Chicago Financial Mathematics Seminar
The Workshop met on Fridays at 5:30 p.m. in Downtown Chicago (1998-1999)
Organizer: Alexander
Adamchuk alex@finmath.com
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The Chicago Financial Mathematics Seminar was a unique project that brought to Chicago 32 Distinguished Invited Speakers in 1998-1999 academic year.We are going to continue our regular FinMath seminars in 1999-2000 academic year and bring to Chicago well known academics, senior executives, and financial professionals actively managing quantitative research, risk, and trading groups at major investment banks and financial firms.
See Invited Speakers.Currently our seminar is not a formal part of any academic program, it is not required for any degree or certificate qualifications. It is open to those who might be interested in professional development and quantitative research in Financial Engineering and Risk Management. We are bringing together professionals from the industry and academia and highly qualified students for an open discussion of applied problems and current research. Our seminar would be a valuable asset to any professional for a successful career in modern financial industry.
In addition to our regular FinMath seminars we are going to organize a couple of one-two day Chicago Financial Engineering and Risk Management Workshops.
Steve Allen, Managing Director, Derivatives Market Risk Management, The Chase Manhattan Bank, N.A., New York; Fellow, Program in Mathematics in Finance, Courant Institute of Mathematical Sciences, New York University (Home page).
Angelo Arvanitis, Head of Quantitative Credit, Insurance and Risk Research, Paribas, London (Biography).
Tanya Styblo Beder, Managing Director, Caxton Corporation, New York; Chairperson, IAFE (International Association of Financial Engineers); Management Fellow, School of Management, Yale University.
Dimitris Bertsimas, Professor of Operations Research, Sloan School of Management, Massachusetts Institute of Technology (MIT) (Home page).
Richard Bookstaber, Head of Risk Management, Moore Capital Management, New York.
Peter Carr, Principal, Banc of America Securities, New York (Home page).
Ashvin B. Chhabra, Vice President, Fixed Income Quantitative Research, J.P.Morgan, New York; Fellow, Program in Mathematics in Finance, Courant Institute of Mathematical Sciences, New York University.
George M. Constantinides, Leo Melamed Professor of Finance, Graduate School of Business, The University of Chicago; Vice President, American Finance Association (Home page).
Christopher L. Culp, Managing Director, CP Risk Management LLC; Adjunct Associate Professor of Finance, Graduate School of Business, The University of Chicago; Managing Editor, Derivatives Quarterly (Home page).
Michel M. Dacorogna, Director of Research & Development, Olsen & Associates, Research Institute for Applied Economics, Zurich (Home page).
Kosrow Dehnad, Managing Director, Head of Hybrid Product Development & Structuring, Citigroup; Adjunct Professor, Department of Industrial Engineering and Operations Research, Columbia University (Biography).
Ron Dembo, President & CEO, Algorithmics, Toronto, Canada (Biography).
Emanuel Derman, Managing Director, Head of Quantitative Strategies/Equity Derivatives, Goldman Sachs Group (New Home page).
Darrell Duffie, Professor of Finance, Graduate School of Business & Program in Financial Mathematics, Stanford University; Co-Editor, Finance and Stochastics (Home page).
Sergei E. Esipov, Assistant Vice President, Centre Solutions, a member of Zurich Financial Services Group.
J. Doyne Farmer, Co-Founder and Chief Scientist, Prediction Company, Santa Fe.
Stephen Figlewski, Professor of Finance, Leonard N. Stern School of Business, New York University; Founding Editor of The Journal of Derivatives (Home page).
Mark Garman, President, Financial Engineering Associates; Professor Emeritus, Haas School of Business, University of California at Berkeley (Home page).
Paul Glasserman, Professor and Chairman, Management Science Division, Graduate School of Business, Columbia University, New York (Home page).
Pat Hagan, Designer, Trading Systems, Numerix; Editor-in-Chief, Applied Mathematical Finance.
Lars Peter Hansen, Professor of Economics, Chairman, Department of Economics, The University of Chicago; Member of the National Academy of Sciences (NAS).
David Heath, Professor, Department of Mathematical Sciences & Program in Computational Finance, Carnegie Mellon University (Home page).
Thomas S.Y. Ho, Founder & President, Global Advance Technology (G.A.T.), New York.
Vince J. Kaminski, Vice President - Research, Enron Corp., Houston.
Jean-Michel Lasry, Global Head of Quantitative Research, Paribas, London.
Alexander Levin, Manager, Treasury Research & Analytics, The Dime Bancorp, New York.
Anlong Li, Senior Vice President, Head of the Structured Product Group for Fixed-Income Derivatives, ABN AMRO North America.
Dilip Madan, Professor, Robert H. Smith School of Business, University of Maryland (Home page).
Michael Ong, Senior Vice President, Head of Enterprise Risk Management, ABN AMRO North America.
Philip Protter, Professor of Mathematics and Statistics, Purdue University (Home page).
Dmitry Pugachevsky, Vice President, Credit Derivatives Research, OTC Derivatives, Deutsche Bank, New York.
Eric Reiner, Managing Director, Equity Structured Products, Warburg Dillon Read LLC., New York.
Stephen A. Ross, Professor of Finance and Economics, Sloan School of Management, Massachusetts Institute of Technology (MIT); Principal, Roll & Ross Asset Management Corp.
Albert N. Shiryaev, Professor of Mathematics, Head of the Department of Probability Theory, Moscow State University; Head of the Laboratory of Statistics of Stochastic Processes of the Department of Probability Theory and Mathematical Statistics, Steklov Mathematical Institute (MIRAN); President, Bachelier Finance Society; Co-Editor, Finance and Stochastics.
William T. Shaw, Quantitative Analysis Group, Nomura International PLC, London; Director, Oxford System Solutions (Home page).
Steven E. Shreve, Professor, Department of Mathematical Sciences & Program in Computational Finance, Carnegie Mellon University; Co-Editor, Finance and Stochastics (Home page).
Kenneth J. Singleton, Professor of Finance, Graduate School of Business & Program in Financial Mathematics, Stanford University (Home page).
Nassim Taleb, Senior Advisor, Banque Paribas, New York; Fellow, Program in Mathematics in Finance, Courant Institute of Mathematical Sciences, New York University (Home page).
Klaus Bjerre Toft, Vice President, Fixed Income Research, Goldman Sachs Group.
J. Gregg Whittaker, Managing Director, Global Head of Credit Derivatives, Chase Securities, Inc.
Invited Speakers in 1998-1999 Academic Year
Marco Avellaneda, Professor, Courant Institute of Mathematical Sciences, New York University; Managing Editor, International Journal of Theoretical and Applied Finance (IJTAF) (Home page). April 16, 1999
Ilia Bouchouev, Head of Derivatives Research, Koch Industries, Inc. December 4, 1998
Peter Carr, Vice President, Equity Derivatives Research, Morgan Stanley & Co (Home page). October 23, 1998
George M. Constantinides, Professor of Finance, Graduate School of Business, The University of Chicago; Vice President, American Finance Association (Home page). May 28, 1999
Kosrow Dehnad, Managing Director, Head of Hybrid Product Development & Structuring, Citigroup; Adjunct Professor, Department of Industrial Engineering and Operations Research, Columbia University (Biography). April 9, 1999
Emanuel Derman, Managing Director, Head of Quantitative Strategies/Equity Derivatives, Goldman, Sachs & Co (Home page). May 7, 1999
David F. DeRosa, President, DeRosa Research and Trading (Home page); Adjunct Professor of Finance, School of Management, Yale University. March 12, 1999
Sergei E. Esipov, Quantitative Analyst, Centre Solutions, A Member of the Zurich Group. November 13, 1998
Alex Eydeland, Vice President, Head of Research, Southern Company Energy Marketing. January 22, 1999
Bjorn Flesaker, Managing Director, Bear, Stearns & Co. February 19, 1999
Paul Glasserman, Professor and Chairman, Management Science Division, Graduate School of Business, Columbia University, New York (Home page). June 4, 1999
Les Gulko, Vice President, General Reinsurance Corporation. February 5, 1999
Pat Hagan, Senior Global Advisor, Paribas Capital Markets, NY; Editor-in-Chief, Applied Mathematical Finance. January 22, 1999
David Heath, Professor, Department of Mathematical Sciences & Program in Computational Finance, Carnegie Mellon University (Home page). October 30, 1998
Lester Ingber, Director of Research, DRW Investments LLC, Chicago (Home page). November 20, 1998
Yan Jin, Quantitative Research Group, Goldman Sachs Asset Management. March 5, 1999
Ioannis Karatzas, Professor of Applied Probability, Department of Mathematics and Department of Statistics, Columbia University; Director, Program in Mathematics of Finance (Home page). May 14, 1999
Anlong Li, Senior Vice President, Head of the Structured Product Group for Fixed-Income Derivatives, ABN AMRO North America. June 4, 1999
Alexander Lipton-Lifschitz, Vice-President, Global Analytics, Bankers Trust; Professor, Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago (Home page). February 26, 1999
Dilip Madan, Professor, Robert H. Smith School of Business, University of Maryland (Home page). October 23, 1998
A. G. (Tassos) Malliaris, Professor, School of Business Administration, Loyola University (Home page). February 19, 1999
William Margrabe, President, The William Margrabe Group, Inc. (Biography). April 30, 1999
John F. Marshall, Professor of Finance, St. John's University; Executive Director, IAFE (International Association of Financial Engineers) 1992-1998 (Biography). October 16, 1998
Bill Morokoff, Firmwide Risk, Goldman Sachs, New York. November 20, 1998
George Papanicolaou, Professor, Department of Mathematics, Stanford University (Home page); Director, Program in Financial Mathematics. March 12, 1999
Philip Protter, Professor of Mathematics and Statistics, Purdue University (Home page). February 26, 1999
Eric Reiner, Managing Director, Equity Structured Products, Warburg Dillon Read LLC. April 16, 1999
Dan Rosen, Director of Research, Algorithmics, Inc. (Home page). May 28, 1999
Larry Shepp, Professor of Statistics, Rutgers University; Member of the National Academy of Science (NAS); Member of the Academy of Arts and Science (Home Page). February 12, 1999
David C. Shimko, Principal and Head, Risk Management Advisory Group, Bankers Trust (Home page). May 21, 1999
Srdjan Stojanovic, Professor, Department of Mathematical Sciences, University of Cincinnati (Home page). December 4, 1998
Andreas S. Weigend, Associate Professor, Information Systems Department, Leonard N. Stern School of Business, New York University (Home page). March 12, 1999
John F. Marshall, Professor of Finance, St. John's University; Executive Director, IAFE (International Association of Financial Engineers) 1992-1998 (Biography)."Option adjusted spread analysis in evaluating bonds with embedded optionality."
This presentation will develop the logic behind the option adjusted spread (OAS) approach to value fixed income securities having embedded optionality. The presentation will use callable bonds for illustrative purposes, but it is equally applicable to puttable bonds and mortgage backed securities. OAS analysis employs the forward rate method for valuing bonds but introduces volatility into the analysis. After developing the logic of OAS, the method will be extended to demonstrate how effective durations are obtained from the OAS approach and why these duration measures are superior to modified duration obtained using the traditional approach for valuing callable bonds.
Peter Carr, Vice President, Equity Derivatives Research, Morgan Stanley & Co (Home page)."Determining volatility surfaces and option values from an implied volatility smile"
Using only the implied volatility smile of a single maturity T and an assumption of path-independence, we analytically determine the risk-neutral stock price process and the local volatility surface up to an arbitrary horizon T'>=T. Our path-independence assumption requires that each positive future stock price Si is a function of only time t and the level W_{i} of the driving standard Brownian motion (SBM) for all t in (0,T'). Using the T-maturity option prices, we identify this stock pricing function and thereby analytically determine the risk-neutral process for stock prices. Our path-independence assumption also implies that local volatility is a function of the stock price and time which can be explicitly represented in terms of the known stock pricing function. Finally, we derive analytic valuation formulae for standard and exotic options which are consistent with the observed T-maturity smile.Dilip Madan, Professor, Robert H. Smith School of Business, University of Maryland (Home page)."Variance Gamma model and its application to optimal positioning in derivatives"
"The measurement, management and control of financial risk"
To manage and control financial risk, one must first be able to measure it. Some risk measures have very peculiar properties; for example (as discovered by Albanese) the use of VaR discourages diversification in important cases. We examine a set of properties a good risk measure should have and obtain a way to construct all risk measures having these properties. We show that some risk measures in current use do have these desirable properties, and we obtain one VaR-like risk measure which has these desirable properties.
"Trading probabilities"
Part 1. The merger of financial and actuarial approaches requires dynamic handling of basis risk. We present a rigorous method of analyzing hybrid contracts, and price the resulting profit and loss distribution on a portfolio basis.
Part 2. This method is successfully applied to S&P500 options, and can be used as a new technology to predict volatility smiles.
Part 3. Fixed-income applications of this method are based on a risk-objective model of term structure. We show how to construct and calibrate such a model.Brief theoretical versions could be downloaded from
- A.N.Adamchuk, S.Adamchuk and S.E.Esipov "Arbitrage relaxation of instruments with temporal constraints"
- A.N.Adamchuk and S.E.Esipov, "Collectively fluctuating assets in the presence of arbitrage opportunities and option pricing"
- S.E.Esipov and I.Vaysburd, "On the profit and loss distribution of dynamic hedging strategies"
- D.Guo and S.E.Esipov, "Portfolio based risk pricing: Pricing long term put options with GJR-GARCH (1,1)/Jump Diffusion Process"
Lester Ingber, Director of Research, DRW Investments LLC, Chicago (Home page)."Some applications of statistical mechanics of financial markets"
A modern calculus of multivariate nonlinear multiplicative Gaussian-Markovian systems provides models of many complex systems faithful to their nature, e.g., by not prematurely applying quasi-linear approximations for the sole purpose of easing analysis. To handle these complex algebraic constructs, sophisticated numerical tools have been developed, e.g., methods of adaptive simulated annealing (ASA) global optimization and of path integration (PATHINT). In-depth application to three quite different complex systems have yielded some insights into the benefits to be obtained by application of these algorithms and tools, in statistical mechanical descriptions of neocortex (short-term memory and electroencephalography), financial markets (interest-rate and trading models), and combat analysis (baselining simulations to exercise data). Papers.Bill Morokoff, Firmwide Risk, Goldman Sachs, New York."Applications of the Brownian Bridge to derivatives pricing and risk"
The Brownian Bridge, also known as tied down Brownian motion, is an extension of the standard Wiener process for which both the initial value and the value at some later time are prescribed. Such processes have many applications in mathematical finance, three of which are described here. First, the distribution of the maximum of a Brownian bridge is used to help reduce bias in Monte Carlo simulations of path dependent options such as a lookback put under a stochastic volatility model; Second, the Brownian bridge is applied to path generation in quasi-Monte Carlo simulation to reduce the effective dimension of the simulation, leading to improved convergence. Finally, the problem of covariance estimation from cumulative returns in the presence of missing or asynchronous data is addressed using a modification of the Expectation-Maximization (E-M) algorithm to account for Brownian Bridge constraints. Paper.
Ilia Bouchouev, Head of Derivatives Research, Koch Industries, Inc."From conductivity to volatility: The inverse problem in financial markets"
Valuation of financial derivatives critically depends on the specification of the stochastic process for the underlying assets. At the same time, market prices of options are directly observable and can be used to recover an unobservable local volatility function of the underlying asset. The problem is known in the industry as the model calibration. We present a mathematical formulation of this inverse problem, discuss important uniqueness and stability issues, and review various approaches to the numerical solution. Two simple iterative algorithms that utilize an adjoint problem are introduced. The reconstructed local volatility function is then used to compute a market-implied probability distribution.
The presentation summarizes the results of the following three papers:
- "The Inverse Problem of Option Pricing" (with V. Isakov), Inverse Problems (1997)
- "Derivatives Valuation for General Diffusion Processes", IAFE Proceedings (1998)
- "Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets" (with V. Isakov), topical review to appear in Inverse Problems (1999).
Srdjan Stojanovic, Professor, Department of
Mathematical Sciences, University of Cincinnati (Home
page).
"Optimal portfolio diversification under budget and short-selling constraints via numerical solutions of Monge-Ampere partial differential equations"
The portfolio consists of large number of stocks initially chosen for consideration. The evolution of their prices is modeled by the system of stochastic differential equations, whose parameters are estimated on the basis of, e.g., historical data. One chooses limited number of indexes (such as Dow Jones, NASDAQ, or any other index well correlated with the chosen stocks, or one chooses factors, such as, short-term interest rates, etc.; furthermore, very important stocks can be called indexes, as well, i.e., indexes can be traded - it is only important that the total number of indexes is quite limited) for tracking. Furthermore, the investors attitude towards risk-taking is quantified by the selection of the utility function of the terminal cash value of the portfolio. Trading strategy is a vector function of the current cash value of the whole portfolio, the current values of indexes, and of time, and whose components are the dollar investments in each considered stock. Trading strategy can be constrained in two ways: budget constraint, and constraints on short-selling. The goal is to maximize, on average, the (utility function of the) cash value of the whole portfolio at some future date. The above is formulated as a stochastic control problem, the corresponding Hamilton-Jacobi-Bellman equation is derived, and transformed into the corresponding Monge-Ampere type equation, which happens to be backward parabolic, possibly degenerate, fully nonlinear, partial differential equation, in variables: wealth, indexes, and time. The further difficulty is that the equation admits multiple solutions. The numerical method is designed, which computes the proper solution, i.e., the one that solves completely the above stochastic control problem.
Alex Eydeland, Vice President, Head of Research, Southern Company Energy Marketing.
"Pricing power derivatives" In this presentation we will introduce a variety of power derivatives and discuss unique challenges one faces while modeling evolution of power prices. Non-storability of electricity resulting in extremely high price volatility, seasonality, geographical, operational and regulatory constraints, lack of liquidity, volumetric risk, etc. -- all this underscores the necessity for developing new approaches to pricing and hedging power derivatives. We will present several pricing models and discuss their advantages and limitations. Finally, we will show how to use derivatives for power asset management. (Handout in MS Word Format)Pat Hagan, Senior Global Advisor, Paribas Capital Markets, NY; Editor-in-Chief, Applied Mathematical Finance."Capturing swaption skews and smiles using equivalent volatility techniques"
We use singular perturbation techniques to analyze vanilla swaptions, caplets, and floorlets under the CEV, Derman-Kani, stochastic volatility, and jump models. This analysis yields equivalent vol formulas: closed form formulas for the implied Black-Scholes volatility. These formulas shows that CEV models capture market skews in a realistic manner, but using the more general Derman-Kani (Dupire) models to capture market smiles leads to disturbingly unrealistic market predictions and hedge instabilities. These formulas also suggest that using either stochastic volatility or jump models is the simplest way of capturing the skews and smiles without creating the unrealistic behavior.
"The entropy theory of bond option pricing" The talk will introduce the Entropy Pricing Theory (EPT) and will apply it to bond option pricing. An informationally efficient price keeps investors as a group in the state of maximum uncertainty about the next price change. The EPT captures this intuition and suggests that, in informationally efficient markets, perfectly uncertain market beliefs must prevail. When the entropy functional is used to index the collective market uncertainty, then the entropy-maximizing consensus beliefs must prevail. The EPT offers asset valuation in incomplete markets. To examplify, we obtain a new bond option pricing model that is similar to Black-Scholes with the lognormal distribution replaced by a beta distribution. Unlike alternative models, the beta model does not restrict the evolution of bond prices or interest rates and does not restrict the underlying bond to being default-free. We present empirical evidence indicating that the beta model is more accurate than alternative option pricing models. Paper in 8 parts (PDF format): 12345678.
Larry Shepp, Professor of Statistics, Rutgers University; Member of the National Academy of Science (NAS); Member, Academy of Arts and Science (Home Page)."Some problems and solutions in mathematical finance"
Simple probabilistical models of financial decision making are solved explicitly and the solutions exhibit unexpected insight for corporate policy makers. Examples of corporate decisions aided by analysis include profit taking, corporate policy direction, and hiring and firing policy. We show that down-sizing is part of optimal corporate policy in a natural formulation and that aggressive maximization of profit inevitably leads to eventual bankruptcy inside of a rigorous mathematical analysis. The method of solution uses Kolmogorov's principle of optimization.
A. G. (Tassos) Malliaris, Professor, School of Business Administration, Loyola University (Home page).Bjorn Flesaker, Managing Director, Bear, Stearns & Co."Methodological issues in asset pricing: Random walk or chaotic dynamics"
We analyze the theoretical foundations of the efficient market hypothesis by stressing the efficient use of information and its effect upon price volatility. The "random walk" hypothesis assumes that price volatility is exogenous and unexplained. Randomness means that a knowledge of the past cannot help to predict the future. We accept the view that randomness appears because information is incomplete. The larger the subset of information available and known, the less emphasis one must place upon the generic term randomness. We construct a general and well accepted intertemporal price determination model, and show that price volatility reflects the output of a higher order dynamic system with an underlying stochastic foundation. Our analysis is used to explain the learning process and the efficient use of information in our archetype model. We estimate a general unrestricted system for financial and agricultural markets to see which specifications we can reject. What emerges is that a system very close to our archetype model is consistent with the evidence. We obtain an equation for price volatility which looks a lot like the GARCH equation. The price variability is a serially correlated variable which is affected by the Bayesian error, and the Bayesian error is a serially correlated variable which is affected by the noisiness of the system. In this manner we have explained some of the determinants of what has been called the "randomness" of price changes.
"Derivative credit risk management" The talk will address some modeling issues in measuring and managing the credit risk arising from running a derivatives business. To quantify this risk we require a model of the joint stochastic process followed by the default probability of a counterparty along with the net value of the portfolio of derivatives with that counterparty, taking into account the effect of any applicable credit enhancements. Questions pertaining to credit line management tend to focus on the tail of the resulting distribution of exposures, whereas credit reserve (or credit price adjustment) questions tend to be about expected losses. Especially challenging aspects of the modeling involve properly accounting for netting, margining, and dynamically changing collateral value; as well as handling the correlation between exposure changes and default events. The rapidly growing market for credit derivatives has opened up new venues for actively managing credit exposures, as well as focusing more effort on credit modeling.
Philip Protter, Professor of Mathematics and Statistics, Purdue University (Home page)."Hedging in incomplete markets"
For a general contingent claim in a Brownian motion paradigm, hedging may not be easy, but it is well understood. In many useful cases the hedging strategy has explicit formulas. We address a more general situation: that of a Markovian setting. This corresponds to most examples of incomplete markets, where not all contingent claims can be replicated. They can nevertheless still be hedged. We will discuss the hedging strategies and concentrate on situations where explicit hedging strategies can be formulated. We will then turn to the question of robustness: under a small change in the model specification, how is the hedging strategy affected? We will indicate how this is interesting even in the Brownian setting when one uses numerical methods to estimate the hedging strategy. My talk is based on joint work with Jean Jacod and Sylvie Meleard.Alexander Lipton-Lifschitz, Vice-President, Global Analytics, Bankers Trust; Professor, Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago (Home page)."Passport options and perfect and imperfect hedging strategies"
Conventional options allow for little (if any) buyer's participation. Passport options (which represent a completely new class of derivative instruments) were invented at Bankers Trust about two years ago in order to address this issue. The buyer of a passport option can choose her own strategy and collect all the gains generated by this strategy (if any) while the seller has to absorb all the losses. Thus, passport options can be considered as call options with zero strike on the trading account generated by the buyer's strategy. In the first part of the talk the speaker will describe the original approach of Hyer, Lipton-Lifschitz and Pugachevsky to the valuation of passport options and their hedging, as well as subsequent results obtained by other investigators. In the second part of the talk the speaker will discuss some relations between passport options and imperfectly hedged conventional options recently studied by Esipov and Vaysburd.
Yan Jin, Quantitative Research Group, Goldman Sachs Asset Management."Equilibrium positive interest rates: A unified view"
This paper develops precise connections among three general approaches to building positive interest rate models: an approach based on direct modeling of the pricing kernel; the Heath, Jarrow, Morton framework based on specifying forward rate volatilities and the market price of risk; and the Flesaker-Hughston framework based on specifying a family of positive martingales. Given the primitive data of any of these formulations we show how to find that of the other two. The connections exploit the observation that a pricing kernel is uniquely determined by its drift, a result of independent interest. Through these connections we provide, for any arbitrage-free term structure model, a representative-consumer real production economy supporting that term structure model in equilibrium. In particular, this gives an explicit construction of an equilibrium supporting any HJM model. Our results provide a means of verifying positivity of interest rates directly from the HJM primitive data and allow us to show explicitly how the Cox, Ingersoll, Ross model fits into the Flesaker-Hughston framework. Finally, we construct a new family of Markovian positive interest rate models that can fit any initial term structure, provide reasonably tractable expressions for bond prices and forward rates, and match elements of the bond return covariance matrix. Paper.
David F. DeRosa, President, DeRosa Research and Trading (Home page); Adjunct Professor of Finance, School of Management, Yale University."Recent market events and popular notions of risk management"
Recent experience in the leveraged fund industry suggest that industry-standard risk management tools are dangerously simplistic. Popular risk management and performance measurement tools were derived from basic portfolio theory as applied to unleveraged portfolios of common stocks. Yet banks, bank regulators, and consultants extend the framework to highly leveraged portfolios that include emerging market currencies and assets. Moreover, some practitioners have tried to apply techniques like Value-at-risk to leveraged arbitrage strategies where the results have been financially catastrophic. What can be learned from episodes like emerging market crises and hedge fund melt-downs?George Papanicolaou, Professor, Department of Mathematics, Stanford University (Home page); Director, Program in Financial Mathematics."Modeling of market volatility"
The modeling of market volatility is a central issue in financial mathematics, in pricing options, for interest rates, etc. There is no shortage of models, of course, but what is lacking is some rational way of selecting and evaluating models. We will survey briefly some commonly used volatility models, deterministic and stochastic, and we will argue that fast mean reverting stochastic volatility models have many attractive features. One of them is that they do reflect market behavior, for example the SP-500 index volatility. We will explain carefully how this is deduced from the data and its consequences for options pricing.Andreas S. Weigend, Associate Professor, Information Systems Department, Leonard N. Stern School of Business, New York University (Home page)."Predicting returns distributions and their tails, with application to risk management"
- Nonlinear prediction of conditional percentiles for Value-at-Risk (Paper).
- Predicting daily probability distributions of S&P500 returns. (Paper)
- Computing portfolio risk using gaussian mixtures and independent component analysis (Paper).
- Extracting risk-neutral densities from option prices using mixture binomial trees (Paper).
Joint work with Issac Chang, Elion Chin, and Chris Pirkner (graduate students at NYU's Stern School of Business), Shanming Shi (J. P. Morgan) and Heinz Zimmermann (University of St. Gallen, Switzerland).Fuller description of talk.
In contrast, most traders utilize simple heuristic rules to characterize their beliefs about how implied volatilities should behave as underlying asset prices change. Two very popular rules are "sticky vol by strike" in which implied volatilities for a given fixed strike price are assumed independent of the underlying's level and "sticky vol by moneyness" in which implied volatilities for a given degree of moneyness (i.e., ratio of spot price to strike price) are assumed independent of the underlying's level. Each of these beliefs appears to be borne out under various market conditions. More importantly, these rules are often incorporated deeply into risk management systems and hedging strategies.
In this paper, we explore the inverse problem of identifying single-factor
processes consistent with these rules and compare and contrast their predictions
with those of the implied diffusion theory. In particular, we:
Marco Avellaneda, Professor, Courant Institute
of Mathematical Sciences, New York University; Managing Editor, International
Journal of Theoretical and Applied Finance (IJTAF) (Home
page).
"Weighted Monte Carlo: new techniques for calibrating multifactor asset-pricing models"
A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given valuation model (such as HJM), the algorithm corrects for price-misspecifications and finite-sample effects by computing adjusted "probability weights" on the simulated paths. The resulting ensemble prices the set of market instruments exactly. Concrete applications to the calibration of multi-factor interest rate and stochastic volatility models are discussed. Hedging and sensitivity analysis are discussed as well.
William Margrabe, President, The William Margrabe Group, Inc. (Biography)."The wand and the wizard"
The old saying, "It isn't the wand, but the wizard who waves it," holds just as true for derivatives pricing models as for -- well -- other things. Turning a trainee loose with a powerful pricing tool -- whether the underlying algorithm is "analytical", binomial, finite difference, Monte Carlo, or another numerical quadrature -- is as dangerous as turning a raw Marine recruit loose with an antitank missile system.Market makers and their managers, controllers, and auditors care about misuse and abuse of pricing algorithms, because a perfectly good pricing algorithm can deliver totally misleading bid, ask, and MTM P&L figures, if the user doesn't correctly set all the switches. The misleading figures can lead to lost jobs and shareholder lawsuits. Correct use of the algorithms is a complicated task, because the crucial parameters, such as number of binomial periods or Monte Carlo paths, spatial step size, and bounds of integration, are many and the correct values can depend on contract terms and market conditions.
We demonstrate this point by first abusing, then fine tuning six standard algorithms for pricing European call and put options. The demonstration shows more than a dozen pitfalls that a sophisticated user of pricing models will avoid when pricing more exotic options with the same tools.
"Beware of Greeks" In the context of option pricing and risk management, "Greeks" are sensitivities of price with respect to input variables and parameters. In calculus terms, "Greeks" are first and second derivatives. Numerical methods are notoriously stronger at integrating to compute option prices than they are at differentiating to compute corresponding Greeks. Risk managers care about "unstable" or otherwise untrustworthy Greeks, because they lead to unplanned P&L swings, which can put a desk out of business.This talk demonstrates how bad numerical estimates of Greeks can be, discusses the implications for risk management, and suggests ways to avoid the problems and get useful hedge ratios.
Emanuel Derman, Managing Director, Head of Quantitative Strategies/Equity Derivatives, Goldman, Sachs & Co (Home page)."Life As A Quant: Facts and Fallacies"
Life on Wall St is very different from the way people in academic life imagine it. For most people, most of the time, it involves neither finding miraculous and previously undiscovered arbitrages, nor doing boring mindless "coding." Instead, it involves a middle path, an interdisciplinary mix of inventive computation, not-too-rigorous mathematics, financial understanding, self-education and educating others, all applied in a hectic environment, to help build and maintain a business. Financial modeling as a practitioner involves accentuating simplicity, avoiding complexity, and making many pragmatic compromises.I will talk about the sociology of quantitative life, what we do everyday, which methods seem to work, and which don't.
Ioannis Karatzas, Professor of Applied Probability, Department of Mathematics and Department of Statistics, Columbia University; Director, Program in Mathematics of Finance (Home page)."American barrier options"
We obtain closed-form expressions for the prices and optimal hedging strategies of American put options in the presence of an "up-and-out" barrier, both with and without constraints on the short-selling of stock. The constrained case leads to an interesting stochastic optimization problem of mixed optimal stopping/singular control type; this is reduced to a variational inequality, which is then solved explicitly (Paper).
"Taking all the credit"
Should these fundamental problems be solved with better models, or with different operating procedures? In this talk, I will present the case for integrated and centralized collateral management as a solution to heretofore unsolved credit risk management problems. I will also discuss why current collateral management techniques fail.
While there is no formal paper for this talk, interested participants
can find five Risk Magazine articles closely related to this topic:
The articles are available in Acrobat format on www.covar.com
(original research). Also, see Credit
Risk: Models and Management, to be published by Risk Publications
in May 1999.
"Bounds on Option Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple Securities"
Dan Rosen, Director of Research, Algorithmics,
Inc. (Home
page).
"Beyond VaR: Parametric and Simulation-Based Risk Management Tools"
Measuring risk is a passive activity; knowing ones VaR alone does not provide much guidance for managing risk. In contrast, risk management is a dynamic endeavor and it requires tools that help identify and reduce the sources of risk. These tools should lead to an effective utilization of the wealth of financial products available in the markets to obtain the desired risk profiles.
To achieve this, a comprehensive risk manager's toolkit should provide
the ability to:
Robert Litterman (1996a, 1996b) recently described a comprehensive
set of analytical risk management tools extending some of the insights
originally developed by Markowitz (1952) and Sharpe (1964). These tools
are based on a linear approximation of the portfolio to measure its risk
and generally assume a joint (log)normal distribution of the underlying
market risk factors, similar to the RiskMetrics VaR methodology (J.P. Morgan
1996). Litterman further emphasized the dangers of managing risk using
only such linear approximations. However, in spite of their onerous assumptions,
the insights provided by these tools are very powerful and hence constitute
a solid basis for a risk management toolkit.
In this talk, we presents an extended simulation-based risk management toolkit developed on top of Letterman's analytical tools. Simulation- based tools provide additional insights when the portfolio contains non-linearities, when the market distributions are not normal or when there are multiple horizons. In particular, these tools are very useful not only for market risk, but also for credit risk, where the exposure and loss distributions are generally skewed and far from normal. We further demonstrate that simulation-based tools can be used, sometimes even more efficiently, with other risk measures in addition to VaR. Indeed, they also uncover limitations of VaR as a coherent risk measure, as has been demonstrated by Artzner et al. (1998). Simulation-based methods to measure VaR (historical or Monte Carlo) are generally much more computationally intensive than parametric methods (such as RiskMetrics), but advances in computational simulation methods and hardware have rendered them practical for enterprise-wide risk measurement. However, it is widely believed that risk management tools based on simulation are impractical since they require substantial additional computational work. We demonstrate that efficient computational methods are available which require no additional simulation to obtain risk management analytics. We also discuss the limitations of these tools, resulting from the fact that they derive from a finite number of sampled scenarios, and propose steps for dealing with these issues. (Recommended Readings)
"A Jump-Diffusion Market Model of Interest Rates"
Anlong Li, Senior Vice President, Head of the Structured Product Group for Fixed-Income Derivatives, ABN AMRO North America.
"A One-Factor Volatility Smile Model with Closed-form Solutions for European Options"
Risk Magazine
Journal of Computational Finance
The Journal of Risk
The Journal of Financial Engineering
The Journal of Derivatives
Financial Economics Network (FEN): SSRN's Electronic Journal of Derivatives
Financial Economics Network (FEN): SSRN's Electronic Journal of Capital Markets: Asset Pricing and Valuation
Economics Research Network (ERN): SSRN's Electronic Journal of Monetary Economics
International Journal of Theoretical and Applied Finance
Applied Mathematical Finance
appliederivatives.com (formerly Applied Derivatives Trading)
erivativesreview.com
Review of Derivatives Research
Asia-Pasific Financial Markets (formerly Financial Engineering and the Japanese Markets)
Derivatives Strategy
The Journal of Finance: The Journal of The American Finance Association
Theory of Probability and Its Applications (SIAM Journal)
The Annals of Applied Probability
Advances in Complex Systems (formerly Journal of Complex Systems)
Journal of Financial and Quantitative Analysis
Journal of Futures Markets
The Review of Futures Markets
Studies in Nonlinear Dynamics and Econometrics
Finance and Stochastics
Mathematical Finance
European Finance Review
Scientific American
Quantum
Mathematica in Education and Research
Nature
The Derivatives 'Zine
NET EXPOSURE, The Electronic Journal of Financial Risk
The Economist
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The Wall Street Journal
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