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FINANCIAL MATH PROGRAM |
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1999- 2000 Financial Math Abstracts
Since the 1987 stock market crash, global index options markets have been characterized by a persistent and large implied volatility skew. In recent years this skew has appeared in fixed-income, currency and commodity markets too. A variety of apocryphal traders' rules and theoritical models attempt to describe how this skew changes. I have isolated several distinct periods ("regimes") in which different patterns of change seem to hold. For each period, I try to determine which rule or model the volatility market seemed to be following, the possible reason why, and whether this pattern was appropriate. Some patterns repeat themselves, and I conjecture that these regimes may hold more generally. Emanuel Derman is a Managing Director in Firmwide Risk at Goldman Sachs. He is a co-author of the Black-Derman-Toy interest-rate model and the Derman-Kani local volatility model. Prior to joining Goldman Sachs in 1985, Dr. Derman obtained a Ph.D. in theoretical physics form Columbia University, held a number of academic positions where he did research in theoritical particle physics, and worked for Bell Laboratories. May 31, 2000 This paper proposes and implements an equilibrium framework for valuing weather derivatives. We generalize the Lucas model of 1978 to include the daily temperature as a fundamental variable in the economy. Temperature behavior in the past twenty years is closely studied for five major cities in the U.S. and a model is proposed for the daily temperature variable which incorporates all the key properties of temperature behavior including seasonal cycles and uneven variations throughout the year. The model system is estimated using the 20-year data and numerical analyses are performed for forward and option contracts on HDD's and CDD's. The key advantages of our model include the use of weather forecasts as inputs and the ability to handle contracts of any maturity,for any reason. Numerical analyses show that within our framework, the market price of risk associated with the temperature variable does not appear to impact the weather derivatives' value in a significant way, which indirectly justifies the use of riskfree rate to derive weather derivatives' values as many practioners do in the industry. Finally, we show that the so-called historical simulation method can lead to significant pricing errors due to its erroneous implicit assumptions. Melanie Cao is currently working at Queen's University in the
Department of Economics. She received her PhD in Finance in 1997 from
the Rotman School of Management, University of Toronto and has been
an assistant professor since 1997. During 1998-1999, she worked at the
Chicago Mercantile Exchange as a senior director of financial product
development and was responsible for designing the exchange-traded weather
derivatives. Her research interests are equilibrium asset pricing, derivatives,
initial public offering and banking. Jason Wei is an associate professor of finance at the University
of Toronto. He holds a Ph.D. from the University of Toronto. His past
research mainly focused on pricing cross-currency derivatives instruments
such as quantos and differential swaps. His current research interests
are primarily in credit derivatives and weather derivatives. He has
published numerous articles in both academic and practioners' journals.
He is currently on the editorial board of The Journal of Derivatives. "Rational Pricing of Internet Companies" In this article, we apply real options theory and modern capital budgeting to the problem of valuing an Internet company. We formulate the model in continuous time, form a discrete time approximation, estimate the model parameters, solve the model by simulation, and then perform sensitivity analysis. Depending on the parameters chosen, we find that the value of an Internet stock may be rational given high enough growth rates in revenues. Even with a very real chance that a firm may go bankrupt, if the initial growth rates are sufficiently high, and if there is enough volatility in this growth rate over time, then valuations can be what would otherwise appear to be dramatically high. In addition, we find a large sensitivity of the valuation to initial conditions and exact specification of the parameters. This is also consistent with the observation that the returns of Internet stocks have been strikingly volatile. Eduardo S. Schwartz is the California Professor of Real Estate and Professor of Finance, Anderson Graduate School of Management at the University of California, Los Angeles. He has an Engineering degree from the University of Chile and a Masters and Ph.D. in Finance from the University of British Columbia. He has been in the faculty at the University of British Columbia and visiting at the London Business School and the University of California at Berkeley. His wide-ranging research has focused on different dimensions in asset and securities pricing. Topics in recent years include interest rate models, asset allocation issues, evaluating natural resource investments, pricing Internet companies and the stochastic behavior of commodity prices. His collected works include more than eighty articles in finance and economic journals, two monographs, and a large number of monograph chapters, conference proceedings, and special reports. He is the winner of a number of awards for both teaching excellence and for the quality of his published work. He has been associate editor for more than a dozen journals, including the Journal of Finance, the Journal of Financial Economics and the Journal of Financial and Quantitative Analysis. He is past president of the Western Finance Association and the American Finance Association. He has also been a consultant to governmental agencies, banks, investment banks and industrial corporations. Building on the work of Breeden and Litzenberger (1979) on risk-neutral asset returns distributions implied by option prices, three key papers in the early 1990s by Rubinstein, Derman and Kani, and Dupire introduced the inverse problem of inferring implied asset price processes from option prices as functions of strike price and time to maturity. To date, most effort on this problem has focused on identifying and estimating the unique single risk-factor diffusive process with level- and time- dependent diffusion coefficient that naturally extends the Black-Scholes approach. This "implied diffusion" theory provides testable predictions of the behaviour of options' implied volatilities as time and underlying asset price evolve and consequently specifies a hedging strategy required to replicate any desired payoff. 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: 1. demonstrate that each model leads to distinct predictions about how implied volatilities for options with fixed absolute and relative strike prices change as the underlying spot moves and consequently differing delta-hedging strategies; 2. show, using a toy model with discretised time and level, that each of these rules allows identification of a (different) unique single-factor process consistent with the input data by permitting reduction of the dimensionality of the transition matrix between states at two times from 2 to 1; 3. apply these results in the continuous time and price limit to identify forward conditional transition densities consistent with the specified rules; and 4. display some characteristics of these transition densities in the "weak skew/smile" limit, allowing analytical results to be obtained, as well as in the limit that one time approaches the other: for the two traders' rules, these infinitesimal generators are intrinsically non-diffusive! Eric Reiner is a managing director in the Equities division of Warburg Dillon Read, the investment banking subsidiary of UBS AG. Based at WDR's Stamford, Connecticut trading centre, he is the global head of Equities Quantitative Strategies (EQS), an interdisciplinary group of 21 individuals spanning IT and trading functions, responsible for the development, implementation, and application of all valuation and risk management models for equities products at WDR. Prior to the UBS/SBC merger, he was based in New York and previously held assignments in UBS's Tokyo and London offices. Before joining UBS, Eric was Senior Vice President at the Los Angeles firm of Leland O'Brien Rubinstein Associates (LOR), where his responsibilities included development of LOR's consulting business, construction of option valuation models, and investigation of optimized trading and hedging strategies. Eric's ongoing interests include 1) pricing and hedging exotic options, 2) identifying applications of mathematical and numerical methods to financial problems, and 3) understanding the link between the micro structural and statistical properties of incomplete markets and the pricing operators deducible from the associated derivatives markets. He has published several articles and has presented many invited lectures on these topics and occasionally works toward completing a monograph on exotic options. Eric completed a Ph.D. in Chemical Engineering at the University of California at Berkeley, where his interest in derivatives began. He also holds an S.B. degree in Chemical Engineering and Economics from M.I.T. He is an associate editor of the Journal of Derivatives and the International Journal of Theoretical and Applied Finance. George Papanicolaou
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 briefly how this is deduced from the data and its consequences for options pricing. George Papanicolaou is currently a Professor at Stanford University in the Department of Mathematics. He received his PhD in mathematics in 1969 from Courant Institute, New York University and was an assistant professor and professor at the Courant Institute from 1969-1993. His research interests are Stochastic differential equations, linear and nonlinear wave propagation, signal and image analysis, and financial mathematics. March 29, 2000 Thomas Wilson, Divisional Chief Risk Officer, Swiss Re, New
Markets Tom Wilson is the Divisional Chief Risk Officer at Swiss Re New Markets, a division of Swiss Reinsurance which focuses on providing capital market and insurance-based risk management and financing solutions to corporations, banks and insurance companies globally. Prior to joining SRNM, Tom was a Partner at McKinsey & Company in New York, London and Zurich where he led the firm’s Global Risk Management practice. Tom has written extensively on market, credit and insurance related risk management issues. Tom earned his PhD in economics from Stanford University and his Bachelors in business administration from University of California, Berkeley. January 26, 2000 Jérôme Detemple, Boston University School of Management and CIRANO "A Monte-Carlo Method for Optimal Portfolios" (Joint work with R. Garcia and M. Rindisbacher) This paper provides (i) new results on the structure of optimal portfolios, (ii) economic insights on the behavior of the hedging components and (iii) an analysis of simulation-based numerical methods. The core of our approach relies on closed form solutions for Malliavin derivatives of diffusion processes which simplify their numerical simulation and facilitate the computation and simulation of the hedging components of optimal portfolios. Our approach is flexible and can be used even when the dimensionality of the set of underlying state variables is large. We implement the procedure for a class of bivariate and trivariate models in which the uncertainty is described by diffusion processes for the market price of risk (MPR), the interest (IR) and other relevant factors. After calibrating the models to the data we document the behavior of the portfolio demand and the hedging components relative to the parameters of the model such as risk aversion, investment horizon, speeds of mean-reversion, IR and MPR levels and volatilities. We show that the hedging terms are important and cannot be ignored for asset allocation purposes. Risk aversion and investment horizon emerge as the most relevant factors: they have a substantial impact on the size of the optimal portfolio and on its economic properties for realistic values of the models' parameters. Jérôme Detemple is Professor of Finance and Economics at Boston University School of Management and a Research Associate at CIRANO. He has a Ph.D. in finance from the Wharton School, University of Pennsylvania and a Doctorat d'État in Economics from the Université Louis Pasteur in Strasbourg. He also holds degrees from ESSEC and Université de Paris-Dauphine. Professor Detemple has published articles in leading journals in finance and economics and is currently an associate editor at four journals. His current research interests revolve around the valuation of American options, the pricing of assets in the presence of constraints and the implementation of asset allocation models. Detemple has taught at Wharton, Columbia, Northwestern, MIT, McGill and Lausanne. He was the Visiting Professor of Mathematical Finance at Boston University's School of Management during Spring 1999. January 26, 2000 George Papanicolaou, Department of Mathematics, Stanford University NOTE: TALK WAS CANCELLED 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 briefly how this is deduced from the data and its consequences for options pricing. George Papanicolaou, is currently a Professor at Stanford University in the Department of Mathematics. He received his PhD in mathematics in 1969 from Courant Institute, New York University and was an assistant professor and professor at the Courant Institute from 1969-1993. His research interests are Stochastic differential equations, linear and nonlinear wave propagation, signal and image analysis, financial mathematics
February 23, 2000 Steven E. Shreve, Department of Mathematics, Carnegie Melon University "Options on a Traded Account" In this article we study options on a traded account. In terms of the actions available to the buyer, the options we study are more general than a class of options known as passport options; in terms of the model of the underlying asset they are more restrictive. Using probabilistic techniques, we find the value of these options, the optimal strategy of the buyer, and the hedging strategy the seller should use in response to a (not necessarily optimal)strategy by the buyer. Steven E. Shreve is professor of mathematics at Carnegie Mellon University, where he has supervised Ph.D. students in finance and teaches in the master's program in computational finance. He holds an M.S. degree in Electrical Engineering and a Ph.D. in Mathematics from the University of Illinois, and has been a visiting faculty member at the University of California at Berkeley and Massachusetts Institute of Technology. With I. Karatzas, he has authored the books "Brownian Motion and Stochastic Calculus" and "Methods of Mathematical Finance." He is a fellow the the Institute for Mathematical Statistics. February 23, 2000 Ken Vetzal, Centre for Advanced Studies in Finance, University of Waterloo "Valuing the Option Features of Segregated Funds" One of the most popular investments available in the Canadian market today is a mutual fund with the added feature of a long term maturity guarantee. These types of investments are known as segregated funds. They contain some very complex embedded option features in the form of multiple shout options which permit the holder to reset the guarantee level and the maturity date for which it applies many times during the life of the contract. The contracts also provide mortality benefits if the investor dies prior to the maturity date. This paper explores the valuation and hedging of segregated funds using an approach based on the numerical solution of a set of linear complementarity problems. Our results indicate that the option components of these contracts seem to be underpriced: our estimates of a lower bound for a fair proportionate fee to be levied to fund the option features is about 1%. However, this appears to be an upper bound in terms of what writers are actually charging. We also show that different contract specifications which generate similar present values may require quite different proportionate fees. This is because the expected durations of the contracts can be quite different. Ken Vetzal is an Associate Professor with the Centre for Advanced Studies in Finance at the University of Waterloo. He holds a Ph.D. in Finance from the University of Toronto. His research interests are primarily in the area of numerical valuation of complex derivative instruments. He has recently published work in journals such as Advances in Futures and Options Research, Applied Mathematical Finance, Journal of Banking and Finance, Journal of Computational Finance, and Journal of Fixed Income. November 24, 1999 "Stochastic Differential Geometry, Financial Modelling, and Arbitrage-Free Pricing There appears to be no particular economic reason why asset prices should be Markovian, but the fact remains that most of the financial models used in practice in the derivatives industry incorporate this property. For example, although the general HJM framework for interest rate dynamics is non-Markovian, the models of Vasicek, CIR, Hull and White, and so on, are Markovian. Differential geometric methods can be used to clarify the role of the so-called "state variables" that invariably characterise such models, and through such considerations we are led to the more general idea of an economic state manifold. It is interesting in this context to explore the geometric meaning of well-known financial concepts such as the principle of no arbitrage. And when it comes to the chaotic behaviour of markets, how much is this is due to unhedgeable randomness in the strict probabilistic sense (say, jumps associated with extremal events), and how much might be due to instabilities inherent in the "classical" dynamics associated with the underlying manifold? Lane P. Hughston, currently Visiting Professor at the University of Texas, has recently been elected to the newly established Chair in Financial Mathematics at King's College London with effect from the beginning of the next millenium, where he will be setting up a new research group in mathematical finance and related areas of mathematics. His career has included stints both in academic institutions and financial institutions. His doctorate is from Oxford University, which he attended as a Rhodes Scholar, and subsequently joined the faculty and worked closely for many years with Roger Penrose's group on problems in gravitation and quantum theory. Later his interests turned to mathematical finance, and for the last dozen years before returning full-time to academia he has held positions at Robert Fleming, the British merchant bank, and at Merrill Lynch, the U.S. investment bank. His investment banking experience has touched in one way or another upon many different aspects of the derivatives business, to the development of which he has made a number of significant contributions. November 24, 1999 "What is missing in the Black-Scholes?" A model of an incomplete market with the incorporation of a new notion of "inside information" is proposed. The Black-Scholes exponential Brownian motion model for stock fluctuations is modified by adjoining a hidden Markov process, which represents the state of information in the investors' community. The drift and volatility parameters take different values when the hidden Markov process is in different states. The hidden process is in state 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. We will present explicit closed-form pricing formula for the European option and the Russian option. It provides a clear, simple and insightful understanding of "arbitrage opportunity". Dr. Xin Guo is the Herman Goldstine Postdoc Fellow of IBM, T.J. Watson Research Center and an Assistant Professor in the Department of Mathematics at the University of Alberta. Dilip Madan, Robert H. Smith School of Business, University
of Maryland This paper presents the case for modeling asset price processes as purely discontinuous processes of finite variation with an infinite arrival rate of jumps that have arrival rates completely monotone in the jump size. The arguments address both the empirical realities of asset returns and the implications of the economic principle of no arbitrage. Two classes of economic models meeting these conditions are presented and linked. An important example given by the variance gamma process is studied in detail and used to design optimal derivative investment portfolios that are calibrated to actual portfolios to reverse engineer trader preferences and beliefs and infer personalized risk neutral measures termed position measures. Illustrative comparisons of statistical, risk neutral and position measures are also provided. Dilip B. Madan is a Professor of Finance at the University of Maryland. He holds PhD degrees in Economics and Mathematics and is internationally recognized for his contributions to the field of Mathematical Finance. Recent research thrusts include work on: the nature of the stochastic process driving asset prices, with its implications for option pricing and investment; the pricing of default risk using hazard rate models adapted to observable diffusion based information filtrations; and fundamental investigations into the concepts of no arbitrage. He is a founding member and Treasurer of the Bachelier Finance Society and Associate Editor for Mathematical Finance. Recent contributions have appeared in the European Finance Review, Finance and Stochastics, Joumal of Computational Finance, Joumal of Financial Economics, Mathematical Finance and Review of Derivatives Research.
"Moments in Financial Markets" A general formulation of Financial Markets (Delbaen and Schachermeyer (1994)) exhibits a necessary and sufficient condition (NFLVR: No Free- Lunch with Vanishing Risk) for the existence of a martingale measure, thus permitting to establish current prices as conditional expectations of future values. When this theory is specialized to european-type-call- option-contracts, with one underlying asset, the current prices (c1,...,cN) of these contracts corresponding to the same expiration date and N different exercise prices, can be regarded as the conditional expectations of angle-type functions (Hardy). By assuming additionally, that conditional distributions exist, these numbers (c) can be regarded as generalized moments (General Theory of Moments: Karlin & Studden, Kemperman, Krein) of these conditional distributions. A full description of this moment problem allows one to state a necessary and sufficient condition for the existence of the alluded distributions given a set of quoted prices (c1,...,cN). These moment-condition can be obtained explicitly for any N=1,2,.. If the moment-condition is not satisfied, then the distributions yielding the prices do not exist and consequently, the NFLVR condition is violated. If the moment-condition is satisfied, one can construct quadrature formulae which yield precise bounds for the current values of other contingent contracts depending on the value of the underlying asset at expiration. These bounds become sharper as N increases or as the functions defining these new contracts are closer to the linear span of the angle-functions involved. This theory is applied to actual market data in a sequential fashion as soon as prices are formed. The main empirical finding in such an application allows for the identification of moments (times) at which the moment-condition is alternatively disrupted or satisfied, and thus it yields a methodology for the on-line scrutiny of such markets. José Luis Farah is currently in the Department of Mathematics at ITAM (Mexico). He holds a Ph.D from the Imperial College of Science and Technology in London and has been engaged in diverse projects of Applied Mathematics from the numerical and modelling standpoint like Geothermal Flux reconnaissance via infrared imagery, Simulation of Electrical Power Plants Modelling of Mexico City's Atmosphere (pollution). Since 1994, became interested in Finance as part of the Applied Mathematics Programme here.
September 29,1999 Stanislav Uryasev, University of Florida"Optimization of Conditional Value-at-Risk" A new approach for optimization or hedging of a portfolio of finance instruments to reduce the risks of high losses is suggested and tested with several applications. As a measure of risk, the expected loss exceeding Value-at-Risk (VaR) is used. This measure is called Conditional VaR (CVaR), Mean Excess Loss, Mean Shortfall, or Tail VaR. CVaR is considered a more consistent measure of risk than VaR. Portfolios with low CVaR also have low VaR because CVaR is greater than VaR. The approach is based on a new representation of the performance function which allows simultaneous calculation of the VaR and minimization of the CVaR. It can be used in conjunction with analytical or scenario based optimization algorithms. If the number of scenarios is fixed, the problem is reduced to a Linear Programming or Nonsmooth Optimization Problem. These techniques allow to optimize portfolios with large numbers of instruments. The approach is tested with two examples: (1) portfolio optimization and comparison with the Minimum Variance approach; (2) hedging of an options portfolio. The suggested methodology can be used for optimizing of portfolios by investment companies, brokerage firms, mutual funds, and any business which evaluates risks. Although the approach s used for portfolio analyses, it is very general and can be applied to any financial or non-financial problems involving optimization of percentiles. Dr. Stanislav Uryasev is a professor at the Department of Industrial and Systems Engineering, University of Florida. He has PhD in Applied Mathematics from Glushkov Institute of Cybernetics, Ukraine. He developed methodology and software for efficient optimization, risk/reliability, equilibrium, and computer modeling techniques. Current research is focused, mostly, on Stochastic Optimization Techniques and applications in Finance such as Portfolio Optimization, Trading Algorithms, Value-at-Risk, and Credit Risk. He has published more than fifty papers and a book, and has written various computer codes. September 29, 1999 Dan Rosen, Algorithmics Inc. "An Integrated Market and Credit Risk Portfolio Model" We present a multi-step model to measure portfolio credit risk that integrates exposure simulation and portfolio credit risk techniques. Thus, it overcomes the major limitation shared by current models when measuring portfolio credit risk of portfolios that contain derivatives. Specifically, the model is an improvement over current portfolio credit risk models in three main aspects. First, it defines explicitly the joint evolution of market factors and credit drivers over time. Second, it models directly stochastic exposures through simulation, as in counterparty credit exposure models. Finally, it extends the Merton model of default to multiple steps. The model is computationally efficient because it combines a Mark-to-Future framework of counterparty exposures and a conditional default probability framework. We also demonstrate how the modeling framework handles naturally "wrong way exposures" as a by product. Therefore, it provides meaningful exposure distributions for counterparty limits management when the exposures are correlated to the market. Dr. Dan Rosen is Director of Research at Algorithmics Incorporated. In this role, he is responsible for the company's financial and mathematical research, as well as joint projects with academic institutions. Dr. Rosen joined Algorithmics in 1995 and was promoted to his current position in 1996. He has headed the design of various market risk management tools, credit risk methodologies, advanced simulation and optimization techniques, as well as their application to several industrial settings. Dr. Rosen is also one of the founders of RiskLab, a network of research centers in Mathematics and Computational Finance, initiated by Algorithmics and the University of Toronto. Prior to joining Algorithmics, he was a research associate at the University of Toronto's Centre for Management of Technology, where he initiated and coordinated the Performance Analysis Research Program for the Financial Services Industry. He holds several degrees, including a M.A.Sc. and Ph.D. in Applied Sciences from the University of Toronto. Dr. Rosen has authored numerous papers on applied mathematics and operations research applications to banking and finance, and lectures extensively on market and credit risk and financial engineering.
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