Program in Quantitative Finance

Applied Mathematics and Statistics

Stony Brook University

Graduate Courses

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The Program introduces four core and five advanced courses. Check Graduate Requirements for further details. Students and their advisors can also check Sample Programs of Study for guidance on how the Program in Quantitative Finnace can be further tuned to specific professional and research interests.

   

The courses in Quantitative Finance are also open to all students in Applied Mathematics and Statisitics. All students within the Department are encouraged to consider them as electives.

       

Course listed as under development below have not yet been approved by the Graduate School.

    


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Overview

       

Currently, the AMS catalog lists three graduate courses in quantitative finance: AMS 592 - Mathematical Methods of Finance and Investments I, AMS 593 - Mathematical Theory of Interest and Portfolio Pricing, and AMS 594 - Mathematical Methods of Finance and Investments II. The proposal is that AMS 592 and AMS 594 - Mathematical Methods of Finance and Investments I and II will be retired and their material subsumed into the four core courses below, and that AMS 593 - Mathematical Thoery of Interest and Portfolio Pricing have some of its material subsumed into the core and be redesignated as AMS 518 - Interest Rate Sensitive Securities Theory & Valuation Methods in which this material will be treated at a more advanced level.

    

The suggested texts listed below each class are not all necessicarily appropriate for use as primary course texts. However, both students and instructors will find them to be useful sources of material.

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Core Quantitative Finance Courses

    

The three courses listed below represent the foundation courses in Quantitative Finance. All three are required for a Masters degree.

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AMS 511 - Foundations of Quantitative Finance

   

Introduction to capital markets and modern portfolio theory, including the organization and operation of securities markets, the Efficient Market Hypothesis and it implications, the Capital Asset Pricing Model, the Arbitrage Pricing Theory and more general factor models. Common stocks and their valuation, statistical analysis, and portfolio selection in a single-period, mean-variance context will be explored along with its solution as a quadratic program. Fixed income securities and their valuation, statistical analysis, and portfolio selection. Discussion of the development and use of financial derivatives. Introduction to risk neutral pricing, stochastic calculus and the Black-Scholes Formula. Whenever practical examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.  Prerequisites: AMS 505 or AMS 510, and AMS 507. 3 credits.

    

Suggested texts:

  • Cerny, Ales, Mathematical Techniques in Finance: Tools for Incomplete Markets, Princeton University Press, 2004.
  • Luenberger, David G., Investment Science, Oxford University Press, 1997.

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AMS 512 - Capital Markets & Portfolio Theory

   

Development of capital markets and portfolio theory in both continuous time and multi-period settings. Utility theory and its application to the determination of optimal consumption and investment policies. Asymptotic growth under conditions of uncertainty. Applications to problems in strategic asset allocation over finite horizons and to problems in public finance. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.  Prerequisite: AMS 511. 3 credits.

    

Suggested texts:

  • Campbell, John Y., & Luis M. Viceira, Strategic Asset Allocation, Oxford University Press, 2002.
  • Duffie, Darrell, Dynamic Asset Pricing Thoery, 3rd Ed., Princeton University Press, 2001.
  • Gollier, Christian, The Economics of Risk and Time, MIT Press, 2004.
  • Markowitz, Harry M., Mean-Variance Analysis in Portfolio Choice and Capital Markets, Blackwell, 1987.
  • Meucci, Attilio, Risk and Asset Allocation, Springer, 2005.
  • Rasmussen, Mikkel, Quantitative Portfolio Optimisation, Asset Allocation, and Risk Management, Palgrave MacMiilan, 2003.

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AMS 513 - Financial Derivatives and Stochastic Calculus

   

Further development of derivative pricing theory including the use of equivalent martingale measures, the Girsanov Theorem, the Radon-Nikodym Derivative, and a deeper, more general understanding of the Arbitrage Theorem. Numerical approaches to solving stochastic PDE’s will be further developed. Applications involving interest rate sensitive securities and more complex options will be introduced. Whenever practical examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned. Prerequisite: AMS 511. 3 credits.

    

Suggested texts:

  • Baxter, Martin, and Andrew Rennie, Financial Calculus, Cambridge University Press, 1996.
  • Hull, John C., Options, Futures and Other Derivatives (5th ed.), Prentice Hall. 2003.
  • Merton, Robert C., Continuous Time Finance, Blackwell, 1990.
  • Shreve, Steven E., Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer, 2004.
  • Shreve, Steven E., Stochastic Calculus for Finance II: Continuous Time Models, Springer, 2004.
  • Shiriaev, Albert N., Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific Publishing, 1999.
  • Wilmott, Paul, Sam Howison, and Jeff Dewynne The Mathematics of Financial Derivatives-A Student Introduction, Cambridge University Press, 1995.

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Advanced Courses and Electives

   

The following electives represent deeper explorations of the topics and themes developed in the Core Courses. The first two, AMS 514 - Computational Finance and AMS 515 - Case Studies in Quantitative Finance, are strongly recommended.

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AMS 514 - Computational Finance

   

Review of foundations: stochastic calculus, martingales, pricing, and arbitrage. Basic principles of Monte Carlo and the efficiency and effectiveness of simulation estimators. Generation of psuedo- and quasi-random numbers with sampling methods and distributions. Variance reduction techniques such as control variates, antithetic variates, stratified and Latin hypercube sampling, and importance sampling. Discretization methods including first and second order methods, trees, jumps, and barrier crossings. Applications in pricing American options, interest rate sensitive derivatives, mortgage-backed securities and risk management. Whenever practical examples will use real market data. Extensive numerical exercises and projects in a general programming environment will also be assigned. Prerequisites: AMS 512 and AMS 513. 3 credits.

   

Suggested texts:

  • Duffy, Daniel J., Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, Wiley, 2006.
  • Glasserman, Paul, Monte Carlo Methods in Financial Engineering, Springer, 2004.
  • Jaeckel, Peter, Monte Carlo Methods in Finance, Wiley, 2002.
  • Stojanovic, Srdjan, Computational Financial Mathematics using Mathematica, Birkhuser, 2003.
  • Tooper, Juergen, Financial Engineering with Finite Elements, Wiley, 2005.
  • Wellin, Paul, Richard Gaylord and Samuel Kamin, An Introduction to Programming with Mathematica, Cambridge University Press, 2005.

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AMS 515 - Case Studies in Quantitative Finance

  

Actual applications of Quantitative Finance to problems of risk assessment, product design, portfolio management and securities pricing will be covered. Particular attention will be paid to data collection and analysis, the design and  implementation of software, and, most importantly, to differences the occur between "theory and practice" in model application, and to the development of practical strategies for handling cases in which “model failure” makes the nave use of quantitative techniques dangerous. Extensive use of guest lecturers drawn from the industry will be made. Prerequisites: AMS 512 and AMS 513. 3 credits.

    

Suggested texts:

  • Taleb, Nassim, Dynamic Hedging - Managing Vanilla and Exotic Options, Wiley, 1997.
  • Mason, Scott, Robert Merton, Andre Perold and Peter Tufano, Cases in Financial Engineering, Prentice-Hall, 1995.

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AMS 516 - Advanced Portfolio Optimization (under development)

   

Development and formulation of portfolio optimization problems in single and multi-period settings. Exploration of the geometry and economic significance of problems of portfolio selection in an advanced setting. Formulation and solution of optimization problems via a variety of approaches such as vector space and interior point methods. The stability and sensitivity of mathematical programs in portfolio management will be explored. There will be extensive coverage of the analysis of data required to parameterize practical problems. Prerequisites: AMS 540 and AMS 513. 3 credits.

   

Suggested texts:

  • Campbell, John Y., & Luis M. Viceira, Strategic Asset Allocation, Oxford University Press, 2002.
  • Luenberger, David G., Linear and Nonlinear Programming (2nd ed.), Kluwer Academic Publishers, 2003.
  • Luenberger, David G., Optimization by Vector Space Methods, Wiley, 1969.
  • Markowitz, Harry M., Mean-Variance Analysis in Portfolio Choice and Capital Markets, Blackwell, 1987.
  • Nocedal, Jorge, & Stephen J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.

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AMS 517 - Advanced Options Theory & Valuation Methods (under development)

   

Advanced topics in the pricing of options. Detailed study of such exotics as lookback options, ladder options, tigger or knock-in options, basket options, and multi-asset options. Development and pricing of customized structured products. Parameter estimation and pricing under conditions of stochastic volatility, including mixture distributions, GARCH models, and the failure of traditional pricing approaches under volatility explosions. Prerequisite: AMS 514. 3 credits.

    

Suggested texts:

  • Cont, Rama, and Peter Tankov, Financial Modelling with Jump Processes, Chapman & Hall, 2004.
  • Lewis, Alna L., Option Valuation under Stochastic Volatility, Finance Press, 2000.
  • Schoutens, Wim, Lvy Processes in Finance: Pricing Financial Derivatives, Wiley, 2003.
  • Shaw, William, Modeling Financial Derivatives with Mathematica, Cambridge University Press, 1998.

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AMS 518 - Interest Rate Sensitive Securities Theory and Valuation (under development)

 

Organization and operation of the LIBOR and Swap Markets. Theory and implementation of interest rate models including no arbitrage pricing, change of numeraire, and short rate factor models such as Heath-Jarrow-Morton. Advanced techniques in pricing interest rate sensitive securities including numerical methods and the estimation of parameters required to calibrate models to current market rates and implied volatilities. Pricing of equity derivatives under stochastic interest rates. Prerequisites AMS 512 and AMS 513. 3 credits.

   

Suggested text:

  • Brigo, Domiano, and Fabio Mercurio, Interest Rate Models - Theory and Practice, Springer, 2001.

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AMS 519 - Credit Risk Modeling and Credit Derivatives (under development)

   

Development of credit derivatives and their markets. Credit risk modeling, including factor model and copulas for default correlation models. and recovery modeling. Pricing models using a variety of approaches. Prerequisites AMS 512 and AMS 513. 3 credits.

    

Suggested texts:

  • Bluhm, Christian, Ludger Overbeck and Christoph Wagner, An Introduction to Credit Risk Management, Chapman and Hall, 2002.
  • Schonbucher, Philipp J., Credit Derivatives Pricing Models: Models Pricing and Implementation, Wiley, 2003.
  • Tavakoli, Janet M., Credit Derivatives and Synthetic Structures, 2 Ed., Wiley, 2001.

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AMS 691 - Special Topics (Quantitative Finance)

   

Sections within the framework of the existing AMS 691 - Special Topics will be designed for second and third year graduate students with a strong foundation in Quantitative Finance who wish pursue research or independent study in the field. Several topics may be taught in different sections. Prerequisites AMS 512 and AMS 513 and permission of the instructor. 3 credits. May be repeated for credit.

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