1st Edition

Extreme Value Methods with Applications to Finance

By Serguei Y. Novak Copyright 2012
    400 Pages 72 B/W Illustrations
    by CRC Press

    Extreme value theory (EVT) deals with extreme (rare) events, which are sometimes reported as outliers. Certain textbooks encourage readers to remove outliers—in other words, to correct reality if it does not fit the model. Recognizing that any model is only an approximation of reality, statisticians are eager to extract information about unknown distribution making as few assumptions as possible.

    Extreme Value Methods with Applications to Finance concentrates on modern topics in EVT, such as processes of exceedances, compound Poisson approximation, Poisson cluster approximation, and nonparametric estimation methods. These topics have not been fully focused on in other books on extremes. In addition, the book covers:

    • Extremes in samples of random size
    • Methods of estimating extreme quantiles and tail probabilities
    • Self-normalized sums of random variables
    • Measures of market risk

    Along with examples from finance and insurance to illustrate the methods, Extreme Value Methods with Applications to Finance includes over 200 exercises, making it useful as a reference book, self-study tool, or comprehensive course text.

    A systematic background to a rapidly growing branch of modern Probability and Statistics: extreme value theory for stationary sequences of random variables.


    Distribution of Extremes

    Methods of Extreme Value Theory
    Order Statistics
    "Blocks" and "Runs" Approaches
    Method of Recurrent Inequalities

    Maximum of Partial Sums

    Erdős–Rényi Maximum of Partial Sums
    Basic Inequalities
    Limit Theorems for MPS

    Extremes in Samples of Random Size

    Maximum of a Random Number of r.v.s
    Number of Exceedances
    Length of the Longest Head Run
    Long Match Patterns

    Poisson Approximation

    Total Variation Distance
    Method of a Common Probability Space
    The Stein Method
    Beyond Bernoulli
    The Magic Factor

    Compound Poisson Approximation

    Limit Theory
    Accuracy of CP Approximation

    Exceedances of Several Levels

    CP Limit Theory
    General Case
    Accuracy of Approximation

    Processes of Exceedances

    One-level EPPE
    Excess Process
    Complete Convergence to CP Processes

    Beyond Compound Poisson

    Excess Process
    Complete Convergence

    Statistics of Extremes

    Inference on Heavy Tails

    Heavy-tailed distributions
    Estimation Methods
    Tail Index Estimation
    Estimation of Extreme Quantiles
    Estimation of the Tail Probability

    Value-at-Risk and Expected Shortfall
    Traditional Methods of VaR Estimation
    VaR and ES Estimation from Heavy-Tailed Data
    VaR over Different Time Horizons
    Technical Analysis of Financial Data

    Extremal Index

    Estimation of the Extremal Index

    Normal Approximation
    Accuracy of Normal Approximation
    Stein’s Method
    Self-Normalized Sums of r.v.s

    Lower Bounds

    Preliminary Results
    Fréchét–Rao–Cramér Inequality
    Information Index
    Continuity Moduli
    Tail Index and Extreme Quantiles


    Probability Distributions
    Properties of Distributions
    Probabilistic Identities and Inequalities
    Large Deviations
    Elements of Renewal Theory
    Point Processes
    Slowly Varying Functions
    Useful Identities and Inequalities




    Dr S.Y. Novak earned his Ph.D. at the Novosibirsk Institute of Mathematics under the supervision of Dr S.A. Utev in 1988. The Novosibirsk group forms a part of Russian tradition in Probability & Statistics that extends its roots to Kolmogorov and Markov.

    Dr S.Y. Novak began his teaching carrier at the Novosibirsk Electrotechnical Institute (NETI) and Novosibirsk Institute of Geodesy, held post-doctoral positions at the University of Sussex and Eurandom (Technical University of Eindhoven), and taught at Brunel University in West London, before joining the Middlesex University (London) in 2003. He published over 40 papers, mostly on the topic of Extreme Value Theory, in which he is considered an expert.

    "The book can be recommended to EVT specialists and Ph.D. students in probability and statistics who wish to specialize in that field. It can provide a useful complement to more practically oriented textbooks …"
    —Christian Genest, Journal of the American Statistical Association, September 2013

    "Each chapter is well structured with the main propositions, lemmas, theorems, and subsequent corollaries outlined and discussed first, with the proofs given towards the end of the chapter to keep the reader from getting bogged down in details. Each chapter has a set of exercises, useful to an advanced graduate class … Some chapters provide a handful of open questions, which will be of interest to new researchers in the field. … The book provides a useful complement to Resnick (1987, 2007) and De Haan & Ferreira (2006). For those interested in financial applications, it provides the next stage of depth compared to Embrechts, Kluppelberg, & Mikosch (2003) and the wide range of application-oriented books of extremes for finance applications. This book will be of interest to researchers interested in the asymptotic probability theory underlying univariate extreme value theory, including non-parametric tail index estimation."
    —Carl Scarrott, Australian & New Zealand Journal of Statistics, 2013

    "The book covers modern topics in EVT such as processes of exceedances, compound Poisson approximation, Poisson cluster approximation, nonparametric estimation methods, extremes in samples of random size, methods of estimating extreme quantiles and tail probabilities, self-normalized sums of random variables and measures of market risk. The novelty of this book in comparison to others on the EVT area is detailed coverage of the above-mentioned topics. The author is an expert on the topic of EVT and many results from his own scientific papers are included in the book. … Theoretical results in the book are illustrated by examples and applications to particular problems of financial risk management. Exercises and open problems are given in all chapters. The list of references includes 407 items and can serve as an excellent source of new results on the topics presented in the book."
    —Pavle Mladenović, Mathematical Reviews, January 2013

    "Though the first part of the book covers the well-known asymptotic theory for extremes, there are many new techniques and results which do not exist in other books on extreme value theory. These chapters will be particularly interesting to probabilists and other experts working on extreme value theory. … Those who want to learn extreme value theory and in particular, those who want to study in detail the non-parametric methods for heavy tailed distributions, will find this book a very valuable contribution. … I would strongly recommend this book to PhD students working on extreme value theory [and] to mathematicians, probabilists and statisticians who want to know about extreme value theory and non-parametric methods of inference for extremes."
    —K.F. Turkman, Journal of Times Series Analysis, March 2012