Nonparametric estimation of copulas of financial time series

This thesis considersusing the kernel smoothing method to estimate bivariate copulas and apply them forfinancial time series.In the thesis, we present the theoretical inference on how to

CAO JIANFEI

NATIONAL UNIVERSITY OF SINGAPORE

2003

CAO JIANFEI

(B.Sc Nankai University)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY

NATIONAL UNIVERSITY OF SINGAPORE

2004

For the completion of this thesis, I would like to express my heartfelt gratitude to mysupervisor Associate Professor Chen Song Xi I deeply appreciate his patient guidance,help and encouragement throughout these two years His careful reading and valuablecomments are necessary to this thesis.

I wish to contribute the thesis to my dearest girlfriend and my family, who give methe enough support and understanding to complete the study in NUS I would also like

to thank all my friends for their help and support

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Summary iv

1.1 Problem and Purpose 1

1.2 Methodology 3

1.3 Main Results and Contents Distribution 4

2 Copulas and Extreme Value Theory 6 2.1 Copulas 6

2.2 Archimedean Copulas 13

2.3 Method of Moment to Estimate θ 23

2.4 Extreme Value Theory 27

3 Kernel Smoothing Method 31 3.1 Kernel Estimator 31

3.2 Joint Distribution Function Estimation 34

3.3 Estimate K C (t), λ(t) and ϕ(t) 35

3.4 Copulas Estimation 36

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4 Simulation and Discussion 43

5.1 Case Study 55

5.1.1 Case 1: Daily Returns of Dow Jones and Hang Seng 57

5.1.2 Case 2: Daily Returns of S&P 500 and NASDAQ 59

5.2 Conclusion 60

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Copulas, especially Archimedean copulas, are fit for modelling non-elliptically distributedmultivariate data with different dependent structure, such as multivariate financial returnseries They are very useful tools for Integrated Risk Management This thesis considersusing the kernel smoothing method to estimate bivariate copulas and apply them forfinancial time series.

In the thesis, we present the theoretical inference on how to use the kernel ing method to estimate the joint distribution functions, the generator functions and thecopulas functions Bandwidth selection is important for the kernel estimators, we useplug-in method to select optimal global bandwidths Numerical simulations are presented

smooth-to verify the theoretical results

Applications concern modelling the daily return of S&P 500 composite index and DAQ composite index and the daily return of Dow Jones Industrials index and Hang Sengindex with the Gumbel family of Archimedean copulas Extending bivariate Archimedeancopulas to multivariate cases is discussed at the end

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2.1 One parameter families of Archimedean copulas 17

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2.1 An example of a copula C(u, v) . 29

4.1 λ(t) with the 95% variability bands under the Gumbel family .ˆ 47

4.3 ϕ(t) with the 95% variability bands under the Gumbel family .ˆ 49

C(u, v) and F (x, y) . 52

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with 95% variability bands 67

5.1.5 The one dimensional estimators of C(u, v): the method of moment estima-tor of C(u, v) under the Frank family and the kernel estimaestima-tor of C(u, v) with 95% variability bands 68

5.1.6 The one dimensional estimators of C(u, v): the method of moment estima-tor of C(u, v) under the Clayton family and the kernel estimaestima-tor of C(u, v) with 95% variability bands 69

5.1.7 The method of moment estimator of C(u, v) under the Gumbel family and the kernel estimator of C(u, v). 70

5.1.8 The method of moment estimator of C(u, v) under the Frank family and the kernel estimator of C(u, v). 71

5.2.1 The kernel estimator of λ(t) with the 95% variability bands . 72

5.2.2 The kernel estimator of ϕ(t) with 95% variability bands . 73

5.2.3 The kernel estimators of C(u, v) and F (x, y) . 74

5.2.4 The one dimensional estimators of C(u, v): the method of moment estima-tor of C(u, v) under the Gumbel family and the kernel estimaestima-tor of C(u, v) with 95% variability bands 75

5.2.5 The one dimensional estimators of C(u, v): the method of moment estima-tor of C(u, v) under the Frank family and the kernel estimaestima-tor of C(u, v) with 95% variability bands 76

5.2.6 The one dimensional estimators of C(u, v): the method of moment estima-tor of C(u, v) under the Clayton family and the kernel estimaestima-tor of C(u, v) with 95% variability bands 77

5.2.7 The method of moment estimator of C(u, v) under the Gumbel family and the kernel estimator of C(u, v). 78

5.2.8 The method of moment estimator of C(u, v) under the Frank family and the kernel estimator of C(u, v). 79

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Chapter 1

Introduction

Recently, multivariate rare events are interested to integrated risk management(IRM),such as domino effects of different financial markets, the systemic risk of different assets

in one or different markets IRM is a process to learn and manage risk at the wide level It requires assessing the potential risks of an organization at every level andthen aggregating the results at the organization-wide level to improve decision-making.IRM is interested in the following issues: measuring multivariate financial risk in realworld, measuring the inter-series dependence of several series of financial observations,estimating stress testing value and so on

organization-Value-at-Risk(VaR) methodology has a strong effect on IRM Let α be close to zero, the 1-α level VaR is:

where F (x) is the profits and losses distribution function, X is a strict stationary return

series In the elliptically distributed world, VaR is a coherent risk measure This is cause if financial data are from elliptical distributions, such as Normal distribution ort-distribution, VaR for the data set has the property of sub-additivity , which is one im-portant criteria for a coherent risk measure But in practice, financial data are often non-elliptically distributed For non-elliptically distributed data, VaR has not sub-additiveproperty and is not coherent In the non-elliptically distributed world, the extreme valuetheory(EVT) and AVaR give us a practical way to estimate coherent risk in the VaR frame-

α

 α

focuses on the extreme part and exceedances of the data and formulates the asymptoticdistribution, which is very fit for describing the heavy tailed(non-elliptically distributed)

through the EVT because VaR can be treated as an extreme quantile(Chavez-Demoulinand Embrechts, 2001) These are only for univariate case Copula is a joint function whichcouples the marginal functions together without knowing the dependent structure Formultivariate non-elliptical financial data, copulas are good tools for risk assessment anddependence evaluation For example, we can use copulas to bound the VaR of dependent

The purposes of this thesis is to apply Archimedean copulas and kernel ing method for estimating the distribution functions of multivariate non-elliptically dis-tributed data without knowing their dependent structure

smooth-1.2 Methodology

Generally speaking, there are two main methods to estimate copulas — parametricand nonparametric methods The Maximum likelihood method(MLE) and the momentmethod are popular parametric approaches We will discuss the moment method in Chap-ter 2 For the MLE to be applicable, we should assume the copula belong to a parametric

families, where θ is the parameter of C The full likelihood is maximized with respect to

θ to obtain the MLE of θ (Shi, Louis, 1995) If we do not know the marginal distribution

functions, we can substitute the empirical cumulative marginal distribution functions intothe full likelihood, this is semiparametric estimation method(Genest, Ghoudi and Rivest,1993)

gave an nonparametric method — empirical distribution method to estimate bivariate

Archimedean copulas Let C be a bivariate copula function The bivariate Archimedean

copula is a special type of the copula family such that:

C(u, v) = ϕ [−1] (ϕ(u) + ϕ(v)),

function of ϕ (See Chapter 2 for details) In their paper, they used empirical distribution

to estimate bivariate joint distribution function F (x, y), which is less smooth than using

estimation of generator function ϕ(t) and the copula function ϕ(t) is defined as ϕ(t) =

t0

1

λ(s) ds }, where t0 is a constant in (0, 1).

Fermaian and Scaillet(2003) presented the kernel smoothing method to estimate

n( ˆ C − C)

is the sample size But they just estimated the joint distribution function F (x, y) and

directly Furthermore, they select bandwidth using Normal reference rule, which meansthat they assume marginal distributions are close to normal distributions, it is not alwaystrue in practice

In this thesis, we use the kernel smoothing method to estimate a bivariate Archimedeancopula, which includes estimating marginal distribution functions, bivariate joint distri-

bution function, λ(t), ϕ(t) and the copula function directly Numerical simulations based

on the Gumbel family of Archimedean copulas are carried out to verify the performance

of the estimators, the sample size ranged from 125 to 1000 The simulation results verifythe Sklar’s theorem(see Chapter 2) and show that the proposed estimators are consistent,even when the sample size is 125

The thesis is organized as follows: In Chapter 2, we introduce the definition and somebasic theorems of copulas and Archimedean copulas We also discuss the moment methodestimation on parameters of Archimedean copulas and describe what is EVT briefly InChapter 3, we give the details on kernel smoothing method to estimate bivariate copu-las, which include estimating the marginal distribution functions, the joint distribution

band-width selection, which is very important in kernel smoothing estimation In Chapter 4,

we display the simulations on the kernel smoothing estimation to the Gumbel family ofArchimedean copulas and explain the results In Chapter 5, we apply the kernel smooth-ing method for estimating copulas functions of financial series, which including the dailyreturn of S&P 500 composite index, the daily return of NASDAQ composite index, thedaily return of Dow Jones Industrials index and the daily return of Hang Seng index.Then we give the conclusions and discuss how to extend bivariate Archimedean copulas

to multivariate cases Most of proofs are given in Appendix

Chapter 2

Copulas and Extreme Value Theory

This chapter is divided into three parts In the first part, we describe the definition andproperties of copulas Sklar theorem is introduced, which is the most important theoremfor copulas In the second part, we give the definition and properties of Archimedeancopulas The general expressions of Archimedean copulas and their generator functionsare discussed At the end, we give an algorithm for generating data set from a givenArchimedean copula In the third part, we give a brief description on the extreme valuetheory

Throughout the thesis, we use DomF to denote the domain of a function F and RanF for the range of F

distributed marginal distributions This thesis concerns mainly bivariate copulas, namely

d = 2 Before give the definition of bivariate copulas, we first give the definition of

grounded 2-increasing function A bivariate copula is a grounded 2-increasing function.

If V F (B) ≥ 0, then the function F is said to be 2-increasing Suppose a1 and b1 are the

Thus, if a function F is grounded 2-increasing, which means F is nondecreasing in

each argument The definition of a bivariate copula is as follows:

Definition 2.1 A bivariate copula is a grounded 2-increasing function from I2 to I, that is:

1 For any u, v ∈ I,

C(u, 1) = u, C(1, v) = v;

C(u, 0) = 0, C(0, v) = 0.

2 For any u1, u2, v1, v2 ∈ I and u1 ≤ u2, v1 ≤ v2,

In Figure 2.1 at the end of this chapter, we give an example of bivariate copula Thecopula is from the Gumbel family of Archimedean copula, which we will discuss in Chapter

4 The expression of the copula is:

From Figure 2.1, we can see that when u = 0, C(0, v) = 0 When v = 0, C(u, 0) = 0 When u = 1, C(1, v) = v, for example, C(1, 0.2) = 0.2 when v = 1, C(u, 1) = u, for example, C(0.2, 1) = 0.2 These also can be verified through (2.1).

From bivariate copulas, we can extent to multivariate copulas The definition of amultivariate copula is as follows:

Definition 2.2 An d-dimensional copula is a grounded d-increasing function from I d to

I That is:

1 For every u ∈ I d

, u = (u1, u2, · · · , ud ), if at least one of the component equal to

0, suppose u i = 0, then C(u) = 0 If all of the component except u i equal to 1, then C(u) = u i

2 For any a, b ∈ I d

,a = (a1, · · · , ad ), b = (b1, · · · , bd ), if a i ≤ bi , i = 1, · · · , d, then

The following theorem is the Sklar’s theorem, whose proof is given in Nelsen(1999)and is very important for copulas It gives the relationship among a copula function,marginal distribution and joint distribution functions

Theorem 2.1 (Sklar’s theorem)

that for all x, y ∈ R2,

on RanF1(x) ×RanF2(y).

(ii) Conversely, if C is a copula and F1(x) and F2(y) are marginal distribution tions, then the function F (x, y) is a joint distribution function with marginal distribution functions F1(x) and F2(y).

func-According to the Sklar’s theorem, we can see that a copula produces a dependencestructure that couples univariate marginal distribution functions to the joint distributionfunction

The following are the properties of copulas

Property 1 Let C be a 2-dimensional copula, then for any (u, v) ∈ I2,

W (u, v) ≤ C(u, v) ≤ M(u, v),

where W (u1, · · · , ud ) = max(u1 +· · · + ud − d + 1, 0), M(u1, · · · , ud ) = min(u1, · · · , ud).This property shows that copulas can measure the inter-series dependence of multivariatedata, which will be discussed in section 2.3.

Property 2 If X and Y are continuous random variables with distribution functions

C(u, v) Then X and Y are independent if and only if

Proof:

Property 3 If X and Y are continuous random variables in R with a copula function

β(Y ) have a copula function C α (X)β(Y ) Then

(1) if α and β are strictly increasing,

C α (X)β(Y ) (u, v) = C XY (u, v);

(2) if α is strictly increasing and β is strictly decreasing,

C α (X)β(Y ) (u, v) = u − CXY (u, 1 − v);

(3) if α is strictly decreasing and β is strictly increasing,

C α (X)β(Y ) (u, v) = v − CXY(1− u, v);

(4) if α and β are strictly decreasing,

C α (X)β(Y ) (u, v) = u + v − 1 + CXY(1− u, 1 − v).

Proof:

From the conditions, X and Y are continuous in R and α, β are strictly monotone

X, α(X), Y, β(Y ) respectively So, RanF1=RanF2=RanG1=RanG2=I.

If α and β are strictly increasing, then

C α (X)β(Y ) (F2(x), G2(y)) = P (α(X) ≤ x, β(Y ) ≤ y)

Thus,

C α (X)β(Y ) (u, v) = C X,Y (u, v).

If α is strictly increasing and β is strictly decreasing, then

C α (X)β(Y ) (F2(x), G2(y)) = P (α(X) ≤ x, β(Y ) ≤ y)

= P (α(X) ≤ x) − P (X ≤ α −1 (x), Y ≤ β −1 (y))

= F2(y) − CXY (F1(α −1 (x)), G1(β −1 (y)))

= F2(x) − CXY (F2(x), 1 − G2(y)).

Thus,

C α (X)β(Y ) (u, v) = u − CXY (u, 1 − v).

If α is strictly decreasing and β is strictly increasing, then

C α (X)β(Y ) (F2(x), G2(y)) = P (α(X) ≤ x, β(Y ) ≤ y)

= G2(y) − CXY (F1(α −1 (x)), G1(β −1 (y)))

= G2(y) − CXY(1− F2(x), G2(y)).

Thus,

C α (X)β(Y ) (u, v) = v − CXY(1− u, v).

If α and β are strictly decreasing, then

C α (X)β(Y ) (F2(x), G2(y)) = P (α(X) ≤ x, β(Y ) ≤ y)

= G2(y) − P (X ≤ α −1 (x)) + P (X ≤ α −1 (x), Y ≤ β −1 (y))

= G2(y) − (1 − F2(x)) + C XY (F1(α −1 (x)), G1(β −1 (y)))

Thus,

C α (X)β(Y ) (u, v) = u + v − 1 + CXY(1− u, 1 − v).

trans-formations of X and Y

There are three typical copulas: Marshall-Olkin copulas, elliptical copulas and Archimedeancopulas Marshall-Olkin copulas focus on survival data(Marshall, Olkin, 1967; Nelsen,1999) Elliptical copulas are the functions of elliptical distributed random vectors, such

as Gaussian copulas and t-copulas The marginal distributions of elliptical copulas areelliptical and of the same type For example, the marginal distributions of Gaussian cop-ulas are normal distributions Elliptical copulas do not have close form and are radiallysymmetric So, the non-elliptically distributed data often can not be modelled with el-liptical copulas Archimedean copulas will be discussed at next section, which are fit fornon-elliptical distributed data

In this section, we will give the definition and introduce some important theorems aboutArchimedean copulas, which are useful to later chapters As mentioned above, Archimedeancopulas are fit for non-elliptically distributed data with different dependent structure.Common Archimedean copulas have close form and many parametric families of copulas

are Archimedean copulas(Nelsen, 1999; Embrechts, Lindskog and McNeil, 2001) Thedisadvantage of Archimedean copulas is that there are some difficult on multivariate ex-tensions At the end of the thesis, we describe a multivariate extension But this extensionneeds more strict conditions.

Definition 2.3 Let ϕ be a continuous, strictly decreasing function from I to [0, ∞] and

ϕ(1) = 0 The pseudo-inverse function of ϕ is:

where, ϕ −1 is the inverse function of ϕ.

in Figure 2.3 at the end of this chapter

Theorem 2.2 Let ϕ be a continuous, strictly decreasing function from I to [0, ∞] such that ϕ(1) = 0 Let C be the function from I2 to I given by

Then C is the copula if and only if ϕ is convex.

Copula C from (2.5) is called Archimedean copula The function ϕ is the generator

The copula from (2.6) is called strict Archimedean copula.

Proof:

The following proof is based on Nelsen(1999) with more details

From lemma 2.1, C is grounded 2-increasing So C is a copula.

From theorem 2.2, if we know the convex generator function ϕ(t) and its pseudo-inverse function, we can construct an Archimedean copula according to (2.5) The following table

lists three popular one parameter families of Archimedean copulas This table also can

be found in Nelsen(1999) That one parameter means that the generator function only

has one parameter — θ In the next section, we will discuss using the method of moment

C(u, v) = uv also means that the random variables X and Y are independent.

The generator functions are all convex listed in Table 2.1 For the Gumbel family,

is the expression of Clayton family of Archimedean copula.

For the Frank family,

Properties of Archimedean copulas

Let C be an Archimedean copula with generator function ϕ Then

Theorem 2.3 Let C be an Archimedean copula generated by ϕ For any t ∈ I, let KC (t) denote the C-measure of the set {(u, v) ∈ I2|C(u, v) ≤ t}, that is

Then we have

The proof of this theorem is available in Appendix, which is based on the proof given

where t0 ∈ (0, 1) is a constant(Genest and Rivest, 1993) To appreciate (2.11), we note

Theorem 2.3 gives us a method on how to estimate generator function ϕ, which

to (2.11) A kernel estimator of ϕ based on the above steps will be given in Chapter 3.

Corollary 2.1 Let random variables U and V be uniform U (0, 1) random variables, their

joint distribution function is the Archimedean copula C generated by ϕ Then the function

= P {C(u, v) ≤ t}.

distribution estimator to estimate it More details are presented in Chapter 3

Theorem 2.4 Let U and V be uniform U (0, 1) random variables, whose joint distribution

function is an Archimedean copula C generated by ϕ Let S = ϕ(U )

ϕ(U ) + ϕ(V ) and T =

C(U, V ), then S and T are independent, S is uniformly distributed on I, and the joint distribution function of (S, T ) is H(s, t) = sKC (t).

The proof of this theorem can be found in Nelsen(1999) Theorem 2.4 can be used tocreate an algorithm to generate simulated data from a given Archimedean copula

According to Theorem 2.4, we have the following algorithm for generating a set of

simulation data from a given Archimedean copula with the generator function ϕ More

algorithms can be found in Nelsen(1999) The steps are as follows:

Step 1 Generate two independent uniformly U (0, 1) random variables S and Q.

ϕ(U ) + ϕ(V ) and K

−1

is a pair of the desired observation

Step 4 repeat steps 1-3 for n times, we can obtain n pairs of desired observations (X1, Y1), · · · , (Xn, Yn)

as their distribution function

2.3 Method of Moment to Estimate θ

In this section, we describe the method of moment to estimate the parameter θ, which is

fundamentally based on the Kendall’s tau Kendall’s tau is a type of measurement on the

dependence between the continuous random variables X and Y

(X, Y ), then the Kendall’s tau is defined as

From the definition, we can see that if random variables X and Y are independent, then

Theorem 2.5 Let X and Y be continuous random variables with copula C, F1 be the marginal distribution function of X, F2 be the marginal distribution function of Y , let

u = F1(x) and v = F2(y) Then τ X,Y = 1 if and only if C(u, v) = M (u, v), τ X,Y = −1

if and only if C(u, v) = W (u, v) If random variables X and Y are independent, then

The proof of this theorem can be found in Embrechts, McNeil, and Straumann(1999)

Figure 2.2 at the end of this chapter contains the contour plots of C(u, v) = M (u, v), C(u, v) = W (u, v) and C(u, v) = Π(u, v).

According to the definition of the Kendall’s tau, the range of inter-series dependence

C(u, v) ≤ M(u, v)) and the Theorem 2.5, obviously, there is a suitable choice of the underlying copula to measure the inter-series dependence between X and Y

The relationship between θ and the Kendall’s tau is described in following theorem

and its corollary

Theorem 2.6 Let (X, Y ) be continuous random vector with copula C(U, V ), U and V

are uniform U (0, 1) random variables, then the Kendall’s tau for (X, Y ) is:

The proof of this theorem can be found in Nelsen(1999)

Corollary 2.2 If (X, Y ) are continuous random variables with an Archimedean copula

Let T = C(U, V ), because C is an Archimedean copula, according to Corollary 2.1,

formulate the method of moment estimator of θ.

tlnt

θ dt =

θ − 1 θ

The generator function of the Clayton family is ϕ(t) = 1

ˆ

θ = 2ˆτ X,Y

t n

e t − 1 dt; θ ≥ 0, n = 0, 1, 2, · · · Then, the method of moment estimator of θ can obtain through the following expression:

2.4 Extreme Value Theory

In this section, we just describe how to formulate the extreme value distributions andthe Generalized Parato distributions briefly Traditionally, EVT concerns about the i.i.dextreme losses of original data as follows:

F , the random variable Y t = Max(X1, X2, · · · , Xt ) has distribution function F Y t, then

types:

• Gumbel distribution, γ = 0, G 0,µ,σ (x) = exp( −e −(x−µ)/σ ), for all x;

• Fr´echet distribution, γ > 0, Gγ,µ,σ (x) = exp( −(1 + γ x − µ

Another aspect of the extreme value theory is the distribution function of the

of F [u] (x) is the Generalized Pareto distribution(Balkema and de Haan, 1974, Pickands,

where µ is the location parameter and σ is the scale parameter.

As mentioned in Chapter 1, EVT is a useful tool for describing non-elliptically tributed data In the thesis, we mainly use the extreme value distributions and the Gener-alized Parato distributions to describe the non-elliptically distributed data in simulationprocess

Contour plot of copula C(u,v)

Figure 2.1: An example of a copula C(u, v).

0.6

0.8 1

independent completely positive association completely negative association

Figure 2.2: Contour plots of C(u, v) = Π(u, v), C(u, v) = W (u, v) and C(u, v) = M (u, v).

Figure 2.3: An example of strictly decreasing function and the pseudo-inverse function.

ϕ(t) = (1 − t)3, (d) is its pseudo-inverse function.