1st Edition

# Extreme Value Methods with Applications to Finance

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

399 Pages
by CRC Press

Also available as eBook on:

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.

Introduction

Distribution of Extremes

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

Maximum of Partial Sums

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

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
Proofs

Compound Poisson Approximation

Limit Theory
Accuracy of CP Approximation
Proofs

Exceedances of Several Levels

CP Limit Theory
General Case
Accuracy of Approximation
Proofs

Processes of Exceedances

One-level EPPE
Excess Process
Complete Convergence to CP Processes
Proofs

Beyond Compound Poisson

Excess Process
Complete Convergence
Proofs

Statistics of Extremes

Inference on Heavy Tails

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

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

Extremal Index

Preliminaries
Estimation of the Extremal Index
Proofs

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

Lower Bounds

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

Appendix

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

References

Index

### Biography

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