Showing papers on "Mathematical finance published in 2000"

Introduction to Econophysics: Correlations and Complexity in Finance

Abstract: This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

Pricing Via Utility Maximization and Entropy

Abstract: In a financial market model with constraints on the portfolios, define the price for a claim C as the smallest real number p such that supπ E[U(XTx+p, π−C)]≥ supπ E[U(XTx, π)], where U is the negative exponential utility function and Xx, π is the wealth associated with portfolio π and initial value x. We give the relations of this price with minimal entropy or fair price in the flavor of Karatzas and Kou (1996) and superreplication. Using dynamical methods, we characterize the price equation, which is a quadratic Backward SDE, and describe the optimal wealth and portfolio. Further use of Backward SDE techniques allows for easy determination of the pricing function properties.

Essentials of stochastic finance

Abstract: There was and still is a big "public noise" about applications of stochastics to financial business. At the core lies a deep interconnection between two seemingly quite different areas Stochastic Calculus and Financial Stock Markets. Stochastic Calculus (SC) is a well-developed and well-established branch of contemporary mathematics requiring advanced ideas and techniques. Who could have predicted, say 30-40 years ago, that SC would become the "right" approach to analyzing complicated phenomena occurring in the world stock markets? Not only has SC become a powerful theory but it has also become a powerful tool used in practice at the highest levels of decision-making in the financial world. Many universities throughout the world now have joint graduate programs in mathematics and finance. The field has attracted first-rate probabilists and statisticians, as well as practitioners in finance, and many of these have made great contributions to both the fundamental theory and the actual practice of finance. The author of "Essentials of Stochastic Finance" is so well-known that he needs no

Theory of financial risk and derivative pricing

Abstract: Risk control and derivative pricing have become of major concern to financial institutions, and there is a real need for adequate statistical tools to measure and anticipate the amplitude of the potential moves of the financial markets Summarising theoretical developments in the field, this 2003 second edition has been substantially expanded Additional chapters now cover stochastic processes, Monte-Carlo methods, Black-Scholes theory, the theory of the yield curve, and Minority Game There are discussions on aspects of data analysis, financial products, non-linear correlations, and herding, feedback and agent based models This book has become a classic reference for graduate students and researchers working in econophysics and mathematical finance, and for quantitative analysts working on risk management, derivative pricing and quantitative trading strategies

Continuous-Time Methods in Finance: A Review and an Assessment

Abstract: I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting. Merton ~1973b! also showed how such a framework can be used to develop equilibrium asset pricing implications, thereby significantly extending the asset pricing theory to richer dynamic settings and expanding the scope of applications of continuous-time methods to study problems in financial economics. 1 Within a span of about 30 years from the publication of Merton’s inf luential papers, continuous-time methods have become an integral part of financial economics. Indeed, in certain core areas in finance ~such as, e.g., asset pricing, derivatives valuation, term structure theory, and portfolio selection! continuoustime methods have proved to be the most attractive way to conduct research and gain economic intuition. The continuous-time approach in these areas has produced models with a rich variety of testable implications. The econometric theory for testing continuous-time models has made rapid strides in the last decade and has thus kept pace with the impressive progress on the theoretical front. One hopes that the actual empirical investigations and estimation using the new procedures will follow suit soon.

Paul Wilmott Introduces Quantitative Finance

Abstract: Paul Wilmott Introduces Quantitative Finance, Second Edition is an accessible introduction to the classical side of quantitative finance specifically for university students. Adapted from the comprehensive, even epic, works Derivatives and Paul Wilmott on Quantitative Finance, Second Edition, it includes carefully selected chapters to give the student a thorough understanding of futures, options and numerical methods. Software is included to help visualize the most important ideas and to show how techniques are implemented in practice. There are comprehensive end-of-chapter exercises to test students on their understanding.

Optimal risk and dividend distribution control models for an insurance company

Abstract: The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption/investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models.¶ In a typical model of this sort, in the absence of control, the reserve (surplus) process, which represents the liquid assets of the company, is governed by a Brownian motion with constant drift and diffusion coefficient. This is a limiting case of the classical Cramer-Lundberg model in which the reserve is a compound Poisson process, amended by a linear term, representing a constant influx of the insurance premiums. Risk control action corresponds to reinsuring part of the claims the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company. This translates into controlling the drift and the diffusion coefficient of the approximating process. The dividend distribution policy consists of choosing the times and the amounts of dividends to be paid out to shareholders. Mathematically, the cumulative dividend process is described by an increasing functional which may or may not be continuous with respect to time.¶ The objective in the models presented here is maximization of the dividend pay-outs. We will discuss models with different types of conditions imposed upon a company and different types of reinsurances available, such as proportional, noncheap, proportional in a presence of a constant debt liability, excess-of-loss. We will show that in most cases the optimal dividend distribution scheme is of a barrier type, while the risk control policy depends significantly on the nature of the reinsurance available.

Pricing double barrier options using Laplace transforms

Abstract: In this paper we address the pricing of double barrier options. To derive the density function of the first-hit times of the barriers, we analytically invert the Laplace transform by contour integration. With these barrier densities, we derive pricing formulaefor new types of barrier options: knock-out barrier options which pay a rebate when either one of the barriers is hit. Furthermore we discuss more complicated types of barrier options like double knock-in options.

On mathematical finance

Abstract: 01. Provide a thorough grounding in the mathematics underlying modern finance theory 02. Develop students’ powers of inquiry, critical analysis and logical thinking and to apply theoretical knowledge to current issues of policy and practice. 03. Provide a thorough training in financial mathematics for careers in such areas as financial engineering, risk and investment management and derivative pricing. 04. Provide many of the tools required to undertake high quality research in academic and financial institutions.

Risk sensitive asset management with transaction costs

Abstract: This paper develops a continuous time risk-sensitive portfolio optimization model with a general transaction cost structure and where the individual securities or asset categories are explicitly affected by underlying economic factors. The security prices and factors follow diffusion processes with the drift and diffusion coefficients for the securities being functions of the factor levels. We develop methods of risk sensitive impulsive control theory in order to maximize an infinite horizon objective that is natural and features the long run expected growth rate, the asymptotic variance, and a single risk aversion parameter. The optimal trading strategy has a simple characterization in terms of the security prices and the factor levels. Moreover, it can be computed by solving a {\it risk sensitive quasi-variational inequality}. The Kelly criterion case is also studied, and the various results are related to the recent work by Morton and Pliska.

Markov-functional interest rate models

Abstract: . We introduce a general class of interest rate models in which the value of pure discount bonds can be expressed as a functional of some (low-dimensional) Markov process. At the abstract level this class includes all current models of practical importance. By specifying these models in Markov-functional form, we obtain a specification which is efficient to implement. An additional advantage of Markov-functional models is the fact that the specification of the model can be such that the forward rate distribution implied by market option prices can be fitted exactly, which makes these models particularly suited for derivatives pricing. We give examples of Markov-functional models that are fitted to market prices of caps/floors and swaptions.

Discrete time option pricing with flexible volatility estimation

Abstract: By extending the GARCH option pricing model of Duan (1995) to more flexible volatility estimation it is shown that the prices of out-of-the-money options strongly depend on volatility features such as asymmetry. Results are provided for the properties of the stationary pricing distribution in the case of a threshold GARCH model. For a stock index series with a pronounced leverage effect, simulated threshold GARCH option prices are substantially closer to observed market prices than the Black/Scholes and simulated GARCH prices.

Irreversible investment problems

Abstract: This paper mathematically treats the following economic problem: A company wants to expand its capacity in investments that are irreversible. The problem is to find the best investment strategy taking the fluctuating market into account. We give some implicit conditions for a solution in the case where the market process is n-dimensional and an explicit solution in the one dimensional case.

Loeb Measures in Practice: Recent Advances

Abstract: 1. Loeb Measures 1.1 Introduction 1.2 Nonstandard Analysis 1.2.1 The hyperreals 1.2.2 The nonstandard universe 1.2.3 N1-saturation 1.2.4 Nonstandard topology 1.3 Construction of Loeb Measures 1.3.1 Example: Lebesgue measure 1.3.2 Example: Haar measure 1.3.3 Example: Wiener measure 1.3.4 Loeb measurable functions 1.4 Loeb Integration Theory 1.5 Elementary Applications 1.5.1 Lebesgue integration 1.5.2 Peano's Existence Theorem 1.5.3 Ito integration and stochastic differential equations 2 Stochastic Fluid Mechanics 2.1 Introduction 2.1.1 Function spaces 2.1.2 Functional formulation of the Navier-Stokes equations 2.1.3 Definition of solutions to the stochastic Navier-Stokes equations 2.1.4 Nonstandard topology in Hilbert spaces 2.2 Solution of the Deterministic Navier-Stokes Equations 2.2.1 Uniqueness 2.3 Solution of the Stochastic Navier-Stokes Equations 2.3.1 Stochastic Flow 2.3.2 Nonhomogeneous stochastic Navier-Stokes equations 2.4 Stochastic Euler Equations 2.5 Statistical Solutions 2.5.1 The Foias equation 2.5.2 Construction of statistical solutions using Loeb measures 2.5.3 Measures by nonstandard densities 2.5.4 Construction of statistical solutions using nonstandard densities 2.5.5 Statistical solutions for stochastic Navier-Stokes equations 2.6 Attractors for the Navier-Stokes Equations 2.6.1 Introduction 2.6.2 Nonstandard attractors and standard attractors 2.6.3 Attractors for 3-dimensional Navier-Stokes equations 2.7 Measure Attractors for Stochastic Navier-Stokes Equations 2.8 Stochastic Attractors for Navier-Stokes Equations 2.8.1 Stochastic attractors 2.8.2 Existence of a stochastic attractor for the Navier-Stokes equations 2.9 Attractors for the 3-dimensional Stochastic Navier-Stokes Equations 3. Stochastic Calculus of Variations 3.1 Introduction 3.1.1 Notation 3.2 Flat Integral Representation of Wiener Measure 3.3 The Wiener Sphere 3.4 Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck Process 3.5 Malliavin Calculus 3.5.1 Notation and preliminaries 3.5.2 The Wiener-Ito chaos decomposition 3.5.3 The derivation operator 3.5.4 The Skorohod integral 3.5.5 The Malliavin operator 4. Mathematical Finance Theory 4.1 Introduction 4.2 The Cox-Ross-Rubinstein Models 4.3 Options and Contingent Claims 4.3.1 Pricing a claim 4.4 The Black-Scholes Model 4.5 The Black-Scholes Model and Hyperfinite CRR Models 4.5.1 The Black-Scholes formula 4.5.2 General claims 4.6 Convergence of Market Models 4.7 Discretisation Schemes 4.8 Further Developments 4.8.1 Poisson pricing models 4.8.2 American options 4.8.3 Incomplete markets 4.8.4 Fractional Brownian motion 4.8.5 Interest rates Index

Option pricing impact of alternative continuous time dynamics for discretely observed stock prices

Abstract: In the present paper we construct stock-price processes with the same marginal lognormal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can assume any value in-between the no-arbitrage lower and upper bounds. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants that are relevant for trading. The findings of the paper are inspired by a theoretical result, linking density-evolution of diffusion processes to exponential families. Such result is briefly reviewed in an appendix.

The Mathematics of Finance: Modeling and Hedging

Abstract: This book is ideally suited for an introductory undergraduate course on financial engineering. It explains the basic concepts of financial derivatives, including put and call options, as well as more complex derivatives such as barrier options and options on futures contracts. Both discrete and continuous models of market behavior are developed in this book. In particular, the analysis of option prices developed by Black and Scholes is explained in a self-contained way, using both the probabilistic Brownian Motion method and the analytical differential equations method. The book begins with binomial stock price models, moves on to multistage models, then to the Cox - Ross - Rubinstein option pricing process, and then to the Black - Scholes formula. Other topics presented include Zero Coupon Bonds, forward rates, the yield curve, and several bond price models. The book continues with foreign exchange models and the Keynes Interest Rate Parity Formula, and concludes with the study of country risk, a topic not inappropriate for the times. In addition to theoretical results, numerical models are presented in much detail. Each of the eleven chapters includes a variety of exercises.

Random Evolutions and their Applications: New Trends

Abstract: Preface. List of Notations. Introduction. 1. Random Evolutions (RE). 2. Stochastic Evoluationary Systems. 3. Random Evolution Equations Driven by Space-Time White Noise. 4. Analogue of Dynkin's Formula (ADF) for Multiplicative Operator Functionals (MOF), RE and SES. 5. Boundary Value Problems (BVP) for RE and SES. 6. Stochastic Stability of RE and SES. 7. Stochastic Optimal Control of Random Evolutions and SES. 8. Statistics of SES. 9. Random Evolutions in Financial Mathematics. Incomplete Market. 10. Random Evolutions in Insurance Mathematics. Incomplete Market. 11. Stochastic Stability of Financial and Insurance Stochastic Models. 12. Stochastic Optimal Control of Financial and Insurance Stochastic Models. 13. Statistics of Financial Stochastic Models. Bibliography. Index.

Superreplication in stochastic volatility models and optimal stopping

Abstract: In this paper we discuss the superreplication of derivatives in a stochastic volatility model under the additional assumption that the volatility follows a bounded process. We characterize the value process of our superhedging strategy by an optimal-stopping problem in the context of the Black-Scholes model which is similar to the optimal stopping problem that arises in the pricing of American-type derivatives. Our proof is based on probabilistic arguments. We study the minimality of these superhedging strategies and discuss PDE-characterizations of the value function of our superhedging strategy. We illustrate our approach by examples and simulations.

Price Functionals with Bid-Ask Spreads: An Axiomatic Approach

Abstract: In Jouini and Kallal [Jouini, E., Kallal, H., 1995. Martinagles and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66 (1) 178-197], the authors characterized the absence of arbitrage opportunities for contingent claims with cash delivery in the presence of bid–ask spreads. Other authors obtained similar results for amore general definition of the contingent claims but assuming some specific price processes and transaction costs rather than bid–ask spreadsin general (see for instance, Cvitanic and Karatzas [Cvitanic, J., Karatzas, I., 1996. Hedging andportfolio optimization under transaction costs: a martinangle approach. Mathematical Finance 6,133-166]). The main difference consists of the fact that the bid–ask ratio is constant in this lastreference. This assumption does not permit to encompass situations where the prices are determinedby the buying and selling limit orders or by a (resp. competitive) specialist (resp. market-makers).Wederive in this paper some implications from the no-arbitrage assumption on the price functionalsthat generalizes all the previous results in a very general setting. Indeed, under some minimalassumptions on the price functional, we prove that the prices of the contingent claims are necessarilyin some minimal interval. This result opens the way to many empirical analyses

On the Approximation of Optimal Stopping Problems with Application to Financial Mathematics

Abstract: The determination of the value function associated with a given reward and stochastic process represents an important class of stochastic control problem. In particular, the expectations of such processes may be represented as solutions of variational inequalities of evolutionary type typically characterized by their high number of degrees of freedom, unbounded domains, and lack of "natural" boundary conditions. In this paper, we introduce two methodologies for computing the value function of optimal stopping associated with general stochastic processes. Our results are implemented utilizing finite elements and are validated using problems taken from financial mathematics.

Black–Scholes option pricing within Itô and Stratonovich conventions

Abstract: Options are financial instruments designed to protect investors from the stock market randomness. In 1973, Black, Scholes and Merton proposed a very popular option pricing method using stochastic differential equations within the Ito interpretation. Herein, we derive the Black–Scholes equation for the option price using the Stratonovich calculus along with a comprehensive review, aimed to physicists, of the classical option pricing method based on the Ito calculus. We show, as can be expected, that the Black–Scholes equation is independent of the interpretation chosen. We nonetheless point out the many subtleties underlying Black–Scholes option pricing method.

Modelling of stock price changes: a real analysis approach

Abstract: . In this paper a real analysis approach to stock price modelling is considered. A stock price and its return are defined in a duality to each other provided there exist suitable limits along a sequence of nested partitions of a time interval, mimicking sum and product integrals. It extends the class of stochastic processes susceptible to theoretical analysis. Also, it is shown that extended classical calculus is applicable to market analysis whenever the local 2–variation of sample functions of the return is zero, or is determined by jumps if the process is discontinuous. In particular, an extended Riemann-Stieltjes integral is used in that case to prove several properties of trading strategies.

Robustness of the Black-Scholes approach in the case of options on several assets

Abstract: . In this paper we analyse a stochastic volatility model that is an extension of the traditional Black-Scholes one. We price European options on several assets by using a superstrategy approach. We characterize the Markov superstrategies, and show that they are linked to a nonlinear PDE, called the Black-Scholes-Barenblatt (BSB) equation. This equation is the Hamilton-Jacobi-Bellman equation of an optimal control problem, which has a nice financial interpretation. Then we analyse the optimization problem included in the BSB equation and give some sufficient conditions for reduction of the BSB equation to a linear Black-Scholes equation. Some examples are given.

Pricing Derivative Securities

Abstract: This book presents techniques for valuing derivative securities at a level suitable for practitioners, students in doctoral programs in economics and finance, and those in masters-level programs in financial mathematics and computational finance. It provides the necessary mathematical tools from analysis, probability theory, the theory of stochastic processes, and stochastic calculus, making extensive use of examples. It also covers pricing theory, with emphasis on martingale methods. The chapters are organized around the assumptions made about the dynamics of underlying price processes. Readers begin with simple, discrete-time models that require little mathematical sophistication, proceed to the basic Black-Scholes theory, and then advance to continuous-time models with multiple risk sources. The second edition takes account of the major developments in the field since 2000. New topics include the use of simulation to price American-style derivatives, a new one-step approach to pricing options by inverting characteristic functions, and models that allow jumps in volatility and Markov-driven changes in regime. The new chapter on interest-rate derivatives includes extensive coverage of the LIBOR market model and an introduction to the modeling of credit risk. As a supplement to the text, the book contains an accompanying CD-ROM with user-friendly FORTRAN, C++, and VBA program components.

Computational Finance: A Scientific Perspective

Abstract: Computational finance deals with the mathematics of computer programs that realize financial models or systems. This book outlines the epistemic risks associated with the current valuations of different financial instruments and discusses the corresponding risk management strategies. It covers most of the research and practical areas in computational finance. Starting from traditional fundamental analysis and using algebraic and geometric tools, it is guided by the logic of science to explore information from financial data without prejudice. In fact, this book has the unique feature that it is structured around the simple requirement of objective science: the geometric structure of the data = the information contained in the data.