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Exercises and Solutions

Statistics of Financial Markets

Szymon Borak • Wolfgang Karl Härdle •Brenda López Cabrera

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laws and regulations and therefore free for general use.

DOI 10.1007/978-3-642-11134-1

Carles Casacuberta, Universitat de BarcelonaAngus MacIntyre, Queen Mary, University of London

Springer Heidelberg Dordrecht London New York

Kenneth Ribet, University of California, Berkeley

Vincenzo Capasso, Università degli Studi di Milano

Claude Sabbah, CNRS, École Polytechnique

Editorial board:

Endre Süli, University of Oxford

Sheldon Axler, San Francisco State University

Wojbor Woyczyński, Case Western Reserve University

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

Szymon Borak

Cover design: WMXDesign GmbH, Heidelberg

Springer is part of Springer Science+Business Media (www.springer.com)

ISBN 978-3-642-11133-4 e-ISBN 978-3-642-11134-1

Library of Congress Control Number: 2010929856

EconomicsSchool of Business and Economics

Unter den Linden 610099 Berlin, Germany

Humboldt-Universität zu Berlin

[email protected]

Prof. Dr. Wolfgang Karl Härdle

[email protected] [email protected]

C.A.S.E. Centre for Applied Statistics andLadislaus von Bortkiewicz Chair of StatisticsBrenda López Cabrera

Preface

“Wir behalten von unseren Studien am Ende doch nur das, was wirpraktisch anwenden.”

“In the end, we really only retain from our studies that which weapply in a practical way.”

J. W. Goethe, Gesprache mit Eckermann, 24. Feb. 1824.

The complexity of modern financial markets requires good comprehension ofeconomic processes, which are understood through the formulation of statis-tical models. Nowadays one can hardly imagine the successful performanceof financial products without the support of quantitative methodology. Riskmanagement, option pricing, portfolio optimization are typical examples ofextensive usage of mathematical and statistical modeling. Models simplifycomplex reality; the simplification though might still demand a high level ofmathematical fitness. One has to be familiar with the basic notions of proba-bility theory, stochastic calculus and statistical techniques. In addition, dataanalysis, numerical and computational skills are a must.

Practice makes perfect. Therefore the best method of mastering models isworking with them. In this book we present a collection of exercises andsolutions which can be helpful in the advanced comprehension of Statistics ofFinancial Markets. Our exercises are correlated to Franke, Hardle and Hafner(2008). The exercises illustrate the theory by discussing practical examples indetail. We provide computational solutions for the majority of the problems.All numerical solutions are calculated with R and Matlab. The correspondingquantlets - a name we give to these program codes - are indicated by inthe text of this book. They follow the name scheme SFSxyz123 and can bedownloaded from the Springer homepage of this book or from the authors’homepages.

Financial Markets are global. We have therefore added, below each chaptertitle, the corresponding translation in one of the world languages. We alsohead each section with a proverb in one of those world languages. We startwith a German proverb from Goethe on the importance of practice.

We have tried to achieve a good balance between theoretical illustration andpractical challenges. We have also kept the presentation relatively smooth

VI Preface

and, for more detailed discussion, refer to more advanced text books that arecited in the reference sections.

The book is divided into three main parts where we discuss the issues relatingto option pricing, time series analysis and advanced quantitative statisticaltechniques.

The main motivation for writing this book came from our students of thecourse Statistics of Financial Markets which we teach at the Humboldt-Universitat zu Berlin. The students expressed a strong demand for solvingadditional problems and assured us that (in line with Goethe) giving plenty ofexamples improves learning speed and quality. We are grateful for their highlymotivating comments, commitment and positive feedback. Very special thanksgo to our very active PhD students Mengmeng Guo, Maria Osipenko, WeiningWang. In particular we would like to thank Richard Song, Julius Mungo, VinhHan Lien, Guo Xu, Vladimir Georgescu and Uwe Ziegenhagen for advise andideas on solutions. We are grateful to our colleagues Ying Chen, Matthias Fen-gler and Michel Benko for their inspiring contributions to the preparation oflectures. We thank Niels Thomas from Springer Verlag for continuous supportand for valuable suggestions on writing style and the content covered.

Szymon Borak, Wolfgang Karl Hardle and Brenda Lopez Cabrera

January 2010, Berlin

Language List

Arabic arabisch اللغة العربية Chinesisch 中文 czech čeština hebrew עברית japanese 日本語 latin lingua Latīna korean 한국말 polish język polski romanian român russian русский язык Ukrainisch українська мова Griechisch ελληνική γλώσσα

Chinese

arabisch اللغة العربية Chinesisch 中文 czech čeština hebrew עברית japanese 日本語 latin lingua Latīna korean 한국말 polish język polski romanian român russian русский язык Ukrainisch українська мова Griechisch ελληνική γλώσσα

Croatian Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

Czech

arabisch ا�� ا�� Chinesisch 中文

czech Čeština

hebrew עברית

japanese 日本語 latin lingua Latīna

korean polish ję romanian român russian русский язык Ukrainisch украї'нська мо'ва Griechisch ελληνική γλώσσα tiếng Việt

Dutch

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

English

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

French

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

German(Colognian)

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

Greek

Chinesisch 中文 czech čeština hebrew עברית

japanese 日本語 latin lingua Latīna

korean polish język polski romanian român russian русский язык Ukrainisch украї'нська мо'ва Griechisch ελληνική γλώσσα tiếng Việt

Hebrew


Indonesian

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

Italian

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

Japanese


Korean


Latin

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

Polish


Romanian

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

Russian

arabisch اللغة العربية Chinesisch 中文 czech čeština hebrew עברית japanese 日本語 latin lingua Latīna korean 한국말 polish j romanian român russian русский язык Ukrainisch українська мова Griechisch ελληνική γλώσσα

Spanish

Hrvatski jezik

Nederlands

English

Deutsch (Kölsch)

Indonesia

Italiano

Lingua Latina

român

español

Français

VIII Language List

Ukrainian

romanian român russian русский язык Ukrainisch українська мова Griechisch ελληνική γλώσσα

Vietnamese

Ukrainisch украї'нська Griechisch ελληνική tiếng Việt

Symbols and Notation

Basics

X,Y random variables or vectorsX1, X2, . . . , Xp random variablesX = (X1, . . . , Xp)

> random vectorX ∼ F X has distribution FΓ,∆ matricesΣ covariance matrix1n vector of ones (1, . . . , 1︸︷︷︸

n-times

)>

0n vector of zeros (0, . . . , 0︸︷︷︸n-times

)>

Ip identity matrix1(.) indicator function, for a set M is 1 = 1 on M ,

1 = 0 otherwisei

√−1

⇒ implication⇔ equivalence≈ approximately equal⊗ Kronecker productiff if and only if, equivalenceSDE stochastic differential equationWt standard Wiener processC complex number setR real number setN positive integer setZ integer set(X)+ |X| ∗ 1(X > 0)Positive homogeneity p(av) = |a|p(v)[λ] largest integer smaller than λa.s. almost sure

X Symbols and Notation

Characteristics of Distributions

f(x) pdf or density of Xf(x, y) joint density of X and YfX(x), fY (y) marginal densities of X and YfX1

(x1), . . . , fXp(xp) marginal densities of X1, . . . , Xp

fh(x) histogram or kernel estimator of f(x)F (x) cdf or distribution function of XF (x, y) joint distribution function of X and YFX(x), FY (y) marginal distribution functions of X and YFX1

(x1), . . . , FXp(xp) marginal distribution functions of X1, . . . , Xp

fY |X=x(y) conditional density of Y given X = xϕX(t) characteristic function of Xmk kth moment of Xκj cumulants or semi-invariants of X

Moments

E(X),E(Y ) mean values of random variables or vectors Xand Y

E(Y |X = x) conditional expectation of random variable orvector Y given X = x

µY |X conditional expectation of Y given XVar(Y |X = x) conditional variance of Y given X = xσ2Y |X conditional variance of Y given X

σXY = Cov(X,Y ) covariance between random variables X and YσXX = Var(X) variance of random variable X

ρXY =Cov(X,Y )√Var(X)Var(Y )

correlation between random variables X and Y

ΣXY = Cov(X,Y ) covariance between random vectors X and Y ,i.e., Cov(X,Y ) = E(X − EX)(Y − EY )>

ΣXX = Var(X) covariance matrix of the random vector X

Samples

x, y observations of X and Yx1, . . . , xn = {xi}ni=1 sample of n observations of XX = {xij}i=1,...,n;j=1,...,p (n× p) data matrix of observations of

X1, . . . , Xp or of X = (X1, . . . , Xp)T

x(1), . . . , x(n) the order statistic of x1, . . . , xn

Symbols and Notation XI

Empirical Moments

x = n−1n∑i=1

xi average of X sampled by {xi}i=1,...,n

sXY =n−1n∑i=1

(xi − x)(yi − y) empirical covariance of random variables Xand Y sampled by {xi}i=1,...,n and{yi}i=1,...,n

sXX = n−1n∑i=1

(xi − x)2 empirical variance of random variable Xsampled by {xi}i=1,...,n

rXY =sXY√sXXsY Y

empirical correlation of X and Y

S = {sXiXj} empirical covariance matrix of X1, . . . , Xp orof the random vector X = (X1, . . . , Xp)

>

R = {rXiXj} empirical correlation matrix of X1, . . . , Xp orof the random vector X = (X1, . . . , Xp)

>

Mathematical Abbreviations

tr(A) trace of matrix Adiag(A) diagonal of matrix Arank(A) rank of matrix Adet(A) or |A| determinant of matrix Ahull(x1, . . . , xk) convex hull of points {x1, . . . , xk}span(x1, . . . , xk) linear space spanned by {x1, . . . , xk}

Financial Market Terminology

OTC over the counterself financing a portfolio strategy with no resulting cash flowrisk measure a mapping from a set of random variables

(representing the risk at hand) to R

XII Symbols and Notation

Distributions

ϕ(x) density of the standard normal distributionΦ(x) distribution function of the standard normal

distributionN(0, 1) standard normal or Gaussian distributionN(µ, σ2) normal distribution with mean µ and

variance σ2

Np(µ,Σ) p-dimensional normal distribution withmean µ and covariance matrix Σ

B(n, p) binomial distribution with parameters n and pLN(µ, σ2) lognormal distribution with mean µ and

variance σ2

L−→ convergence in distributionP−→ convergence in probability

CLT Central Limit Theoremχ2p χ2 distribution with p degrees of freedom

χ21−α;p 1− α quantile of the χ2 distribution with p

degrees of freedomtn t-distribution with n degrees of freedomt1−α/2;n 1− α/2 quantile of the t-distribution with n

degrees of freedomFn,m F -distribution with n and m degrees of

freedomF1−α;n,m 1− α quantile of the F -distribution with n

and m degrees of freedom

Some Terminology

Кто не рискует, тот не пьёт шампанского.

No pain, no gain.

This section contains an overview of some terminology that is used throughoutthe book. The notations are in part identical to those of Harville (2001). Moredetailed definitions and further explanations of the statistical terms can befound, e.g., in Breiman (1973), Feller (1966), Hardle and Simar (2007), Mardia,Kent and Bibby (1979), or Serfling (2002).

adjoint matrix The adjoint matrix of an n × n matrix A = {aij} is thetranspose of the cofactor matrix of A (or equivalently is the n×n matrixwhose ijth element is the cofactor of aji).

asymptotic normality A sequenceX1, X2, . . . of random variables is asymp-totically normal if there exist sequences of constants {µi}∞i=1 and {σi}∞i=1

such that σ−1n (Xn − µn)L−→ N(0, 1). The asymptotic normality means

that for sufficiently large n, the random variable Xn has approximatelyN(µn, σ

2n) distribution.

bias Consider a random variable X that is parametrized by θ ∈ Θ. Supposethat there is an estimator θ of θ. The bias is defined as the systematicdifference between θ and θ, E{θ−θ}. The estimator is unbiased if E θ = θ.

characteristic function Consider a random vector X ∈ Rp with pdf f . Thecharacteristic function (cf) is defined for t ∈ Rp:

ϕX(t)− E[exp(it>X)] =

∫exp(it>X)f(x)dx.

The cf fulfills ϕX(0) = 1, |ϕX(t)| ≤ 1. The pdf (density) f may be recov-ered from the cf: f(x) = (2π)−p

∫exp(−it>X)ϕX(t)dt.

characteristic polynomial (and equation) Corresponding to any n × nmatrix A is its characteristic polynomial, say p(.), defined (for −∞ < λ <∞) by p(λ) = |A− λI|, and its characteristic equation p(λ) = 0 obtainedby setting its characteristic polynomial equal to 0; p(λ) is a polynomial inλ of degree n and hence is of the form p(λ) = c0 + c1λ+ · · ·+ cn−1λ

n−1 +cnλ

n, where the coefficients c0, c1, . . . , cn−1, cn depend on the elements ofA.

conditional distribution Consider the joint distribution of two randomvectors X ∈ Rp and Y ∈ Rq with pdf f(x, y) : Rp+1 −→ R. The marginal

XIV Some Terminology

density of X is fX(x) =∫f(x, y)dy and similarly fY (y) =

∫f(x, y)dx.

The conditional density of X given Y is fX|Y (x|y) = f(x, y)/fY (y). Sim-ilarly, the conditional density of Y given X is fY |X(y|x) = f(x, y)/fX(x).

conditional moments Consider two random vectors X ∈ Rp and Y ∈ Rqwith joint pdf f(x, y). The conditional moments of Y given X are definedas the moments of the conditional distribution.

contingency table Suppose that two random variables X and Y are ob-served on discrete values. The two entry frequency table that reports thesimultaneous occurrence of X and Y is called a contingency table.

critical value Suppose one needs to test a hypothesis H0 : θ = θ0. Considera test statistic T for which the distribution under the null hypothesis isgiven by Pθ0 . For a given significance level α, the critical value is cα suchthat Pθ0(T > cα) = α. The critical value corresponds to the thresholdthat a test statistic has to exceed in order to reject the null hypothesis.

cumulative distribution function (cdf) Let X be a p-dimensional ran-dom vector. The cumulative distribution function (cdf) of X is defined byF (x) = P(X ≤ x) = P(X1 ≤ x1, X2 ≤ x2, . . . , Xp ≤ xp).

eigenvalues and eigenvectors An eigenvalue of an n× n matrix A is (bydefinition) a scalar (real number), say λ, for which there exists an n × 1vector, say x, such that Ax = λx, or equivalently such that (A−λI)x = 0;any such vector x is referred to as an eigenvector (of A) and is said tobelong to (or correspond to) the eigenvalue λ. Eigenvalues (and eigenvec-tors), as defined herein, are restricted to real numbers (and vectors of realnumbers).

eigenvalues (not necessarily distinct) The characteristic polynomial, sayp(.), of an n× n matrix A is expressible as

p(λ) = (−1)n(λ− d1)(λ− d2) · · · (λ− dm)q(λ) (−∞ < λ <∞),

where d1, d2, . . . , dm are not-necessarily-distinct scalars and q(.) is a poly-nomial (of degree n−m) that has no real roots; d1, d2, . . . , dm are referredto as the not-necessarily-distinct eigenvalues of A or (at the possible riskof confusion) simply as the eigenvalues of A. If the spectrum of A hask members, say λ1, . . . , λk, with algebraic multiplicities of γ1, . . . , γk, re-spectively, then m =

∑ki=1 γi, and (for i = 1, . . . , k) γi of the m not-

necessarily-distinct eigenvalues equal λi .

empirical distribution function Assume that X1, . . . , Xn are iid observa-tions of a p-dimensional random vector. The empirical distribution func-tion (edf) is defined through Fn(x) = n−1

∑ni=1 1(Xi ≤ x).

empirical moments The moments of a random vectorX are defined throughmk = E(Xk) =

∫xkdF (x) =

∫xkf(x)dx. Similarly, the empirical mo-

ments are defined through the empirical distribution function Fn(x) =n−1

∑ni=1 1(Xi ≤ x). This leads to mk = n−1

∑ni=1X

ki =

∫xkdFn(x).

Some Terminology XV

estimate An estimate is a function of the observations designed to approxi-mate an unknown parameter value.

estimator An estimator is the prescription (on the basis of a random sample)of how to approximate an unknown parameter.

expected (or mean) value For a random vector X with pdf f the meanor expected value is E(X) =

∫xf(x)dx.

Hessian matrix The Hessian matrix of a function f , whose value is an mdimension real vector, is the m ×m matrix whose ij−th element is theij−th partial derivative ∂2f/∂xi∂xj of f .

kernel density estimator The kernel density estimator f of a pdf f , basedon a random sample X1, X2, . . . , Xn from f , is defined by

f(x) = (nh)−1

n∑i=1

K

(x−Xi

h

).

The properties of the estimator f(x) depend on the choice of the kernelfunction K(.) and the bandwidth h. The kernel density estimator canbe seen as a smoothed histogram; see also Hardle, Muller, Sperlich andWerwatz (2004).

likelihood function Suppose that {xi}ni=1 is an iid sample from a popula-tion with pdf f(x; θ). The likelihood function is defined as the joint pdfof the observations x1, . . . , xn considered as a function of the parame-ter θ, i.e., L(x1, . . . , xn; θ) =

∏ni=1 f(xi; θ). The log-likelihood function,

`(x1, . . . , xn; θ) = logL(x1, . . . , xn; θ) =∑ni=1 log f(xi; θ), is often easier

to handle.

linear dependence or independence A nonempty (but finite) set of ma-trices (of the same dimensions (n× p)), say A1,A2, . . . ,Ak, is (by defini-tion) linearly dependent if there exist scalars x1, x2, . . . , xk, not all 0, such

that∑ki=1 xiAi = 0n0>p ; otherwise (if no such scalars exist), the set is lin-

early independent. By convention, the empty set is linearly independent.

marginal distribution For two random vectors X and Y with the jointpdf f(x, y), the marginal pdfs are defined as fX(x) =

∫f(x, y)dy and

fY (y) =∫f(x, y)dx.

marginal moments The marginal moments are the moments of the marginaldistribution.

mean The mean is the first-order empirical moment x =∫xdFn(x) =

n−1∑ni=1 xi = m1.

mean squared error (MSE) Suppose that for a random vector C with a

distribution parametrized by θ ∈ Θ there exists an estimator θ. The meansquared error (MSE) is defined as EX(θ − θ)2.

XVI Some Terminology

median Suppose that X is a continuous random variable with pdf f(x).The median x lies in the center of the distribution. It is defined as∫ x−∞ f(x)dx =

∫ +∞x

f(x)dx− 0.5.

moments The moments of a random vector X with the distribution functionF (x) are defined through mk = E(Xk) =

∫xkdF (x). For continuous

random vectors with pdf f(x), we have mk = E(Xk) =∫xkf(x)dx.

normal (or Gaussian) distribution A random vector X with the multi-normal distribution N(µ,Σ) with the mean vector µ and the variancematrix Σ is given by the pdf

fX(x) = |2πΣ|−1/2 exp

{−1

2(x− µ)>Σ−1(x− µ)

}.

orthogonal matrix An (n×n) matrixA is orthogonal ifA>A = AA> = In.

probability density function (pdf) For a continuous random vector Xwith cdf F , the probability density function (pdf) is defined as f(x) =∂F (x)/∂x.

quantile For a random variable X with pdf f the α quantile qα is definedthrough:

∫ qα−∞ f(x)dx = α.

p-value The critical value cα gives the critical threshold of a test statistic Tfor rejection of a null hypothesis H0 : θ = θ0. The probability Pθ0(T >cα) = p defines that p-value. If the p-value is smaller than the significancelevel α, the null hypothesis is rejected.

random variable and vector Random events occur in a probability spacewith a certain even structure. A random variable is a function from thisprobability space to R (or Rp for random vectors) also known as the statespace. The concept of a random variable (vector) allows one to elegantlydescribe events that are happening in an abstract space.

scatterplot A scatterplot is a graphical presentation of the joint empiricaldistribution of two random variables.

singular value decomposition (SVD) An m × n matrix A of rank r isexpressible as

A = P(D1 00 0

)Q> = P1D1Q>1 =

r∑i=1

sipiq>i =

k∑j=1

αjU j ,

where Q = (q1, . . . , qn) is an n× n orthogonal matrix and D1 = diag(s1,

. . . , sr) an r× r diagonal matrix such that Q>A>AQ =

(D2

1 00 0

), where

s1, . . . , sr are (strictly) positive, where Q1 = (q1, . . . , qr), P1 = (p1, . . . ,pr) = AQ1D−11 , and, for any m× (m−r) matrix P2 such that P>1 P2 = 0,P = (P1,P2), where α1, . . . , αk are the distinct values represented among

s1, . . . , sr , and where (for j = 1, . . . , k) U j =∑{i : si=αj} piq

>i ; any of

these four representations may be referred to as the singular value decom-position of A, and s1, . . . , sr are referred to as the singular values of A.In fact, s1, . . . , sr are the positive square roots of the nonzero eigenvaluesof A>A (or equivalently AA>), q1, . . . , qn are eigenvectors of A>A, andthe columns of P are eigenvectors of AA>.

spectral decomposition A p× p symmetric matrix A is expressible as

A = ΓΛΓ> =

p∑i=1

λiγiγ>i

where λ1, . . . , λp are the not-necessarily-distinct eigenvalues of A, γ1, . . . ,γp are orthonormal eigenvectors corresponding to λ1, . . . , λp, respectively,Γ = (γ1, . . . , γp), D = diag(λ1, . . . , λp).

subspace A subspace of a linear space V is a subset of V that is itself a linearspace.

Taylor expansion The Taylor series of a function f(x) in a point a is the

power series∑∞n=0

f(n)(a)n! (x−a)n. A truncated Taylor series is often used

to approximate the function f(x).

Some Terminology XVII

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Language List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII

Part I Option Pricing

1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Introduction to Option Management . . . . . . . . . . . . . . . . . . . . . . . 13

3 Basic Concepts of Probability Theory . . . . . . . . . . . . . . . . . . . . . 27

4 Stochastic Processes in Discrete Time . . . . . . . . . . . . . . . . . . . . . 37

5 Stochastic Integrals and Differential Equations . . . . . . . . . . . . 45

6 Black-Scholes Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 61

7 Binomial Model for European Options . . . . . . . . . . . . . . . . . . . . . 81

8 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

XX Contents

9 Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10 Models for the Interest Rate and Interest Rate Derivatives 111

Part II Statistical Model of Financial Time Series

11 Financial Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

12 ARIMA Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

13 Time Series with Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . 155

Part III Selected Financial Applications

14 Value at Risk and Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

15 Copulae and Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

16 Statistics of Extreme Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

17 Volatility Risk of Option Portfolios . . . . . . . . . . . . . . . . . . . . . . . . 213

18 Portfolio Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

universitext3a978-3-642...preface \wir behalten von unseren studien am ende doch nur das, was wir...

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