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Infinitesimal Operator Based Methods for Continuous-time Finance Models

Infinitesimal Operator Based Methods for Continuous-time Finance Models PDF Author: Zhaogang Song
Publisher:
ISBN:
Category :
Languages : en
Pages : 376

Book Description
Continuous time Markov processes, including diffusion, jump-diffusion and Levy jump-diffusion models, have become an essential tool of modern finance over the past three decades. Nowadays, they are widely used in modeling dynamics of, for instance, interest rates, stock prices, exchange rates and option prices. However, data are always recorded at discrete points in time, e.g., monthly, weekly, and daily, although these models are formulated in continuous time. This feature makes most econometric inferential procedures developed for discrete time econometrics unsuitable for continuous time models and complicates the econometric analysis considerably. For example, estimators obtained by applying discrete time econometric methods to the discretized version of continuous time models are not consistent for a fixed sampling interval. More seriously, although the maximum likelihood method is a very appealing econometric procedure due to its nice properties like efficiency, the transition density and hence likelihood function of most continuous time Markov models have no analytic expressions. This poses a serious impediment for the implementation of likelihood procedures. Many approaches have been proposed to deal with this problem but they either incur substantive computation burdens especially for multivariate cases or involve complicated approximation formulas with limited applicability. Consequently, there is a strong need for convenient econometric methodologies designed for continuous time mod- els given discrete sampled data. Unlike the transition density, the infinitesimal operator, as an important mathematical tool in probability theory, enjoys the nice property of being a closed-form expression of drift, diffusion and jump terms of the process. As a result, no approximated formulas or simulation based implementations are needed. Furthermore, it is equivalent to the transition density in characterizing the complete dynamics of the processes. Based on this convenient infinitesimal operator, this dissertation proposes a sequence of econometric procedures for continuous time Markov models with applications to affine jump diffusion (AJD) term structure models of interest rates. It is divided into four chapters. In the first chapter, "Infinitesimal Operator Based Estimation for Continuous Time Markov Processes", I propose an estimation method based on the infinitesimal operator for general multivariate continuous-time Markov processes, which cover diffusion, jump-diffusion and Levy-driven jump models as special cases. A conditional moment restriction is first obtained via the infinitesimal operator based identification of the process. Then an empirical likelihood type estimator is constructed by a kernel smoothing approach. Unlike the transition density which is rarely available in closed-form, the infinitesimal operator has an analytic form for all continuous time Markov models. As a result, different from the maximum likelihood estimator (MLE) which involves either numerical or simulated transition densities, the proposed estimator can be conveniently implemented by plugging in parametric components of the models. Furthermore, I prove that the proposed estimator attains the semi-parametric efficiency bound for conditional moment restrictions models of Markov processes and hence is asymptotically efficient. Simulation studies show that the proposed estimator has good finite sample performances comparable to the MLE. In the empirical application, I estimate Levy jump diffusion models for daily Euro/Dollar (2000-2010) and Yen/Dollar (1990-2000) rates. Results show that Levy jumps are important components in exchange rate dynamics and Poissontype jump diffusion models cannot capture them. In the second chapter, "Expectation Puzzles, Time Varying Conditional Volatility, and Jumps in Affine Term Structure Models", I study how jumps in interest rates, which are well documented in the literature, affect the term structure dynamics of the LIBOR-Swap curve in a multivariate AJD model. The motivation is that affine diffusion (AD) term structure models, as the major framework for interest rate dynamics, face two empirical challenges: first, they ignore well-documented jumps in interest rates as the state variables follow affine diffusions; second, they fail to capture simultaneously time variations in risk premiums implied by the violations of the "expectation hypothesis" and time variations in volatilities which are critical for pricing fixed-income derivatives. In this paper, I develop a multivariate AJD term structure model that overcomes these two challenges. Using LIBOR-Swap yields from 1990 to 2008, I estimate three-factor AJD models with infinitesimal operator methods and examine the contributions of jumps to term structure dynamics. I find that jumps are state dependent and negative. The risk premium is positive for jump size risk and negative for jump time risk, while the total jump risk premium is positive. Jump risk premiums lead to flexible time-varying market prices of risks without restricting time variations in conditional volatilities. As a result, two models in the three-factor AJD class capture time variations in both the risk premium and conditional volatility of LIBOR-Swap yields simultaneously. In the third chapter (part of this chapter has been published as Song (2011) in Journal of Econometrics, 162-2, 189-212.), "A Martingale Approach for Testing Diffusion Models Based on Infinitesimal Operator", I develop an omnibus specification test for diffusion models based on the infinitesimal operator instead of the transition density extensively used in literature. The infinitesimal operator based identification of the diffusion process is equivalent to a "martingale hypothesis" for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the "martingale hypothesis" via a multivariate generalized spectral derivative based approach which delivers an N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process enjoys the nice property of being a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications and about their interactions. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performances. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift mis-specification plays an important role in such rejections. In the fourth chapter, "Estimating Semi-Parametric Diffusion Models with Unrestricted Volatility via Infinitesimal Operator", two generalized method of moments estimators are proposed for the drift parameters in both univariate and multivariate semi-parametric diffusion models with unrestricted volatility based on the infinitesimal operator. The first estimator is obtained by integrating out the diffusion function via the quadratic variation (co-variation), which is estimated by the realized volatility (covariance) in a first step using high frequency data. The second is constructed based on the separate identification condition and is actually applicable for a general instantaneous conditional mean model in continuous time, which covers the stochastic volatility and jump diffusion models as special cases. Simulation studies show that they possess fairly good finite sample performances.

Infinitesimal Operator Based Methods for Continuous-time Finance Models

Infinitesimal Operator Based Methods for Continuous-time Finance Models PDF Author: Zhaogang Song
Publisher:
ISBN:
Category :
Languages : en
Pages : 376

Book Description
Continuous time Markov processes, including diffusion, jump-diffusion and Levy jump-diffusion models, have become an essential tool of modern finance over the past three decades. Nowadays, they are widely used in modeling dynamics of, for instance, interest rates, stock prices, exchange rates and option prices. However, data are always recorded at discrete points in time, e.g., monthly, weekly, and daily, although these models are formulated in continuous time. This feature makes most econometric inferential procedures developed for discrete time econometrics unsuitable for continuous time models and complicates the econometric analysis considerably. For example, estimators obtained by applying discrete time econometric methods to the discretized version of continuous time models are not consistent for a fixed sampling interval. More seriously, although the maximum likelihood method is a very appealing econometric procedure due to its nice properties like efficiency, the transition density and hence likelihood function of most continuous time Markov models have no analytic expressions. This poses a serious impediment for the implementation of likelihood procedures. Many approaches have been proposed to deal with this problem but they either incur substantive computation burdens especially for multivariate cases or involve complicated approximation formulas with limited applicability. Consequently, there is a strong need for convenient econometric methodologies designed for continuous time mod- els given discrete sampled data. Unlike the transition density, the infinitesimal operator, as an important mathematical tool in probability theory, enjoys the nice property of being a closed-form expression of drift, diffusion and jump terms of the process. As a result, no approximated formulas or simulation based implementations are needed. Furthermore, it is equivalent to the transition density in characterizing the complete dynamics of the processes. Based on this convenient infinitesimal operator, this dissertation proposes a sequence of econometric procedures for continuous time Markov models with applications to affine jump diffusion (AJD) term structure models of interest rates. It is divided into four chapters. In the first chapter, "Infinitesimal Operator Based Estimation for Continuous Time Markov Processes", I propose an estimation method based on the infinitesimal operator for general multivariate continuous-time Markov processes, which cover diffusion, jump-diffusion and Levy-driven jump models as special cases. A conditional moment restriction is first obtained via the infinitesimal operator based identification of the process. Then an empirical likelihood type estimator is constructed by a kernel smoothing approach. Unlike the transition density which is rarely available in closed-form, the infinitesimal operator has an analytic form for all continuous time Markov models. As a result, different from the maximum likelihood estimator (MLE) which involves either numerical or simulated transition densities, the proposed estimator can be conveniently implemented by plugging in parametric components of the models. Furthermore, I prove that the proposed estimator attains the semi-parametric efficiency bound for conditional moment restrictions models of Markov processes and hence is asymptotically efficient. Simulation studies show that the proposed estimator has good finite sample performances comparable to the MLE. In the empirical application, I estimate Levy jump diffusion models for daily Euro/Dollar (2000-2010) and Yen/Dollar (1990-2000) rates. Results show that Levy jumps are important components in exchange rate dynamics and Poissontype jump diffusion models cannot capture them. In the second chapter, "Expectation Puzzles, Time Varying Conditional Volatility, and Jumps in Affine Term Structure Models", I study how jumps in interest rates, which are well documented in the literature, affect the term structure dynamics of the LIBOR-Swap curve in a multivariate AJD model. The motivation is that affine diffusion (AD) term structure models, as the major framework for interest rate dynamics, face two empirical challenges: first, they ignore well-documented jumps in interest rates as the state variables follow affine diffusions; second, they fail to capture simultaneously time variations in risk premiums implied by the violations of the "expectation hypothesis" and time variations in volatilities which are critical for pricing fixed-income derivatives. In this paper, I develop a multivariate AJD term structure model that overcomes these two challenges. Using LIBOR-Swap yields from 1990 to 2008, I estimate three-factor AJD models with infinitesimal operator methods and examine the contributions of jumps to term structure dynamics. I find that jumps are state dependent and negative. The risk premium is positive for jump size risk and negative for jump time risk, while the total jump risk premium is positive. Jump risk premiums lead to flexible time-varying market prices of risks without restricting time variations in conditional volatilities. As a result, two models in the three-factor AJD class capture time variations in both the risk premium and conditional volatility of LIBOR-Swap yields simultaneously. In the third chapter (part of this chapter has been published as Song (2011) in Journal of Econometrics, 162-2, 189-212.), "A Martingale Approach for Testing Diffusion Models Based on Infinitesimal Operator", I develop an omnibus specification test for diffusion models based on the infinitesimal operator instead of the transition density extensively used in literature. The infinitesimal operator based identification of the diffusion process is equivalent to a "martingale hypothesis" for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the "martingale hypothesis" via a multivariate generalized spectral derivative based approach which delivers an N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process enjoys the nice property of being a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications and about their interactions. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performances. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift mis-specification plays an important role in such rejections. In the fourth chapter, "Estimating Semi-Parametric Diffusion Models with Unrestricted Volatility via Infinitesimal Operator", two generalized method of moments estimators are proposed for the drift parameters in both univariate and multivariate semi-parametric diffusion models with unrestricted volatility based on the infinitesimal operator. The first estimator is obtained by integrating out the diffusion function via the quadratic variation (co-variation), which is estimated by the realized volatility (covariance) in a first step using high frequency data. The second is constructed based on the separate identification condition and is actually applicable for a general instantaneous conditional mean model in continuous time, which covers the stochastic volatility and jump diffusion models as special cases. Simulation studies show that they possess fairly good finite sample performances.

Principles of Infinitesimal Stochastic and Financial Analysis

Principles of Infinitesimal Stochastic and Financial Analysis PDF Author: Imme van den Berg
Publisher: World Scientific
ISBN: 9789810243586
Category : Mathematics
Languages : en
Pages : 156

Book Description
There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level. The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude -- in the sense of nonstandard analysis -- are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.

Handbook of Financial Econometrics

Handbook of Financial Econometrics PDF Author: Yacine Ait-Sahalia
Publisher: Elsevier
ISBN: 0080929842
Category : Business & Economics
Languages : en
Pages : 809

Book Description
This collection of original articles—8 years in the making—shines a bright light on recent advances in financial econometrics. From a survey of mathematical and statistical tools for understanding nonlinear Markov processes to an exploration of the time-series evolution of the risk-return tradeoff for stock market investment, noted scholars Yacine Aït-Sahalia and Lars Peter Hansen benchmark the current state of knowledge while contributors build a framework for its growth. Whether in the presence of statistical uncertainty or the proven advantages and limitations of value at risk models, readers will discover that they can set few constraints on the value of this long-awaited volume. - Presents a broad survey of current research—from local characterizations of the Markov process dynamics to financial market trading activity - Contributors include Nobel Laureate Robert Engle and leading econometricians - Offers a clarity of method and explanation unavailable in other financial econometrics collections

Hypermodels in Mathematical Finance

Hypermodels in Mathematical Finance PDF Author: Siu-Ah Ng
Publisher: World Scientific
ISBN: 9810244282
Category : Business & Economics
Languages : en
Pages : 313

Book Description
At the beginning of the new millennium, two unstoppable processes are taking place in the world: (1) globalization of the economy; (2) information revolution. As a consequence, there is greater participation of the world population in capital market investment, such as bonds and stocks and their derivatives. Hence there is a need for risk management and analytic theory explaining the market. This leads to quantitative tools based on mathematical methods, i.e. the theory of mathematical finance.Ever since the pioneer work of Black, Scholes and Merton in the 70's, there has been rapid growth in the study of mathematical finance, involving ever more sophisticated mathematics. However, from the practitioner's point of view, it is desirable to have simpler and more useful mathematical tools.This book introduces research students and practitioners to the intuitive but rigorous hypermodel techniques in finance. It is based on Robinson's infinitesimal analysis, which is easily grasped by anyone with as little background as first-year calculus. It covers topics such as pricing derivative securities (including the Black-Scholes formula), hedging, term structure models of interest rates, consumption and equilibrium. The reader is introduced to mathematical tools needed for the aforementioned topics. Mathematical proofs and details are given in an appendix. Some programs in MATHEMATICA are also included.

Change of Time Methods in Quantitative Finance

Change of Time Methods in Quantitative Finance PDF Author: Anatoliy Swishchuk
Publisher: Springer
ISBN: 331932408X
Category : Mathematics
Languages : en
Pages : 140

Book Description
This book is devoted to the history of Change of Time Methods (CTM), the connections of CTM to stochastic volatilities and finance, fundamental aspects of the theory of CTM, basic concepts, and its properties. An emphasis is given on many applications of CTM in financial and energy markets, and the presented numerical examples are based on real data. The change of time method is applied to derive the well-known Black-Scholes formula for European call options, and to derive an explicit option pricing formula for a European call option for a mean-reverting model for commodity prices. Explicit formulas are also derived for variance and volatility swaps for financial markets with a stochastic volatility following a classical and delayed Heston model. The CTM is applied to price financial and energy derivatives for one-factor and multi-factor alpha-stable Levy-based models. Readers should have a basic knowledge of probability and statistics, and some familiarity with stochastic processes, such as Brownian motion, Levy process and martingale.

Time-Inconsistent Control Theory with Finance Applications

Time-Inconsistent Control Theory with Finance Applications PDF Author: Tomas Björk
Publisher: Springer Nature
ISBN: 3030818438
Category : Mathematics
Languages : en
Pages : 328

Book Description
This book is devoted to problems of stochastic control and stopping that are time inconsistent in the sense that they do not admit a Bellman optimality principle. These problems are cast in a game-theoretic framework, with the focus on subgame-perfect Nash equilibrium strategies. The general theory is illustrated with a number of finance applications. In dynamic choice problems, time inconsistency is the rule rather than the exception. Indeed, as Robert H. Strotz pointed out in his seminal 1955 paper, relaxing the widely used ad hoc assumption of exponential discounting gives rise to time inconsistency. Other famous examples of time inconsistency include mean-variance portfolio choice and prospect theory in a dynamic context. For such models, the very concept of optimality becomes problematic, as the decision maker’s preferences change over time in a temporally inconsistent way. In this book, a time-inconsistent problem is viewed as a non-cooperative game between the agent’s current and future selves, with the objective of finding intrapersonal equilibria in the game-theoretic sense. A range of finance applications are provided, including problems with non-exponential discounting, mean-variance objective, time-inconsistent linear quadratic regulator, probability distortion, and market equilibrium with time-inconsistent preferences. Time-Inconsistent Control Theory with Finance Applications offers the first comprehensive treatment of time-inconsistent control and stopping problems, in both continuous and discrete time, and in the context of finance applications. Intended for researchers and graduate students in the fields of finance and economics, it includes a review of the standard time-consistent results, bibliographical notes, as well as detailed examples showcasing time inconsistency problems. For the reader unacquainted with standard arbitrage theory, an appendix provides a toolbox of material needed for the book.

Specification Analysis of Continuous Time Models in Finance

Specification Analysis of Continuous Time Models in Finance PDF Author: A. Ronald Gallant
Publisher:
ISBN:
Category :
Languages : en
Pages : 31

Book Description
The paper describes the use of the Gallant-Tauchen efficient method of moments (EMM) technique for diagnostic checking of stochastic differential equations (SDEs) estimated from financial market data. The EMM technique is a simulation-based method that uses the score function of an auxiliary model as the criterion to define a generalized method of moments (GMM) objective function. The technique can handle multivariate SDEs where the state vector is not completely observed. The optimized GMM objective function is distributed as chi-square and may be used to test model adequacy. Elements of the score function correspond to specific parameters and large values reflect features of data that a rejected SDE specification does not describe well. The diagnostics are illustrated by estimating a three-factor model to weekly, 1962-1995, term structure data comprised of short (3 month), medium (12 month), and long (10 year) Treasury rates. The Yield-Factor Model is sharply rejected, although an extension that permits the local variance function to be a convex function of the interest rates comes much closer to describing the data.

Continuous-time Asset Pricing Models in Applied Stochastic Finance

Continuous-time Asset Pricing Models in Applied Stochastic Finance PDF Author: P. C. G. Vassiliou
Publisher: Wiley-ISTE
ISBN: 9781848211599
Category : Mathematics
Languages : en
Pages : 0

Book Description
Stochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging this kind of risk. These two volumes aim to provide a foundation course on applied stochastic finance. They are designed for three groups of readers: firstly, students of various backgrounds seeking a core knowledge on the subject of stochastic finance; secondly financial analysts and practitioners in the investment, banking and insurance industries; and finally other professionals who are interested in learning advanced mathematical and stochastic methods, which are basic knowledge in many areas, through finance. In Volume 2 we study continuous time models by presenting the necessary material from continuous martingales, measure theory and stochastic differential equations as models for various assets, such as the Wiener process, Brownian motion, etc. We then build, with many examples and intuitive explanations, the necessary stochastic analysis background i.e. Itô’s lemma, stochastic integration, Girsanovís theorem, etc. The book then guides the reader into the pricing of vanilla options in continuous time i.e. the continuous time models of Black and Scholes, followed by interest rate models and the models of Heath-Jarrow-Morton and the forward Libor model. The final part of the book presents the pricing of credit derivatives.

Continuous-Time Finance

Continuous-Time Finance PDF Author: Robert C. Merton
Publisher: Wiley-Blackwell
ISBN: 9780631185086
Category : Business & Economics
Languages : en
Pages : 754

Book Description
Robert C. Merton's widely-used text provides an overview and synthesis of finance theory from the perspective of continuous-time analysis. It covers individual finance choice, corporate finance, financial intermediation, capital markets, and selected topics on the interface between private and public finance.

Lévy Processes

Lévy Processes PDF Author: Ole E Barndorff-Nielsen
Publisher: Springer Science & Business Media
ISBN: 1461201977
Category : Mathematics
Languages : en
Pages : 414

Book Description
A Lévy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. Martingales, Markov processes, and diffusions are extensions and generalizations of these processes. In the past, representatives of the Lévy class were considered most useful for applications to either Brownian motion or the Poisson process. Nowadays the need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general Lévy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Lévy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts present surveys of recent developments, or focus on some most promising applications. Despite its special character, every topic is aimed at the non- specialist, keen on learning about the new exciting face of a rather aged class of processes. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for futurearch. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lévy processes.