cox-Ross-Rubinstein binomial lattice for the natural gas spot price to derive an algebraic expression for the value of swing with a non-zero take-or-pay obligation.

H. M. Taylor and Samuel Karlin. An Introduction to Stochastic Modeling. Academic Press, third edition edition, 1998.
A simplified version of the classic two-volume set from Karlin and Taylor. This volume would be suitable for a senior-level class on stochastic processes and applied probability. Another excellent model of mathematical good taste, excellent exposition and the right level of mathematical sophistication. Does not require measure theoretic ideas.

S. R. S. Varadhan. Probability Theory. Number 7 in Courant Lecture Notes. American Mathematical Society, 2001.

S. R. S. Varadhan. Stochastic Processes, volume 16 of Courant lecture Notes. American Mathematical Society, 2007.

Johannes Voit. Statistical Mechanics of Financial Markets, Study Edition. Texts and Monographs in Physics. Springer Verlag, 2003.
From the advertising blurb: ``This introductory treatment describes parallels between statistical physics and finance, both long established and new research results on capital markets. Forming the core of it's treatment are the concepts of random walks, scaling of data, and risk control. Voit discuses the underlying assumptions using empirical financial data and analogies to physical models such as fluid flow and turbulence. This corrected edition has been updated with several new and significant developments e.g. the dynamics of volatility smiles and implied volatility surfaces, path integral approaches to option pricing, a new simulation scheme for options, multifractals, the application of nonextensive statistical mechanics to financial markets and the minority game.

P. Wilmott, S. Howison, and J. Dewynne. The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, 1995. HG 6024 A3W554.

Paul Wilmott, S. Howison, and J. Dewynne. The Mathematics of Financial Derivatives. Cambridge University Press, 1995.

R. J. Williams. Introduction to the Mathematics of Finance, volume 72 of Graduate Studies in Mathematics. American Mathematical Society, 2006.

Marc Yor. On some exponential functionals of brownian motion. Advances in Applied Probability, 24(3):509–531, September 1992. Brownian motions with drift, Bessel functions, Bessel processes, last-exit times.

R. Zagst. Interest Rate Management. Spring-Verlag, 2002.

Y.-L. Zhu and I.-L. Chern. Derivative Securities and Difference Methods. Springer Finance. Springer-Verlag, 2004.
From the publisher's blurb: ``This book is devoted to determining the prices of financial derivatives using a partial differential equation approach. In the first part, the authors describe the formulation of the problems, including related free-boundary problems, and derive the closed form solutions if they have been found. The second part discusses how to obtain their numerical solutions efficiently for both European-style and American style derivatives and for both stock option s and interest rate derivatives.''

Peter G. Zhang. Exotic Options. World Scientific, Singapore, 1997.
HG6024 A 3Z 1997