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Most financial and investment decisions are based on considerations of possible future changes and require forecasts on the evolution of the financial world. Time series and processes are the natural tools for describing the dynamic behavior of financial data, leading to the required forecasts. This book presents a survey of the empirical properties of financial time series, their descriptions by means of mathematical processes, and some implications for important financial applications used in many areas like risk evaluation, option pricing or portfolio construction. The statistical tools used to extract information from raw data are introduced. Extensive multiscale empirical statistics provide a solid benchmark of stylized facts (heteroskedasticity, long memory, fat-tails, leverage…), in order to assess various mathematical structures that can capture the observed regularities. The author introduces a broad range of processes and evaluates them systematically against the benchmark, summarizing the successes and limitations of these models from an empirical point of view. The outcome is that only multiscale ARCH processes with long memory, discrete multiplicative structures and non-normal innovations are able to capture correctly the empirical properties. In particular, only a discrete time series framework allows to capture all the stylized facts in a process, whereas the stochastic calculus used in the continuum limit is too constraining. The present volume offers various applications and extensions for this class of processes including high-frequency volatility estimators, market risk evaluation, covariance estimation and multivariate extensions of the processes. The book discusses many practical implications and is addressed to practitioners and quants in the financial industry, as well as to academics, including graduate (Master or PhD level) students. The prerequisites are basic statistics and some elementary financial mathematics.
- Published on: 2014-10-15
- Released on: 2014-10-15
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .81" w x 6.10" l, 1.06 pounds
- Binding: Paperback
- 322 pages
Review
“The layout of the book is well done and very easy to read. From my experience, there are not many books of a similar approach; I believe it is quite unique in its nature. … it provides an incredible amount of information that researchers interested in both mathematical and applied finance will find it a useful resource to learn basic asset behavior. The level is right for all researchers in the area with a master’s degree in statistics.” (Stergios B. Fotopoulos, Technometrics, Vol. 58 (3), August, 2016)
“The book aims to synthesize the present status of the field, but it also represents a subjective snapshot of the current situation, as viewed by the author. It is written in a very concise and elegant way, explaining the notation used as it is required. … This book is definitely recommended to anyone (practitioners, quants, academics or graduate students) interested in attaining a deeper understanding of the dynamics of prices, as well as the corresponding stylized facts … .” (Omar Rojas, MAA Reviews, January, 2013)
From the Back Cover
Most financial and investment decisions are based on considerations of possible future changes and require forecasts on the evolution of the financial world. Time series and processes are the natural tools for describing the dynamic behavior of financial data, leading to the required forecasts.
This book presents a survey of the empirical properties of financial time series, their descriptions by means of mathematical processes, and some implications for important financial applications used in many areas like risk evaluation, option pricing or portfolio construction. The statistical tools used to extract information from raw data are introduced. Extensive multiscale empirical statistics provide a solid benchmark of stylized facts (heteroskedasticity, long memory, fat-tails, leverage…), in order to assess various mathematical structures that can capture the observed regularities. The author introduces a broad range of processes and evaluates them systematically against the benchmark, summarizing the successes and limitations of these models from an empirical point of view. The outcome is that only multiscale ARCH processes with long memory, discrete multiplicative structures and non-normal innovations are able to capture correctly the empirical properties. In particular, only a discrete time series framework allows to capture all the stylized facts in a process, whereas the stochastic calculus used in the continuum limit is too constraining. The present volume offers various applications and extensions for this class of processes including high-frequency volatility estimators, market risk evaluation, covariance estimation and multivariate extensions of the processes. The book discusses many practical implications and is addressed to practitioners and quants in the financial industry, as well as to academics, including graduate (Master or PhD level) students. The prerequisites are basic statistics and some elementary financial mathematics.
Gilles Zumbach has worked for several institutions, including banks, hedge funds and service providers and continues to be engaged in research on many topics in finance. His primary areas of interest are volatility, ARCH processes and financial applications.
About the Author
Gilles Zumbach has worked for several institutions, including banks, hedge funds and service providers, while continuing to engage in research on many topics in finance, including tick-by-tick time series, market risk evaluations and option pricing, as well as long-term forecasts, bond portfolio construction, and real-time optimization of market orders. His primary areas of interest are volatility, ARCH processes and financial applications.
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Most helpful customer reviews
3 of 3 people found the following review helpful.
Dark horse
By Dimitri Shvorob
I consider myself a fairly moderate grammar Nazi, and when I see a book littered with typos and grammar errors, I do accuse the author and the publisher of being disrespectful to the reader, and do (rhetorically) ask myself if an author and a publisher who did not bother to hire a proof-reader would bother to involve an editor or a peer reviewer - but do not automatically black-ball the book, and try to keep an open mind.
In this case, it will be appropriate to note that out of the 171 references in the book's bibliography, 25 have been written or co-written by the author (and are split about equally between SSRN links and publications in the (esteemed) Quantitative Finance journal); this helps quantify the author's relative disinterest in the work of others, and a certain eclecticism in the book's composition. The book's central theme isn't discreteness - I have no idea why Springer went with "Discrete time series, processes and applications in finance" - but subjecting a small set of univariate volatility models (several GARCH flavors, and a couple of token SV models) to a diagnostic check based primarily on (required lack of) time reversibility.
Personally, I do not find the exercise convincing - for starters, the author's empirical stylized facts come from a handful of FX series, and the model-implied counterparts are obtained with god-knows-how-selected parameter values, with both statistical and "model-based" robustness never discussed - and do not recommend the book as a reference on volatility models. (Regime-switching models get 1 page (!) of text, and the broader SV approach is not treated much fairer).
1 of 1 people found the following review helpful.
On Discrete Time Series and the construction of financial models
By Omar G. Rojas Altamirano
The main focus of the book Discrete Time Series, Processes and Applications in Finance, as Gilles Zumbach, the author states in the introduction, is the construction of mathematical processes describing financial time series. The construction of such processes relies, first of all, on describing some of the statistical empirical stylized facts exhibited by the financial time series, whereas of returns or volatilities. For this purpose, the most relevant graphs are summarized in the mug shots, which constitutes a collection of graphs that summarize the most important stylized facts of a given time series. Such a name comes from the mug shot, which, in police slang, is the pair of pictures of someone face, one front, and one profile. After presenting such facts in full color graphs, which are also available to the interested reader at the companion website [...], the author gets into constructing mathematical processes that can reproduce some, possibly all, stylized facts.
The construction of processes in finance is similar to the description of the laws of nature by mathematical formulas as pursued in physics. Such is the approach of the book. To construct the mathematical processes describing financial time series, since several chapters of the book are devoted to collecting empirical statistics, trying to extract the leading empirical facts while not using any models or processes. The focus point then is on the mathematical structure, as the parameters in a model can be modified to accommodate quantitatively one or another particular time series.
The description of financial time series by various models followed an explosive growth in the last two or three decades. The book aims to synthesize the present status of the field, but it also represents a subjective snapshot of the current situation, as viewed by the author.
The book is divided in 20 chapters of different lengths and subjects of interest. A brief review of each chapter is as follows:
Chapter 1. Introduction. In here, the author states the main focus of the book, the approach taken, the time scale used, and explains the mathematical and statistical point of view used in the construction of the processes describing financial time series.
Chapter 2. Notation, Naming, and General Definitions. It fixes the notation and gives some general definitions of returns, volatility, time series and operators of time series. A distinction between time and time intervals is made, fixing convenient notations for both of them. The problem of deseasonalizing empirical high-frequency data is also address. For a given time interval [δt], and at a specific moment it time, t, the return time series is denoted by r[δt](t). This notation, though more cumbersome than the usual r(t), becomes handy once one starts changing the time intervals in order to observe different stylized facts of financial time series.
Chapter 3. Stylized Facts. The general statistical properties that are expected to be present in the returns or asset prices of financial time series, regardless of the instruments, markets or time periods, are called stylized facts. Although there is not a general consensus about the complete list (if any) of those facts, the ones that appear most often are related to the probability density function (pdf) and the autocorrelation function (acf). The goal of such facts, extracted from empirical data, is to have enough information in order to build processes that describe accurately financial time series. Color graphs of great detail and quality are much appreciated from the reader.
Chapter 4. Empirical mug shots. In order to summarize the key findings (empirical facts) in one page that characterizes best empirical time series and processes used to model them, a mug shot is introduced. According to a footnote, a mug shot, in police slang, is the pair of pictures of someone face, one front, one profile. These empirical mug shots consist of eight pictures that characterize the heteroscedasticity, the convergence toward a Gaussian distribution, the volatility clustering, the dependency structure between time horizons, the asymmetry with respect to time reversal invariance and the dynamics of the volatility evolution. All these graphs, using foreign exchange data of some main currencies, can be obtained from the website [...]. Even if the reader appreciates the graphs, some computer code in Matlab or R would be of better use. However, as the author kindly replied to me by email when asked about it, the mug shots where created with a large C++ program and then, using scripts, the plots where created using the commercial software Tecplot.
Chapter 5. Process Overview. Having presented some stylized facts, this chapter serves as an introduction to the next goal of the book: to construct mathematical processes that can reproduce some, possibly all, stylized facts. The reasons to use discrete forms over continuous, are presented here. Amongst them, it is argued that the empirical data are known with a finite time increment. On the other hand, the problem of the continuum limit is discussed, as is the discrete nature of Monte Carlo simulations.
Chapter 6. Logarithmic Versus Relative Random Walks. Given that a random walk can be discretized in two forms, logarithmic and geometric, in this chapter a comparison between their similarities and differences is presented. Three definitions of the returns are also given (difference, logarithmic and relative). It is clearly stated that the logarithmic return definition, the most widely used in finance, is better due to its analytical tractability. Two important statements are made in this chapter: that the simplest random walk with constant volatility and Gaussian innovations is nowadays not accurate enough to be satisfactory, using computer power and financial data to describe empirical time series in a better way; secondly, the shift from analytical to simulation tools in financial time series.
Chapter 7. ARCH Processes. The core idea of the ARCH (Auto Regressive Conditional Heteroscedastic) process is to have an effective volatility depending on the recent returns. They where introduced by Robert Engle in 1982, obtaining the Nobel Memorial Prize in Economical Sciences in 2003, "for methods of analyzing economic time series with time-varying volatility (ARCH)". The underlying intuition behind such models is that large moves in the market trigger other market participants to trade, hence creating subsequent volatility. In this chapter, ARCH models and some generalizations, like GARCH, EGARCH, FIGARCH and GARTCH, are discussed.
Chapter 8. Stochastic Volatility Processes. Some processes are introduced, like the Exponential and the Heston Stochastic Volatility Processes, along with their corresponding mug shots. The basic idea for such processes is that the volatility is a genuine independent process.
Chapter 9. Regime-Switching Process. The shortest chapter of the book. The idea behind a regime-switching model is that the market can take a few states, like periods of high and low volatility. The mug shot of a three-state regime-switching process, with Student innovations and dynamics specified by a Markov chain with given transition probabilities is presented. The author concludes that the process is not a good candidate as a model for the empirical price time series, since the properties of the process are quite far from the ones of the empirical time series. However, I believe more research should be done before discrediting such processes and, if a chapter is devoted to such subject, Threshold Auto-Regressive models were in place, which have proved very fruitful in modeling non-linear financial time series of returns and volatility.
Chapter 10. Price and Volatility Using High-Frequency Data. This chapter discusses two basic models for the description at very short time of the prices. Both of these models are based on the distinction between the observed price and a hypothetical true underlying price.
Chapter 11. Time-Reversal Asymmetry. The focus of this chapter is to study various statistics related to the volatility, for time horizons ranging from 3 minutes to 3 months, in order to study the possibility of time-reversal invariance (TRI) in financial time series. The author emphasizes the systematic study of the TRI in finance and shows how the empirical time series are not TRI.
Chapter 12. Characterizing Heteroscedasticity. The goal of the chapter is to find a simple stylized description of the volatility lagged correlation that gives a good description of its shape for a broad set of time series, for lags ranging from a few days to a few days. This is achieved by performing some Monte Carlo simulations. Some very interesting conclusions with regards to the characterization of lagged correlation decay are stated and are worth a second thought from the reader.
Chapter 13. The Innovation Distributions. Another short chapter. It delves into the probability distribution for large price changes, or the tail-behavior of the prices. Such tail-behavior is widely accepted as fat-tailed. Some graphs are shown, comparing the cumulative probability function with a Student of 3 and 5 degrees of freedom, and with the standard normal distribution. I do not consider this a chapter by itself but part of the Stylized Facts chapter.
Chapter 14. Leverage Effect. The stylized fact that designates the negative correlation between historical return and realized volatility, the "leverage effect", is studied in this chapter. After performing a multiscale analysis of the leverage effect in stock time series using Monte Carlo simulations on daily close prices for the stocks of three indexes (SPI, DJ EuroStoxx, S&P 500), it is shown that the leverage effect is quantitatively important for stock and stock indexes.
Chapter 15. Processes and Market Risk Evaluation. A risk methodology is essentially an algorithm to compute the desired probability distribution forecasts, while backtesting measures the performances of the forecasts on historical data. Given the forecasted probability distribution for the returns, the computation of the actual risk measures, like Value at Risk (VaR) and Expected Shortfall (ES), is straightforward. Thus, the chapter focuses on describing the risk methodologies, and presents an example using DJIA data. It also discusses how to measure accurately shocks.
Chapter 16. Option Pricing. One of the more mathematical chapters of the book, since it presents the construction of an equivalent martingale measure in order to price a derivative. Nice computations that shown, for those that like to see the insides of the formulas, as well as Taylor expansions used to price European options.
Chapter 17. The Empirical Properties of Large Covariance Matrices. The purpose of this chapter is to extend the univariate variance to the multivariate covariance, using the stylized fact of the dynamics of the volatility, which exhibits a similar decay for the lagged correlations of all empirical time series. Of course some difficulties are raised by the multivariate extension, and they are properly stated in the introduction. An International Capital Market dataset, consisting of 340 time series that covers major asset classes and world geographical areas is used in order to study the dynamics of the covariance and correlation spectrum, as well as the spectral density of the correlation matrix.
Chapter 18. Multivariate ARCH Processes. This chapter shows how to extend the univariate ARCH processes to the multivariate setting. The natural multivariate extension is to replace the variance by the covariance matrix and some other mathematical changes. The author emphasizes that large multivariate systems are still an area of active research, where significant progresses can be expected in the forthcoming years. These kinds of comments are always well appreciated by the researcher-reader, since they point into areas of interest worth pursuing.
Chapter 19. The Processes Compatible with the Stylized Facts. This chapter presents a summary of the models or processes that are compatible with the data or which can be falsified (to reject them as descriptions of financial time series). For example, Student innovations are compatible, whereas Normal innovations are falsified, since innovations should have a fat-tailed distribution, regardless of the model for the volatility.
Chapter 20. Further Thoughts. The final chapter of the book rummages on some topics like Multi-time horizon analysis, Slow decay for the lagged correlation, Definition of volatility, the Importance of heteroscedasticity, Fat-tailed distributions, Convergence towards a Gaussian distribution, Temporal aggregation and Mean reversion and Ornstein-Uhlenbeck processes. This chapter opens the door to some questions, from the research point of view, as from the philosophical one.
This book is definitely recommended to anyone (practitioners, quants, academics or graduate students) interested in attaining a deeper understanding of the dynamics of the prices, as well as the corresponding stylized facts, which are central to many applications like portfolio optimizations, risk evaluations, or the valuation of contingent claims.
About the reviewer:
Omar Rojas (orojas@up.edu.mx) is a mathematician turned into finance, working as a research professor at the School of Business and Economy at Panamerican University, Guadalajara, Mexico. His areas of interest are Mathematical Finance, Operations Research and Applied Dynamical Systems.
0 of 0 people found the following review helpful.
one star for effort
By QFin
I don't know where to start: the lack of clarity, the million typos, the bias towards the author's own work, I was not able to find a single good aspect about this book. I barely understood that the book is about time series just because I knew a thing or two about them. If you, whether practitioner or academic, would like to study time series in finance, this exposition is guaranteed to confuse you. What a waste
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