Luận văn toán học INTRODUCTION TO COPULA AND APPLICATIONS

In probability, it is used in the sense of "connec-tion function" of the probability distribution of a random variable with eachother more.. The n-dimensional copula is the function n va

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCE

FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Tran Duc Anh

INTRODUCTION TO COPULA AND APPLICATIONS

Undergraduate ThesisUndergaraduate Advanced Program In Mathematics

Ha Noi - 2012

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCE

FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Tran Duc Anh

INTRODUCTION TO COPULA AND APPLICATIONS

Undergraduate ThesisUndergaraduate Advanced Program In Mathematics

Advisor: Dr.Tran Manh Cuong

Ha Noi - 2012

In the last financial modelling yields estimates of the assets in the portfolio isone of the most important issues When researching a portfolio, they usuallydetermine the performance of the portfolio by the traditional method is to op-timize the average variance, to measure the certainty of the problem

The former method is commonly assumed property yields in the list havethe same distribution, which is easily identified and their distribution How-ever, in actual yields of the string properties usually abide by the rules ofdistribution of different boundary conditions, so it is difficult to determinethe distribution of assets yielding string However, we can handle by means

of copula Copula function is the link between the one-dimensional boundaryconditions for them to become multi-dimensional distribution function

Abe Sklar copula concept is put into the statistical probability in 1959, butonly in recent decades about 2 copula theory just developed, the demand forapplications in financial risk management The dictionary of statistics to 1997appeared to distance this term According to English, copula means "parts areconnected, the connection" In probability, it is used in the sense of "connec-tion function" of the probability distribution of a random variable with eachother more

The n-dimensional copula is the function n variables, from on, showing thedependence on each of a set of n random variables The copula is the specialfunction with many interesting properties, and as we know, the copula canalso calculate the dependence of random variables on the covariance and cor-relation The theory of investment and risk covariance and correlation usingonly (of the index, price, etc.) alone is not enough, but to examine their copula

In the financial sector, the events: such as the bankruptcy case, the nomic crisis, financial market changes, many investors are interested This

eco-is an eco-issue related to financial reco-isk measurement The essence of the financialrisk measurement is to study the consistency of the results, so people oftenuse the probability distribution for risk measurement Currently, the copula

is used mainly for research in the field of financial risk measurement For theabove reasons I choose the topic "Introduction to copula and applications" asthe thesis

Outline of this research project report

Chapter 1: In this chapter, we prepare to construct the definitions aboutcopulas and some tool to work with copulas We will consider the notationsand the knowledges about random variables, random vectors with their dis-tribution ie definitions, theorem, notations

Chapter 2: Introduction to the copula Presents the definition, nature andthe theorems of the copula, the dependence, copula function profile couple Extends the technical tools to a multivariate setting Here we can find detailabout some copula functions This chapter is also devote to statistical infer-ence theory applied to copula models

Chapter 3: Presents applications and give examples in risk managementand finance

Because the acknowledge of author is limited and the problem for time

to do then the thesis has some mistakes Please, give me your ideas or youropinion that help me complete the thesis

Thank you very much !

Before presenting the content thesis I would like to express deep gratitude toDr.Tran Manh Cuong, University of Natural Sciences, Hanoi National Uni-versity, who has instructed me during study and research, for his patience,motivation, enthusiasm, and immense knowledge He has taught me so muchabout conducting academic research and writing career planning The advice,support and friendship of his have been invaluable on both an academic and

a personal level, for which I am extremely grateful

I also would like to express sincere gratitude to all teachers in the Faculty ofMathematics-Mechanics-Informatics, University of Natural Sciences, HanoiNational University, has dedicated teaching me during learning I take thisopportunity sincerely to thank family, friends were always cheering, encour-aging, enabling you to complete this thesis

Tran Duc Anh

Ha Noi,December, 2012

1.1 Introduction some notations 1

1.2 Random variables, Random vectors and Their Distributions 1

1.2.1 Joint and marginal distributions 2

1.2.2 Conditional distributions and independence 3

1.2.3 Moments and characteristic function 4

1.2.4 Empirical Distribution 4

1.3 The notion of a n-increasing function 5

2 Introduction to Copula 8 2.1 Definitions and properties 8

2.2 Sklar’s theorem 10

2.3 Copulas and random variables 14

2.4 Introduction some copula functions 17

2.4.1 Elliptical copulas 17

2.4.2 Archimedean copulas 19

2.5 Statistic inference for copula 20

2.5.1 Exact Maximum Likelihood Method 20

2.5.2 CML Method 21

2.5.3 Non-parametric Estimation 22

3 Applications of Copula 24 3.1 Pricing multi-asset options using Lévy copulas 24

3.2 Compute in risk management 26

3.2.1 Discrete case 26

3.2.2 Continuous case 26

3.3 Dependence 29

C HAPTER 1

Preliminaries

We will let R denote the ordinary real line (−∞,+∞), R denote the

ex-tended real line[−∞,+∞)], andRn denote the extendedn-spaceR×R× ×

R We will use vector notation for point in Rn, e.g., a =(a1, a2, , an), and wewill writea≤b when ak ≤bk for all k; a<b when ak <bk ∀ k

A n-box in Rn is the Cartesian product of n closed intervals: B= [a1, b1] ×[a2, b2] × × [an, bn] The vertices of ann-box B are point c =(c1, c2, , cn)whereeachck is equal to theirak orbk

Ann-place real function H is a function whose domain, DomH is subset of

Rn and whose range, RanH is a subset ofR The unit n-cube In is the product

Example 1.2.1 Suppose that our experiment consists of tossing 3 fair coins If we let

Y denote the number of heads appearing, then Y is random variables taking on one ofthe values 0, 1, 2, 3 with respective probabilities

1.2 Random variables, Random vectors and Their Distributions

P{Y =3} = P{(H, H, H)} = 1

8Since Y must take on one of the values 0 through 3, we must have:

relation-1.2.1 Joint and marginal distributions

Thus far, we have only concerned ourselves with probability distributions forsingle random variables However, we often interested in probability state-ments concerning two or more random variables.In order to deal with suchproperties, we define the joint cumulative probability distributions f unction ofrandom vector X LetX= (X1, X2, , Xn)withXi,i=1, 2, , n are one-dimensionrandom variables, then X is called n−dimension random vector The distribu-tion function of X is completely described by the joint distribution function:

FX(x) = P{X1 <x1, X2<x2, , Xn <xn}, xi ∈R, i =1, , n (1.2)Where no ambiguity arises we simply write F, omitting the subscript

In specially, Xi, i=1, 2, , n are independent We have:

FX =P{X1 <x1}P{X2< x2} P{Xn < xn} = FX1(x1)FX2(x2) FXn(xn) (1.3)

In additions,when Xi, i = 1, 2, , n are discrete random variables, it is nient to define thejoint probability mass f unction:

conve-pX =P{X1 =x1}P{X2= x2} P{Xn = xn} (1.4)

Example 1.2.2 Suppose that 3 balls are randomly selected from an urn containing

3 red, 4 white and 5 blue balls If we let X and Y denote, respectively, the number

of red and white balls chosen, then the joint probability mass functions of X and Y,

p(i, j) = P{X=i, Y =j}, is given by:

p(0, 0) =

53



123

 = 10

220, p(0, 1) =

41

 52



123

 = 40

220,

p(0, 2) =

42

 51



123

 = 30

220, p(0, 3) =

43

 123



123

220,

1.2 Random variables, Random vectors and Their Distributions

p(1, 0) =

31

 52



123

 = 30

220, p(1, 1) =

31

 41



123

 = 60

220,

p(1, 2) =

31

 42



123

 = 18

220, p(2, 0) =

32

 51



123

 = 15

220,

p(2, 1) =

32

 41



123

 = 12

220, p(3, 0) =

33



123

 = 1

220.

The marginal distribution function of Xi , written FXi or often simply Fi,

is the distribution function of that risk factor considered individually and iseasily calculated from the joint distribution function For alli we have:

Fx1 =P(Xi) = F(∞, , ∞, xi,∞, , ∞) (1.5)

1.2.2 Conditional distributions and independence

If we have a multivariate model for risks in the form of a joint distributionfunctions, survival function or density, then we have implicitly described theirdependence structure We can make conditional probability statements aboutthe probability that certain components take certain values given that othercomponents take other values For example, consider again our partition of

X into (X10, X20) and assume absolute continuity of the distribution function of

X Let fX1 denote the joint density of the k-dimensional marginal distribution

FX1 Then the conditional distribution ofX2 given X1=x1has density

con-F(x) = FX1(x1)FX2(x2),∀x,

or, in the case where X possesses a joint density, f(x) = fX1(x1)fX2(x2)

1.2 Random variables, Random vectors and Their Distributions

1.2.3 Moments and characteristic function

The mean vector of X, when it exists, is given by

E(X):= (E(X1), , E(Xd)).The covariance matrix, when it exists, is the matrix cov(X) defined by

cov(X) = E((X−E(X))(X−E(X)),where the expectation operator acts componentwise on matrices If we write

∑ for cov(X), then the (i,j)th element of this matrix is

σij =cov(XiXj) = E(XiXj) −E(Xi)E(Xj),the ordinary pairwise covariance between Xi and Xj The diagonal elements

σ11, , σdd are the variances of the components of X

The correlation matrix of X, denoted by ρ(X), can be defined by introducing

a standardized vector Y such that Yi = Xi

√

varXi for all for all i and taking

ρ(X) = cov(Y) If we writeP for ρ(X), then the (i,j)th element of this matrix is

ρij =ρ(Xi, Xj) = cov(Xi, Xj)

qvar(Xi)varXj

extracts a correlation matrix The covariance and correlation matricesΣ and

1.3 The notion of a n-increasing function

where f{xi < x is number of sample values xi which is less than x When

x change, we obtain function Fn(x) to real variable x The function is calledempiric distribution

By the different samples we realize any different empiric distribution tions They have the trapezium graphs, and all of them have a common prop-erty is: They come a fiducial distribution function, when the size of sampleincreasing to infinity

To accomplish what we have outline about copula, we need to generalize thenotion of "nondecreasing" for univariate functions to a concept applicable tomultivariate function

Definition 1.3.1 Let S1, S2, , Sn be non-empty subsets of R, and let H be an

n-place real function such that DomH= S1×S2× ×Sn Let B=[a,b] be an n-box all

of whose vertices are in DomH Then the H−Volume of B is given by

VH(B) =∑sgn(c)H(c), (1.15)where the sum is taken over all vertices c of B, and sgn(c) is given by

sgn(c)=

(

1 ck =ak for an even number of k0s

−1 ck =ak for an odd number of k0s (1.16)Equivalently, the H-volume of an n-box B=[a,b] is the nth order difference of

∆b

aH(t) = H(t1, , tk− 1, bk, tk+ 1, , tn) −H(t1, , tk− 1, ak, tk+ 1, , tn) (1.18)For example, if H(x1, , xn) = P(X1 6 x1, , Xn 6 xn) is probability simulatedistribution function ofn random variables X1, , Xn then we have:

VH(B) = P(a16X1 6b1, , an 6Xn 6bn) (1.19)

Definition 1.3.2 An n-place real function H is n−increasing if VH(B) ≥ 0 for alln-boxes B whose vertices lie in DomH

Suppose that the domain of ann-place real function H is given by DomH=S1×

S2× ×Sn where each Sk has a least element ak We say that H is grounded

if H(t) = 0 for all t in DomH such that tk = ak for at least onek If each Sk isnon-empty and has a greatest elementbk, then we say thatH has margins, andthe one− dimensional margins of H are the functions Hk given by DomHk=Skand Hk(x) = H(b1, , bk − 1, xk, bk + 1, , bn) for all x in Sk Higher dimensionalmargins are defined by fixing fewer places in H

1.3 The notion of a n-increasing function

In bivariate case, instead by boxes we use a rectangle B and then we canhave:

The volume of rectangle B (the second order difference of H on B) is

rectan-Example 1.3.3 Let H be the function defined on I2 by H(x, y) =max(x, y) Then

H is a nondecreasing function of x and y However, VH( 2) = −1, so that H is not2-incerasing

We will close this section with two important lemmas concerning grounded2-increasing functions with margin

Lemma 1.3.4 Let S1, S2 be non-empty subsets of R, and let H be a grounded

2-increasing function with domain S1×S2 Then H is nondecreasing in each argument.

PROOF Let a1,a2denote the least elements of S1, S2,respectively,and set(x1 =

a1, x2) be in S1 with x1 ≤ x2, (y1 = a2, y2) be in S2 with y1 ≤ y2 Then thefunction t 7→ H(t, y2) −H(t, y1) is nondecreasing on S1 and the function t 7→

H(x2, t) −H(x1, t) is nondecreasing onS2

Lemma 1.3.5 Let S1, S2 be non-empty subsets of R, and let H be a grounded

2-increasing function with margins whose domain is S1×S2 Let (x1, y1) and

(x2, y2) be any points in S1×S2 Then

|H(x2, y2) −H(x1, y1)| 6 |F(x2) −F(x1)| + |G(y2) −G(y1)| (1.20)where the functions F and G are margin of the grounded 2-increasing functionH

PROOF

|H(x2, y2) −H(x1, y1)| 6 |H(x2, y2) −H(x1, y2)| + |H(x1, y2) −H(x1, y1)|

(1.21)Now assume x1 ≤ x2 Because H is grounded, 2-increasing, and has mar-gins, above lemma yield 0 ≤ H(x2, y2) − H(x1, y1) ≤ F(x2) −F(x1 An anal-ogous inequality holds when x2 ≤ x1, hence it follows that for any x1, x2

in S1, |H(x2, y2) − H(x1, y2)| ≤ |F(x2) − F(x1| Similarly for any y1, y2 in S2,

|H(x1, y2) −H(x1, y1)| ≤ |G(y2) −G(y1| Which completes the proof

Example 1.3.6 Let H be the function with domain [0, 1] × [2, 4] given by

1.3 The notion of a n-increasing function

and

G(y) = H(1, y) = (y−2)(e−1)

C HAPTER 2

Introduction to Copula

In section we discus about definitions and basic properties of the Copulas

To give the definition of copulas we need to define subcopulas as a certainclass of grounded 2-increasing functions with margins And then we talk someproperties about continuous and bounds

We define copulas as subcopulas with domain I2

Definition 2.1.1 A 2-dimensional subcopula (2-subcopula) is a function C’ with the

following properties:

1.DomC’=S2×S2, where S1 and S2 are subsets of I2 containing 0 and 1;

2.C’ is grounded and 2-increasing;

3.For every u in S1 and every v in S2,

C0(u, 1) =u and C0(1, v) =v

Where for every (u,v) in DomC’, 0 ≤ C0(u, v) ≤ 1, so that RanC’ is also a

subsets of I Now, we define the Copula be the definition.

Definition 2.1.2 An dimensional copula (copula or briefly, a copula) is a subcopula C whose domain I2

2-Equivalent, an copula is a functionC from I2to I with the following properties:

1 For everyu, v in I,

C(u, 0) = 0=C(0, v) (2.1)and

2.1 Definitions and properties

to be a minor one, but it will be rather important in the next section when

we discuss Sklar’s theorem In addition, many of the important properties ofcopulas are actually properties of subcopulas

Theorem 2.1.3 Let C’ be a subcopula Then for every (u,v) in DomC’,

max(u+v−1, 0) ≤ C0(u, v) ≤ min(u, v) (2.4)

PROOF Let (u,v) be an arbitrary point in DomC’ Now C0(u, v) ≤ C0(u, 1) = uandC0(u, v) ≤ C0(1, v) = v yield C0(u, v) ≤min(u, v) Furthermore,VC0([u, 1] ×[v, 1]) ≥0 implies C0(u, v) ≥u+v−1, which when combined with C0(u, v) ≥ 0yieldsC0(u, v) ≥ max(u+v−1, 0)

Because every copula is a subcopula, the inequality in the above theoremholds for copulas, then we can have the Frechet-Hoeffding Theorem (afterMaurice Rene Frechet and Wassily Hoeffding) states that for any Copula C:

IN −→ I and any(u1, u2, , uN) in IN the following bounds hold:

W(u1, u2, , uN) ≤C(u1, u2, , uN) ≤ M(u1, u2, , uN) (2.5)The function M is called upper Frechet-Hoeffding bound and is definedas:

M(u1, u2, , uN) = min(u1, u2, , uN) (2.6)The upper bound is sharp: M is always a copula, random variables withcopula M are often called comonotonic

The functionW is called lower Frechet-Hoeffding bound and is defined as:

The functionM is not a copula for all N ≥3, random variables with copula

M are often called countermonotonic

In two variables case we easy to realize :M(u, v) = min(u, v)andW(u, v) =

Hence, if n≥3 W is not n-increasing and then is not a copula

Theorem 2.1.5 For every n≥3 and for all u⊂ [0, 1n]has n-copula C then:

2.2 Sklar’s theorem

The theorem in the title of this section is central to the theory of copulas and isthe foundation of many, if not most, of the applications of that theory to statis-tics Sklar’s theorem elucidates the role that copulas play in the relationshipbetween multivariate distribution functions and their univariate margins

Theorem 2.2.1 (Skalar’s theorem in n-dimensions) Let H be an n-dimensional distribution function with margins F1, , FN Then there exist an n-copula C such that for all x inRn

a “multilinear interpolation” of the subcopula to a copula similar to dimensional version in second lemma The proof in the n-dimensional case,however, is somewhat more involved (Moore and Spruill 1975; Deheuvels 1978;Sklar 1996)

two-We rewrite the theorem in bivariate case

Theorem 2.2.2 Let H be a joint distribution function with margins F and G Then there exists a copula C such that for all x,y in R,

H(x, y) = C(F(x), G(y)) (2.12)

If F and G are continuous, then C is unique; otherwise, C is uniquely mined on RanF×RanG Conversely, if C is a copula and F and G are distribu- tion functions, then the function H defined by above form is a joint distribution function with margins F and G.

deter-Now, consider the first lemma

Lemma 2.2.3 Let H be a joint distribution function with margins F and G Then there exists a unique subcopula C’ such that

1 DomC’ = RanF×RanG,

2 For all x,y in R, H(x,y) = C’(F(x),G(y)).

PROOF The joint distribution H satisfies the hypotheses of Lemma 1.3.5 with

S1 =S2= R Hence for any points(x1, y1)and(x2, y2)inR2 ,

|Hx−Hy| 6 |F(x1) −F(x2)| + |G(y1) −G(y2)| (2.13)

to show that the definition holds, we first note that for each u in RanF, there

is an x in R such that F(x) = u Thus C0(u, 1) = C0(F(x), G(∞)) = H(x,∞) =

F(x) =u Verifications of the other conditions in Definition are similar

The second lemma

Lemma 2.2.4 Let C’ be a subcopula Then there exists a copula C such that C(u,v) = C’(u,v) for all (u,v) in DomC’; i.e., any subcopula can be extended to a copula The extension is generally non-unique.

PROOF LetDomC0 =S1×S2 Using the theorem:

Theorem 2.2.5 Let C’ be a subcopula Then for every(u1, u2),(v1, v2)in DomC’,

|C0(u2, v2) −C0(u1, v1)| ≤ |u2−u1| + |v2−v1|

Hence C’ is uniformly continuous on its domain and the fact that C’ is creasing in each place, we can extend C’ by continuity to a function C" withdomainS1×S2, whereS1is the closure ofS1andS2is the closure ofS2 ClearlyC" is also a subcopula We next extend C" to a function C with domain I2 Tothis end, let (a,b) be any point in I2, let a1and a2be, respectively, the greatestand least elements ofS1that satisfy a1 ≤a≤ a2; and letb1 andb2be, respec-tively, the greatest and least elements ofS2that satisfyb1≤b ≤b2 Note that

nonde-if a is inS1, thena1 =a= a2; and if b is in S2, thenb1=b =b2 Now let

C(a, b) = (1−λ1(1−µ1)C”(a1, b1) + (1−λ1)µ1C”(a1, b2)

+λ1(1−µ1)C”(a2, b1) +λ1µ1C”(a2, b2)

Because λ1 and µ1are linear in a and b, respectively Notice that the lation defined for C(a, b) is linear in each place And we have the DomC=I2,thatC(a, b) = C”(a, b)for any(a, b)in DomC” By using the definition to provethis lemma, then now we must to show that the number so assigned must benonnegative follow (2.3) To accomplish this, let (c, d) be another point in I 2

interpo-such that c ≥ a and d ≥ b, and let c , d , c , d , λ , µ be related to c and d as

2.2 Sklar’s theorem

a1, b1, a2, b2, λ1, µ1 are related to a and b In evaluating VC(B) for the gle B = [a, c] × [b, d], there will be several cases to consider, depending uponwhether or not there is a point in S1 strictly between a and c, and whether

rectan-or not there is a point in S2 strictly between b and d In the simplest of thesecases, there is no point inS1strictly betweena and c, and no point in S2strictlybetween b and d, so that c1 = a1, c2 = a2, d1 = b1, and d2 = b2 Substituting

C(a, b) that we just defined in above and the corresponding terms for C(a, d),

C(c, b)andC(c, d)into the expression given by (1.17) forVC(B)and simplifyingyields

VC(B) = VC([a, c] × [b, d]) = (λ2−λ1)(µ2−µ1)VC([a1, a2] × [b1, b2]),from which it follows that VC(B) ≥ 0 in this case, as c ≥ a and d ≥ b imply

— which is illustrated in Fig 2.1 — substituting C(a, b)in above and the responding terms for C(a, d), C(c, b) and C(c, d) into the expression given by(1.17) forVC(B) and rearranging the terms yields

cor-VC(B) = (1−λ1)µ2VC([a1, a2] × [d1, d2]) +µ2VC([a2, c1] × [d1, d2])

+λ2µ2VC([c1, c2] × [d1, d2]) + (1−λ1)VC([a1, a2] × [b2, d1])

+VC([a2, c1] × [b2, d1]) +λ2VC([c1, c2] × [b2, d1])+(1−λ1)(1−µ1VC([a1, 12] × [b1, b2])+(1−µ1)VC([aa, c1] × [b1, b2]) +λ2(1−µ1)VC([c1, c2] × [b1, b2])

The right-hand side of the above expression is a combination of nine negative quantities (the C-volumes of the nine rectangles determined by thedashed lines in Fig 2.1) with nonnegative coefficients, and hence is nonnega-tive The remaining cases are similar, which completes the proof

non-2.2 Sklar’s theorem

We are now ready to prove Sklar’s theorem,

PROOF The existence of a copula C such that definition holds for all x,y in Rfollows from Lemmas 2.2.3 and 2.2.4 IfF and G are continuous, then RanF =

RanG = I, so that the unique subcopula C’ is a copula The converse is amatter of straightforward verification

Corollary 2.2.6 Let H,C,F1, , FN be as in Sklar’s theorem, and let F1ư1, , FNư1

be quasi-inverse of F1, , FN respectively.

Then for any u in[0, 1]n,

C(u1, u1, , un) =C(F1ư1(x1), F2ư1(x2), , Fnư1(xn) (2.14)

Remark 2.2.7 : Sklar’s theorem elucidates the role that copula play in the relationship

between multivariate distribution functions and their univariate margin

Example 2.2.8 Let (a, b) be any point in R 2, and consider the following distributionfunction H:

H(x, y) =

(

0, x< a or y<b,

1, x≥ a and y≥b,The margins of H are the unit step functions Fa and Gb Applying Lemma 2.2.3 yields thesubcopula C0 with domain{0, 1} × {0, 1}such that C0(0, 0) =C0(0, 1) = C0(1, 0) = 0and C0(1, 1) = 1 The extension of C0 to a copula C via Lemma 2.2.4 is the copula

C = Π, i.e., C(u,v) = uv Notice however, that every copula agrees with C0 on itsdomain, and thus is an extension of this C0

Example 2.2.9 Let H be the function with domain R2 given by

Quasi-inverses of F and G are given by Fư1(u) =2uư1 and Gư1(v) = ưln(1ưv)

for u, v in I Because RanF = RanG = I, (corollary 1.3.3) yields the copula C given by

C(u, v) = uv

u+vưuv.