financial mathematics dissertation

Modelling Tail Dependence of Stock

Indexes by GARCH-Copula Model

Chenghao Li

September 2019

School of Mathematics,

Cardiff University

A dissertation submitted in partial fulfilment of the

requirements for MSc (in Operational Research, Applied Statistics and Financial Risk) by taught programme.

CANDIDATE’S ID NUMBER

1869787

CANDIDATE’S SURNAME

Please circle as appropriate Mr / Miss / Ms/ Mrs / Rev / Dr / Other ……Li….......

CANDIDATE’S FULL FORENAMES

Chenghao

DECLARATION This work has not previously been accepted in substance for any degree and is not concurrently submitted in candidature for any degree. Signed ……………………………………………. (candidate) Date ………………………… STATEMENT 1 This dissertation is being submitted in partial fulfilment of the requirements for the degree of ………MSc…………(insert MA, MSc,MBA, etc, as appropriate) Signed ……………………………………………. (candidate) Date ………………………… STATEMENT 2 This dissertation is the result of my own independent work/investigation, except where otherwise stated. Other sources are acknowledged by footnotes giving explicit references. A Bibliography is appended. Signed ……………………………………………. (candidate) Date ………………………… STATEMENT 3 – I hereby give consent for my dissertation, if accepted, to be available for photocopying and for public viewing, and for the title and summary to be made available to outside organisations. Signed ……………………………………………. (candidate) Date ………………………… STATEMENT 4 - BAR ON ACCESS APPROVED I hereby give consent for my dissertation, if accepted, to be available for photocopying and for public viewing after expiry of a bar on access approved by the Graduate Development Committee. Signed ……………………………………………. (candidate) Date …………………………

Executive Summary

Abstract

Economic globalization has increased the linkage between financial markets. Estimating

the correlation between the log returns of the global stock market has a strong practical

significance for financial asset pricing and financial risk management. In this paper, the

GARCH-copula model is used to estimate the correlation and tail dependence between the

log returns of FTSE100, S&P500, Nikkei225 and HS300.

Methodology

In this paper, the GARCH(1,1) model is used to fit the log return of each stock index and

obtain the marginal distributions of the log returns. The two indexes distributions are then

joined as a bivariate joint distribution using the Gaussian copula, T copula and Clayton copula

functions, respectively. The correlation between the log returns is examined according to the

parameters of the copula function. The tail dependence coefficients between the log returns are

investigated by the nature of tail dependence in T copula and Clayton copula.

Results

As a result, there is a clear correlation between the log returns of the stock indexes, and

there is strong tail dependence correlation between each pair of log returns. This shows that

there is linkage between global financial markets, and in extreme cases, its correlation will

increase. It was also found that the correlation between FTSE100 and S&P500's log returns is

the strongest, and the correlation between HS300 and log returns of other indices is relatively

weaker. To a certain extent, this partly reflects the greater freedom of capital flow between the

United Kingdom and the United States, while China has the control of capital flows.

Acknowledgements

Throughout the writing of this dissertation I have received a great deal of support and assistance. I

would first like to thank my supervisor, Dr. Anqi Liu, whose expertise was invaluable in the formulating of

the research topic and methodology in particular.

I would like to acknowledge my friend Jiliang Zhu who provided me with a cloud server account, which

enabled my code to run for a long time on the cloud server to get the research results.

Contents Abstract ............................................................................................................................................................... 1

1 Introduction ................................................................................................................................................... 1

2 Literature Review ........................................................................................................................................ 4

2.1 Techniques Used in Modelling Financial Returns ..................................................................... 4

2.2 Copula Function Estimation Methods .......................................................................................... 6

2.3 Tail Dependence .................................................................................................................................... 7

3 The Model ....................................................................................................................................................... 9

3.1 Model Index Returns .......................................................................................................................... 9

3.2 Estimate Copula Function ............................................................................................................... 10

3.2.1 Estimate Gaussian Copula ...................................................................................................... 10

3.2.2 Estimate T Copula ..................................................................................................................... 11

3.2.3 Estimate Clayton Copula ......................................................................................................... 11

3.3 Tail Dependence .................................................................................................................................. 12

4 Empirical Calibration and Results ....................................................................................................... 13

4.1 The Data ................................................................................................................................................ 13

4.2 Results .................................................................................................................................................... 15

4.2.1 Marginal Distributions ............................................................................................................. 15

4.2.2 Estimated Copula Functions .................................................................................................. 17

4.2.3 Tail Dependence .......................................................................................................................... 17

5 Conclusion .................................................................................................................................................... 19

Reference .......................................................................................................................................................... 21

1

Modelling Tail Dependence of Stock Indexes by

GARCH-Copula Model

Abstract With the increasing degree of economic globalization, the linkage between

national stock markets is also growing stronger. Especially in the case of extremely

bad conditions, the global stock market is more likely to fall at the same time, which

is the so-called tail dependence. This paper uses the stock market indexes to establish

copula models. The fact of tail dependence between markets will be researched.

1 Introduction Financial markets are interrelated, and portraying the relationship between

financial products or financial markets can help solve financial asset pricing problems

and financial risk management issues. To understand the correlation between financial

markets, traditional correlation measures (such as Pearson’s rho) are often insufficient,

see, e.g., Embrechts et al. (2002). The best way to characterize the correlation between

the variables of financial markets or financial assets is to obtain a joint distribution of

the variables. But directly estimating the joint distribution between variables is

difficult. The Copula theory proposed by Sklar (1959) provided a solution for this.

Sklar's theorem states that any joint distribution can be represented by its marginal

distributions and an appropriate copula function. When the joint distribution is

continuous, the copula function is unique. To illustrate this theorem，let’s consider a

2-dimention joint distribution function H(𝑋1, 𝑋2) . 𝐹1(𝑋1 ) and 𝐹2(𝑋2) are the

marginal distributions corresponding to H(𝑋1, 𝑋2). Then, a copula function can be

found to join the univariate marginal distributions to be the multivariate joint

distribution.

H(𝑋1, 𝑋2) = 𝐶(𝐹1(𝑋1), 𝐹2(𝑋2))

If H(𝑋1, 𝑋2) is continuous, the copula function 𝐶(𝑈1, 𝑈2) is unique. Conversely,

2

if the 2-dimention copula function 𝐶(𝑈1, 𝑈2) and the marginal distributions 𝐹1(𝑋1)

and 𝐹2(𝑋2) are known, we can find a bivariate distribution function H(𝑋1, 𝑋2) such

that 𝐹1(𝑋1) and 𝐹2(𝑋2) are margins of H(𝑋1, 𝑋2).

Therefore, with the help of copula function, the estimation of joint distribution

function can be generally separated into two steps: 1) estimate the marginal

distributions; 2) chose the copula function and estimate the parameter of the copula

function. It can be said that copula function contains all the dependence information.

Although the copula theory was proposed early, the application of copula theory

in the financial field was in the early 21st century, 60 years later than the time proposed

by the copula theory. In 1999, David X. Li first proposed the use of the copula function

to model the relevance of loan defaults and how to apply it to the pricing of credit

derivatives. But it can be said that its first application in the financial field failed. Wall

Street referenced the method proposed by Li and apply Gaussian Copula to the pricing

of Collateralized Debt Obligations (CDOs) which speeded up the issuance of CDOs.

As a result, we all know that the CDOs market has collapsed, and a large number of

mortgage defaults have caused the value of CDOs to be devastating, and investors

have suffered heavy losses. Felix Salmon (2009) described Gaussian Copula as a

‘recipe for disaster’ in his article. The mainly reason for this failed application is that

Wall Street did not choose the right copula function. Gaussian Copula has no tail

dependence, but in the extreme case of financial markets, the price of assets will show

an increase in correlation. Therefore, if you want to apply copula to finance, whether

it is financial risk management or financial asset pricing, it is necessary to consider

the tail dependence. It is very important to choose the appropriate copula function.

This paper will establish GARCH-Copula models for the log returns of stock

market indexes. Why use the GARCH model to fit log returns is due to the

phenomenon of fat tail and volatility clustering in financial time series. The traditional

econometric models cannot solve these problems well until Engle (1982) proposed

autoregressive conditionally heteroscedastic (ARCH) and Bollerslev (1986) proposed

the problem of generalized ARCH (GARCH). Therefore, this paper will use the

3

GARCH model to fit log returns. In section 2, we will introduce the method of fitting

financial time series in more detail, and introduce the expression of GARCH model in

detail. The copula model is to build joint distribution and better describe the

correlation between log returns, especially to measure its tail dependence. As

mentioned earlier, the correlation between financial assets will increase significantly

during the financial crisis. This is called “correlation breakdown”. If financial asset

pricing or financial risk management does not take into account the increase in the

correlation of financial assets in an extreme market environment, it will bring losses.

Tail dependence coefficients can capture the correlation between financial assets in

extreme market conditions, so calculating tail dependence has a strong practical

significance.

This paper researches four indexes’ log returns including FTSE100, S&P500,

Nikkei225 and HS300, and then examine their tail dependence. This paper finds that

there is a lower tail dependence between the indices, which indicates that the pricing

of financial products and the management of financial risks should take into account

the existence of tail dependence. In addition, the lower tail dependence coefficient of

FTSE100 and S&P500 is the largest, and the lower tail dependence between HS300

and the other three indexes is relatively weak. This reflects to a certain extent the

relatively weak linkage between China's capital market and the world's major capital

markets.

This paper is organized as follows. Section 2 is a literature review. The literature

review includes three aspects: 1) commonly used models in financial time series

modelling; 2) estimation methods of copula functions; 3) tail dependence. In section

3, the model is described. Section 4 shows the empirical results. Section 5 concludes

this paper.

4

2 Literature Review

2.1 Techniques Used in Modelling Financial Returns

Before using the copula function to establish a joint distribution, a good estimate

of the marginal distribution is required. In financial time series, such as stock returns,

there are often effects of volatility clustering, fat tail and financial leverage. Volatility

clustering is the variability in time of the conditional variance. In financial time series,

it can be usually observed that the high volatility is clustered in some time interval.

Fat tail is another common phenomenon in financial returns which means the spread

of returns is significantly larger than that corresponding to the normal distribution.

Financial leverage effect is the phenomenon that positive and negative information

have asymmetric influence on the whole variance of time series.

It is significant that many financial time series of returns cannot be assumed to be

normal. The key point to precise modelling of financial returns is the volatility

modelling. Currently, a broad class of GARCH processes with fat-tailed innovations

are used to model the volatility of financial returns. Some well know GARCH-type

processes are listed below.

The GARCH model was created by Bollerslev (1986). The financial time series

of returns {𝑅𝑡 } is said to be modelled with GARCH(p,q) when

{ 𝑅𝑡 = 𝜇 + 𝜀𝑡

𝜀𝑡 = 𝜎𝑡 𝑣𝑡 𝜎𝑡2 = 𝜔 + ∑ 𝛼𝑖 𝜀𝑡−𝑖

2𝑝 𝑖=1 + ∑ 𝛽𝑗 𝜎𝑡−𝑗

2𝑞 𝑗=1

（2.1）

In the above formula, 𝑣𝑡 denotes the innovations. The advantage of the GARCH

model is it can deal with conditional heteroscedasticity. In formula (2.1) the

conditional variance is described by the function 𝜎𝑡2 = 𝜔 + ∑ 𝛼𝑖 𝜀𝑡−𝑖 2𝑝

𝑖=1 +

∑ 𝛽𝑗 𝜎𝑡−𝑗 2𝑞

𝑗=1 . Formula (2.1) is a constant mean GARCH model and it can be extended

to be AR-GARCH model or ARMA-GARCH model when autoregressive effect is

considered. Time series {𝑅𝑡 } modelled with AR(k)-GARCH(p,q) model can be

expressed as

5

{ 𝑅𝑡 = ∑ 𝜑𝑙 𝑅𝑡−𝑙𝑘𝑙=1 + 𝜀𝑡

𝜀𝑡 = 𝜎𝑡 𝑣𝑡 𝜎𝑡2 = 𝜔 + ∑ 𝛼𝑖 𝜀𝑡−𝑖

2𝑝 𝑖=1 + ∑ 𝛽𝑗 𝜎𝑡−𝑗

2𝑞 𝑗=1

（2.2）

The basic GARCH model does not take the financial leverage effect into account.

The effect of financial leverage can be understood as good information and bad

information will have asymmetric effects on the variance of financial returns.

Obviously, good information corresponds to a positive return, and bad information

corresponds to a negative return. Specific to the return of stock market indexes, a

negative return will make the variance of the index returns larger. Several ways can be

found in the literature to handle the financial leverage effect. An EGARCH model

invented by Nelson(1991) can deal with the disadvantage of the GARCH model. In

EGARCH(p,q) model, the noise process {𝜀𝑡 } satisfies 𝜀𝑡 = 𝜎𝑡 𝑣𝑡 and below

equation:

{ ln(𝜎𝑡2) = 𝛼0 + ∑ 𝛼𝑖 𝑔(𝑣𝑡−𝑖 )

𝑝 𝑖=1 + ∑ 𝛽𝑗 ln (𝜎𝑡−𝑗

2𝑞 𝑗=1 )

g(𝑣𝑡 ) = 𝜃𝑣𝑡 + 𝛿(|𝑣𝑡 | − 𝐸|𝑣𝑡 |) (2.3)

GJR-GARCH proposed by Glosten et.al. (1993) is another model to deal with

financial leverage effect and it is very popular. The noise process {𝜀𝑡 } follows the

GJR-GARCH(p,q) model when {𝜀𝑡 } satisfies 𝜀𝑡 = 𝜎𝑡 𝑣𝑡 and

𝜎𝑡2 = 𝛼0 + ∑ 𝛼𝑖 𝜀𝑡−𝑖 2𝑝

𝑖=1 + ∑ 𝛽𝑗 𝜎𝑡−𝑗 2𝑞

𝑗=1 + ∑ 𝛾𝑖 𝐈{𝜀𝑡−𝑖<0}(𝜀𝑡−𝑖 2𝑝

𝑖=1 ) (2.4)

In the GJR-GARCH model, the 𝐈{𝑥<0} denotes the indicator function that is

𝐈{𝑥<0}(𝑥) = 1 while x < 0 and 𝐈{𝑥<0}(𝑥) = 0 while x ≥ 0 . Through equation

(2.4), it can be found that parameter 𝛾𝑖 determines the sensitivity of conditional

volatility function with respect to negative returns.

Regarding to the distribution of {𝑣𝑡 }，the commonly used distributions are normal,

t-Student (as in Bollerslev, 1987), skewed t (as in Patton, 2004) and GED (generalized

error distribution). The latter three distributions are more suitable to fit {𝑣𝑡 } since

they have heavier tail than normal distributions. Ferenstein and Gasowski(2004)

modelled stock returns by AR-GARCH model and found GED and Student-t

distribution are the best to fit the innovation.

6

In general, fitting the financial returns requires taking into account the three

characteristics of the financial time series and then selecting the appropriate model to

characterize the three features.

2.2 Copula Function Estimation Methods

Generally, copula function estimation method can be classified into two types,

parametric estimation and non-parametric estimation (Cherubini et al., 2004).

There are three parametric estimation methods:1) Exact maximum likelihood

method; 2) Inference for the margins (IFM) method; 3) Canonical maximum

likelihood method.

Exact maximum likelihood method is based on the canonical representation:

f(𝑥1, 𝑥2, … , 𝑥𝑛) = 𝑐(𝐹1(𝑥1), 𝐹2(𝑥2), … , 𝐹𝑛 (𝑥𝑛)) × ∏ 𝑓𝑗 (𝑥𝑗 ) 𝑛 𝑗=1 (2.5)

Let ℵ = {𝑥1𝑡 , 𝑥2𝑡 , … , 𝑥𝑛𝑡 }𝑡=1𝑇 be the sample data matrix. Thus, the expression for

the log-likelihood function is

𝑙(θ) = ∑ 𝑙𝑛𝑐(𝐹1(𝑥1𝑡 ),𝑇𝑡=1 𝐹2(𝑥2𝑡 ), … , 𝐹𝑛 (𝑥𝑛𝑡 )) + ∑ ∑ ln 𝑓𝑖 (𝑥𝑗𝑡 ) 𝑛 𝑗=1

𝑇 𝑡=1 (2.6)

By maximizing the above log-likelihood function the maximum likelihood

estimator can be found:

𝜃𝑀𝐿𝐸 = 𝑚𝑎𝑥𝜃∈Θ𝑙(𝜃) (2.7)

The exact maximum likelihood method estimate the parameter marginal

distribution and the parameter of copula function at the same time, but this could be

very computationally intensive, especially in the high dimension case. According to

(2.6), it can be found the log-likelihood function is composed by two terms: one term

involving the copula function parameters and one term involving the parameters of the

marginal distributions, so the parameters can be estimated separately not

simultaneously to reduce computational load. Based on this idea, the IFM method (Joe

and Xu, 1996) was proposed. IFM method estimates the parameters in two steps:

Step 1: The parameter 𝜽𝟏 of the marginal distribution is estimated by

�̂�𝟏 = ArgMax𝜽𝟏 ∑ ∑ 𝑙𝑛𝑓𝑗 (𝑥𝑗𝑡 ; 𝜽𝟏 𝑛 𝑗=1

𝑇 𝑡=1 ) (2.8)

7

Step 2: Then, given �̂�𝟏, the estimation of 𝜽𝟐 is performed

�̂�𝟐 = 𝐴𝑟𝑔𝑀𝑎𝑥𝜽𝟐 ∑ ln 𝑐(𝐹1(𝑥1𝑡 ), 𝐹2(𝑥2𝑡 ), … , 𝐹𝑛 (𝑥𝑛𝑡 ); 𝜽𝟐, 𝑇 𝑡=1 �̂�𝟏) (2.9)

Compared to ML method, IFM method is highly efficient (Joe, 1997).

Canonical maximum likelihood method estimate the copula parameters without

specifying the marginal distribution. This method can be described as following steps:

Step 1: Estimate the marginal distributions by using empirical distribution,

namely �̂�𝑖 (𝑥𝑖𝑡 ) with i = 1, … , n.

Step 2: Estimate the copula parameters via MLE

�̂�𝟐 = 𝐴𝑟𝑔𝑀𝑎𝑥𝜽𝟐 ∑ ln 𝑐( 𝑇 𝑡=1 �̂�1(𝑥1𝑡 ), �̂�2(𝑥2𝑡 ), … , �̂�𝑛 (𝑥𝑛𝑡 ); 𝜽𝟐) (2.10)

Non-parametric estimation no longer assumes a particular parametric copula.

Empirical copula and Kernel copula are two commonly used non-parametric

estimation methods.

2.3 Tail Dependence

Whether financial markets will become more interdependent during the financial

crisis is a concern because it relates to asset allocation and risk management. Many

literatures have pointed out that there is a correlation breakdown between financial

markets, namely, in crash period, there exist a statistically significant increase in

correlation between financial markets. Bertero and Mayer (1989) and King and

Wadhwani (1990) find the correlation of stock returns at the time of the 1987 crash.

Calvo and Reinhardt (1996) find evidence that correlation shifts in the Mexican crisis.

Baijn and Goldfajn (1999) shows evidence of contagion in the currency and equity

markets between Malaysia, Indonesia, Korea and Philippine during the Asian crisis.

Since there is such a phenomenon in the financial market, we want to understand

the dependence between financial markets in extreme cases. For example, in an

extreme case, when a financial asset has a huge loss, we want to know whether another

asset will also suffer a huge loss, what is the degree of correlation between the two

assets and which method should be used to accurately measure the correlation. Some

8

empirical studies, such as Ane and Kharoubi (2003), indicate that tail dependence is a

useful tool to describe correlations in extreme cases.

The most common definition of tail dependence, discussed by Sibuya (1960) and

Joe (1997) among others, is the following approach. Let (X, Y) be a random vector,

and the joint distribution function is F, and the marginal distribution functions of X

and Y are G and H, respectively.

Then its upper tail dependence coefficient is

𝜆𝑈 = lim 𝑡→1−

𝑃{G(X) > t|𝐻(𝑌) > 𝑡} (2.11)

Lower tail dependence coefficient is

𝜆𝐿 = lim 𝑡→0+

𝑃{G(X) ≤ t|𝐻(𝑌) ≤ 𝑡} (2.12)

By this definition, it can be found that the tail dependence coefficient is exactly

equal to the probability that one variable exceeds one high/low threshold and the other

variable also exceeds one high/low threshold. As Juri (2002) pointed out that the aim

to researching the tail dependence between random variables is to know the probability

that a random variable will change similarly when one random variable changes.

Tail dependence coefficient can also be defined by copula function

𝜆𝑈 = lim 𝑢→1−

1−2𝑢+𝐶(𝑢,𝑢) 1−𝑢

(2.13)

𝜆𝐿 = lim 𝑢→0+

𝐶(𝑢,𝑢) 𝑢

(2.14)

The tail dependence coefficient defined by the copula function depends only on

the form and the parameters of the copula function itself, and it is independent of the

marginal distributions. Therefore, the copula function can be easily used to study the

tail dependence. Just select the appropriate copula function and estimate the

parameters of the copula function to get the tail dependence coefficient. Patton (2006)

considered an extension of the theory of copulas and found evidence of asymmetric

exchange rate dependence. Rodriguez（2007）utilized copula approach and found that

in times of financial turmoil, tail dependence will be more prevalent. Aloui et al. (2011)

employed a multivariate copula approach to capture the tail dependence of four

emerging markets and the US markets.

9

This paper will use the copula function to model the stock market indexes and

examine their tail dependence.

3 The Model The modelling steps are divided into three steps: 1) fit the log return of each index

with the GARCH(1,1) model to obtain the conditional distribution of log return; 2)

Estimate the parameters of the copula function after knowing marginal distributions

and selecting certain parametric copula family; 3) Calculate the tail dependence based

on the obtained copula model. These three steps will be described separately below.

3.1 Model Index Returns

Let {𝑆𝑡 } denote the index close price and {𝑅𝑡 } denote the log return, so

{𝑅𝑡 } = {𝑙𝑛 𝑆𝑡−1

𝑆𝑡 } (3.1)

When using the GARCH(1,1) model to fit the {𝑅𝑡 }, this can be expressed as

{ 𝑅𝑡 = 𝜇 + 𝜀𝑡

𝜀𝑡 = 𝜎𝑡 𝑣𝑡 𝜎𝑡2 = 𝛼0 + 𝛼1𝜀𝑡−12 + 𝛽𝜎𝑡−12

(3.2)

In formula (3.2), {𝑣𝑡 } are i.i.d. random variables with zero mean and unit

variance. The distribution form of {𝑣𝑡 } determines the distribution of the marginal

distribution. The commonly used distribution of {𝑣𝑡 } is Gaussian distribution, T

distribution, skewed T distribution and GED (generalized error distribution). The latter

three distribution types can better characterize the fat tail characteristic of financial

time series. This article will use the Students t distribution as the distribution form of

{𝑣𝑡 }, namely 𝑣𝑡 ~𝑡𝑑𝑓 . Therefore, there is one more parameter to be estimated, i.e. the

degree of freedom.

The parameters to be estimated for the t-GARCH(1,1) model are 𝜇, df, 𝛼0, 𝛼1

and 𝛽. After estimating these parameters, the conditional distribution of 𝑅𝑡+1 can be

obtained.

F(r) = P(𝑅𝑇+1 ≤ 𝑟 | 𝐼𝑇 ) = 𝑃(𝜀𝑡+1 ≤ 𝑟 − 𝜇 | 𝐼𝑇 ) = 𝑃(𝜎𝑡+1𝑣𝑡+1 ≤ (𝑟 − 𝜇)|𝐼𝑇 )

10

= P (𝑣𝑡+1 ≤ 𝑟−𝜇

√𝛼0+𝛼1𝜀𝑡 2+𝛽𝜎𝑡

2 ) = 𝑡𝑑𝑓 (

𝑟−𝜇 𝛼0+𝛼1𝜀𝑡

2+𝛽𝜎𝑡 2) (3.3)

In formula (3.3), 𝐼𝑇 is the set of information up to time T.

3.2 Estimate Copula Function

After having the conditional distribution and historical observations of the four

indexes log returns, we can get {𝑢𝑖,𝑡 = 𝑡𝑑𝑓 ( 𝑟𝑖,𝑡−𝜇

𝛼𝑖,0+𝛼𝑖,1𝜀𝑖,𝑡 2 +𝛽𝑖𝜎𝑖,𝑡

2 )} 𝑡=1

𝑇 𝑖 = 1,2,3,4 and

estimate bivariate copula functions.

There is one point to note before estimating the copula function. The thing is that

since the indexes may not be traded for some days, the samples are not as 𝒓𝑡 =

{𝑟i,𝑡 , 𝑟j,𝑡 }𝑡=1𝑇 , 1 ≤ i ≠ j ≤ 4 which is a complete observation vector at each day. Since

the estimate of the marginal distribution involves only the sample of each index itself,

the day of non-transaction can be removed as a holiday. However, this cannot be done

when estimating the copula function, and the data needs to be pre-processed. The

approach taken in this paper is to select the days when both indices have transactions,

that is, to exclude those sample points that any one of the indexes has no data.

In this paper, the maximum likelihood estimation method is used to estimate the

three different parametric copula functions, namely Gaussian copula, T copula and

Clayton copula.

3.2.1 Estimate Gaussian Copula The density of bivariate Gaussian copula is:

𝑐(𝑢1, 𝑢2, … , 𝑢𝑛 ) =

1

(2𝜋) 𝑛 2 |𝑅|

1 2

exp(−1 2

𝒙′𝑅−1𝒙)

∏ ( 1 √2𝜋

𝑛 𝑗=1 exp (−

1 2

𝑥𝑗 2))

(3.4)

In formula (3.4), 𝑥𝑗 = Φ−1(𝑢𝑗 ). Then we can get

𝑐(𝑢1, 𝑢2, … , 𝑢𝑛 ) = 1

|𝑅| 1 2

exp (− 1 2

𝝇′(𝑅−1 − 𝐼)𝝇) (3.5)

In formula (3.5), 𝝇 = (Φ−1(𝑢1), Φ−1(𝑢2), … , Φ−1(𝑢𝑛))′ . Then the log

11

likelihood function is

𝑙(𝛉) = − T 2

ln|𝑅| − 1 2

∑ 𝝇𝑡′ (𝑅−1 − 𝐼)𝝇𝑡𝑇𝑡=1 (3.6)

In the bivariate case the only parameter is the correlation coefficient ρ . The

specific log likelihood function in bivariate case is

𝑙 = − T 2

ln(1 − ρ2) − 1 2

∑[ (Φ−1(𝑢1,𝑡 ))2 + (Φ−1(𝑢2,𝑡 ))2 − 2𝜌Φ−1(𝑢1,𝑡 )Φ−1(𝑢2,𝑡 )

1 − 𝜌2

𝑇

𝑡=1

−(Φ−1(𝑢1,𝑡 ))2 − (Φ−1(𝑢2,𝑡 ))2] (3.7)

The likelihood function values can be obtained by bringing {𝑢1,𝑡 }, {𝑢2,𝑡 } and 𝜌

into the above equation. The parameter 𝜌 can be estimated by iterating the value of

𝜌 such that the value of the log likelihood function is maximized.

3.2.2 Estimate T Copula The density of the bivariate t copula is

𝑐(𝑢1, 𝑢2) = 1

√1−𝜌2

Γ(𝑣+2 2

)Γ(𝑣 2

)(1+ 𝑡𝑣

−1(𝑢1) 2+𝑡𝑣

−1(𝑢2) 2−2𝜌𝑡𝑣

−1(𝑢1)𝑡𝑣 −1(𝑢2)

𝑣(1−𝜌2) )−

𝑣+2 2

Γ2(𝑣+1 2

)(1+ 𝑡𝑣

−1(𝑢1)2

𝑣 )−

𝑣+1 2 (1+

𝑡𝑣 −1(𝑢2)2

𝑣 )−

𝑣+1 2

(3.8)

In the above expression, v is the degree of freedom of T copula functions. It can

be deduced that the log likelihood function is

𝑙(𝜌, 𝑣) = − 𝑇 2

ln(1 − 𝜌2) + T × ln (Γ (𝑣+2 2

)) + 𝑇 × ln (Γ (𝑣 2 )) − 𝑣+2

2 ∑ ln ((1 +

𝑡𝑣−1(𝑢1,𝑡) 2

+𝑡𝑣−1(𝑢2,𝑡) 2

−2𝜌𝑡𝑣−1(𝑢1,𝑡)𝑡𝑣−1(𝑢2,𝑡) 𝑣(1−𝜌2)

))𝑇𝑡=1 −

2 T × ln (Γ (𝑣+1 2

)) + 𝑣+1 2

∑ ln (1 + 𝑡𝑣 −1(𝑢1,𝑡)

2

𝑣 )𝑇𝑡=1 +

𝑣+1 2

∑ ln (1 + 𝑡𝑣 −1(𝑢2,𝑡)

2

𝑣 )𝑇𝑡=1 (3.9)

By iterating the value of 𝜌 and v, the estimated parameters can be obtained by

maximizing the log likelihood function value.

3.2.3 Estimate Clayton Copula Clayton Copula is a kind of Archimedean Copula. In bivariate case, its copula

density is

c(u1, u2) = (1 + α)(u1u2)−𝛼−1(u1−𝛼 + u2−𝛼 − 1) −2−1

𝛼 (3.10)

In formula(3.9), α is the parameter of Clayton Copula.

12

The log likelihood function corresponding to bivariate Clayton Copula is

𝑙(𝛼) = T ∗ ln(1 + α) − (α + 1) ∑(ln (u1,𝑡 + 𝑇

𝑡=1

u2,𝑡 )

−(1 𝛼

+ 2) ∑ ln (𝑇𝑡=1 u1,𝑡 −𝛼 + u2,𝑡 −𝛼 − 1) (3.11)

Similarly to the estimation of Gaussian and T copulas, by iterating the parameter

α and maximizing the log likelihood function (3.10) the parameter 𝛼 can be

estimated.

3.3 Tail Dependence

The calculation of the tail dependence coefficient is based on the calculation

formula of the tail dependence coefficient of each copula, since the tail dependence

coefficient defined by the copula function depends only on the form and parameters

of the copula function itself. The following table shows the formula to calculate the

tail dependence coefficient of Gaussian copula, T copula and Clayton copula.

Table 1: Tail dependence coefficient formula

Copula Function 𝜆𝑢𝑝 𝜆𝑙𝑜𝑤

Gaussian Copula 0 0

T Copula 2𝑡𝑣+1(−√𝑣 + 1√ 1 − 𝜌 1 + 𝜌

) 2𝑡𝑣+1(−√𝑣 + 1√ 1 − 𝜌 1 + 𝜌

)

Clayton Copula 0 2− 1 𝛼

Gumbel Copula 2 − 2− 1 𝛼 0

Frank Copula 0 0

Gaussian copula has no tail dependence, namely its upper and lower tail

dependence are equal to zero. T copula has upper and lower tail dependence and they

are equal. Clayton copula has only lower tail dependence.

13

4 Empirical Calibration and Results In this section, the daily closing price of the stock market indexes will be modelled,

including FTSE100, S&P500, HS300, and Nikkei225. In order to make the fitting

process smooth, log return is scaled by 100, i.e. {100 × 𝑅𝑡 } is the object to be

modelled. The time interval is from January 1, 2009 to July 31, 2019.

4.1 The Data

Financial time series have some characteristics of their own. Through the

following descriptive statistics, it can be found that the log return of all indexes shows

the case of negative skewness. Except that the log return of FTSE100 has a kurtosis

of less than 3, showing a platykurtic, the log returns of other indices exhibit a

leptokurtic, that is, the kurtosis is greater than 3. This indicates that the financial time

series often has fat tail phenomenon.

Table 2: Descriptive statistics

Indexes Mean Median Std Skewness Krutosis

FTSE100 0.000189 0.000341 0.010012 -0.159446 2.839389

S&P500 0.000436 0.000663 0.010293 -0.329508 5.184206

Nikkei225 0.000324 0.000513 0.013499 -0.477822 4.778648

HS300 0.000276 0.000552 0.015492 -0.623943 4.107534

Some basic features of the financial time series can also be found through time

series plots. The following figure shows the time series plot of {𝑅𝑡 } of FTSE100.

14

Figure 1: The left is the time series plot of {𝑅𝑡 } of FTSE100. The right is the time series plot of {𝑅𝑡2} of FTSE100.

Through the time series plot at the top of the left, it can be seen that there is

volatility clustering in the log return sequence of FTSE100 where volatility is

obviously not constant, but changes over time. Through ACF and PACF in (a), it seems

that log return does not have autocorrelation. The QQ plot at the bottom left tells us

that there is a fat tail in the distribution of log return. The ACF and PACF on the right

tell us that there is a significant autocorrelation of {𝑅𝑡2}. Similarly, the log returns of

the S&P500, HS300 and Nikkei225 have similar characteristics.

Figure 2: The left is the time series plot of {𝑅𝑡 } of S&P500. The right is the time series plot of {𝑅𝑡2} of S&P500. We can find there exist volatility clustering and fat tail in the {𝑅𝑡 }. And the {𝑅𝑡2} has autocorrelation.

15

Figure 3: The left is the time series plot of {𝑅𝑡 } of Nikkei225.The right is the time series plot of {𝑅𝑡2} of Nikkei225. We can find there exist volatility clustering and fat tail in the {𝑅𝑡 }. And the {𝑅𝑡2} has autocorrelation.

Figure 4: The left is the time series plot of {𝑅𝑡 } of HS300. The right is the time series plot of {𝑅𝑡2} of HS300. We can find there exist volatility clustering and fat tail in the {𝑅𝑡 }. And the {𝑅𝑡2} has auto- correlation.

According to the above analysis, it can be found that there are fat tail and volatility

clustering in the log return series. Therefore, it is reasonable to use the GARCH model

with the T-distributed innovation to fit the marginal distributions.

4.2 Results

4.2.1 Marginal Distributions The GARCH(1,1) model was established for FTSE100, S&P500, Nikkei225 and

16

HS300 respectively. The estimated parameters are as follows.

Table 3: estimated parameters for GARCH models

Parameters/Indexes FTSE100 S&P500 Nikkei225 HS300

𝜇 0.0415***

(t=2.842)

0.0838***

(t=6.902)

0.0818***

(t=4.117)

0.0507**

(t=2.415)

𝛼0 0.0254***

(t=3.084)

0.0169***

(t=3.364)

0.0465***

(t=3.133)

9.3504e-03**

(t=2.261)

𝛼1 0.1207***

(t=5.389)

0.1407***

(t=6.422)

0.1162***

(t=5.117)

0.0550***

(t=6.017)

𝛽 0.8580***

(t=33.318)

0.8540***

(t=42.385)

0.8635***

(t=35.189)

0.9443***

(t=112.036)

𝑣 6.8268***

(t=8.025)

4.9861***

(9.763)

5.9979***

(t=8.189)

4.7577***

(t=10.532)

Note: The t statistics are in parentheses. ‘*’ means significant at the 10% significance level, ‘**’ means

significant at the 5% significance level and '***’ means significant at the 1% significance level.

According to the parameter estimation obtained in the above table, the marginal

distributions of the four log returns of FTSE100, S&P500, Nikkei225 and HS300 can

be obtained by (3.3).

Through the parameter estimation result table of the above GARCH (1, 1) model,

we can find that the β value of each GARCH model exceeds 0.85. This shows that

there are strong serial correlations in all four log returns. In addition, we can also find

that 𝛼1 + 𝛽 of each GARCH model is close to 1. 𝛼1 + 𝛽 is called the persistence,

as it defines the speed at which shocks to the variance revert to their long-run values.

This shows that the persistence of these model is very strong, that is if the variance is

increased by an impact, it takes a long time to recover the long-run average level.

17

4.2.2 Estimated Copula Functions The parameters of the copula function estimated based on the MLE method are

shown in the following table.

Table 4: Estimated parameters of copula functions

Parameters Gaussian Copula T Copula Clayton Copula

FTSE100 vs. S&P500

ρ 0.670 0.666 / df / 5.406 / α / / 1.112

FTSE100 vs. Nikkei225

ρ 0.374 0.358 / df / 12.043 / α / / 0.450

FTSE100 vs. HS300

ρ 0.308 0.273 / df / 6.506 / α / / 0.349

S&P500 vs. Nikkei225

ρ 0.332 0.286 / df / 5.151 / α / / 0.352

S&P500 vs. HS300

ρ 0.257 0.212 / df / 5.804 / α / / 0.250

Nikkei225 vs. HS300

ρ 0.422 0.411 / df / 18.984 / α / / 0.512

From the estimation results of the copula function, it can be found that the

estimated parameters are within a reasonable interval.

For Gaussian copula and T copula, the greater the ρ, the greater the correlation

between the two log return sequences. For Clayton copula, the larger the α, the greater

the correlation between the two log return sequences. From the parameter table, it can

be found that the parameters of Gaussian copula, T copula and Clayton copula show

consistency, that is, the correlation ranking is consistent in each copula model.

4.2.3 Tail Dependence The tail dependence between the log returns can be visually observed first through

the three-dimensional histogram.

18

Figure 5: 3D histograms

From the above three-dimensional histograms, it can be found that the height of

the bar in the lower-left corner of each graph is higher, that is the two log returns have

a higher frequency of having smaller values at the same time, which indicates that

there is a tail dependence between the two log returns.

The tail dependence coefficient can quantify the magnitude of the tail dependence.

After estimating the parameters of the copula function, the tail dependence coefficient

can be obtained according to the formula of the tail dependence coefficient. The table

below summarizes the tail dependence coefficient.

Table 5: Tail dependence coefficients

T Copula Clayton Copul

19

FTSE100 vs. S&P500 𝜆𝑈 = 𝜆𝐿 = 0.297 𝜆𝑈 = 0, 𝜆𝐿 = 0.536

FTSE100 vs. Nikkei225 𝜆𝑈 = 𝜆𝐿 = 0.027 𝜆𝑈 = 0, 𝜆𝐿 = 0.214

FTSE100 vs. HS300 𝜆𝑈 = 𝜆𝐿 = 0.074 𝜆𝑈 = 0, 𝜆𝐿 = 0.138

S&P500 vs. Nikkei225 𝜆𝑈 = 𝜆𝐿 = 0.113 𝜆𝑈 = 0, 𝜆𝐿 = 0.140

S&P500 vs. HS300 𝜆𝑈 = 𝜆𝐿 = 0.075 𝜆𝑈 = 0, 𝜆𝐿 = 0.063

Nikkei225 vs. HS300 𝜆𝑈 = 𝜆𝐿 = 0.009 𝜆𝑈 = 0, 𝜆𝐿 = 0.258

Since Clayton Copula can better reflect the lower tail dependence, through the

lower tail dependence coefficient of Clayton Copula, we can find that there is a

relatively strong lower tail dependence between the log returns of each index. This

means that one index has a large probability to fall when the other index falls.

The lower tail dependence coefficient between FTSE100 and S&P500 is the

largest. It seems that UK and USA stock markets have the highest level of financial

contagion. HS300 has relatively low tail dependence coefficients with other indexes

compared to other indexes. This may be explained by China's capital control over the

capital market. Capital control makes the circulation of funds unfree, and the linkage

between stock markets declines.

5 Conclusion This paper uses the GARCH-copula method to establish the bivariate joint

distribution model between stock index log returns. First, the log returns of FTSE100,

S&P500, Nikkei225 and HS300 are fitted by GARCH(1,1) respectively. The GARCH

model better solves the problems of volatility clustering and fat tail in the log returns

sequence. From the parameters of the GARCH model, we can find that the Beta value

of each model is relatively large (larger than 0.85), and the persistence of the model is

close to 1, which means that the variance of each log returns sequence takes a long

time to recover to long-run value after being shocked.

Secondly, Gaussian copula, T copula and Clayton copula are estimated. Finally

20

the tail dependence is calculated between each index log return and it is found that

there is a strong lower dependence between the indices. It can be seen from the

analysis results that the copula function of FTSE100 and S&P500 have the largest

correlation coefficient parameters, which indicates a strong correlation. At the same

time, the tail dependence coefficient of the two is also relatively large. In contrast, the

correlation between HS300 and the other three indices is much weaker. This shows

that the linkage between the UK and the US stock market is strong, which is

inseparable from the financial system in which capital flows freely. China’s control

over capital flows has always been strict, which will definitely lead to a decline in the

linkage between its stock market and international developed stock markets. However,

on September 10, 2019, the China Foreign Exchange Administration announced the

cancellation of the investment quota limit for QFII (Qualified Foreign Institutional

Investor) and RQFII (RMB Qualified Foreign Institutional Investor). This move will

certainly enhance the linkage between the Chinese stock market and the international

stock market in the future. Human behavior is guided and restricted by various systems.

Therefore, it is obvious that when we do investment decentralization or pricing of

financial products, we should take into account changes of systems which will cause

human behaviour changes and lead to changes in market correlation.

21

Reference [1] Aloui, R., Aïssa, M.S.B. and Nguyen, D.K., 2011. Global financial crisis, extreme

interdependences, and contagion effects: The role of economic structure?. Journal of

Banking & Finance, 35(1), pp.130-141.

[2] Ane, T. and Kharoubi, C., 2003. Dependence structure and risk measure. The

journal of business, 76(3), pp.411-438.

[3] Baig, T. and Goldfajn, I., 1999. Financial market contagion in the Asian crisis. IMF

staff papers, 46(2), pp.167-195.

[4] Bertero, E., & Mayer, C., 1990. Structure and performance: Global

interdependence of stock markets around the crash of October 1987. European

Economic Review, 34(6), pp.1155-1180.

[5] Bollerslev, T., 1986. Generalized autoregressive conditional

heteroskedasticity. Journal of econometrics, 31(3), pp.307-327.

[6] Bollerslev, T., 1987. A conditionally heteroskedastic time series model for

speculative prices and rates of return. Review of economics and statistics, 69(3),

pp.542-547.

[7] Calvo, S., 1999. Capital flows to Latin America: is there evidence of contagion

effects?. The World Bank.

[8] Cherubini, U., Luciano, E. and Vecchiato, W., 2004. Copula methods in finance.

John Wiley & Sons.

[9]Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of

the variance of United Kingdom inflation. Econometrica: Journal of the Econometric

Society, pp.987-1007.

[10] Ferenstein, E. and Gasowski, M., 2004. Modelling stock returns with AR-

GARCH processes. SORT-Statistics and Operations Research Transactions, 28(1),

pp.55-68.

[11] Frahm, G., Junker, M. and Schmidt, R., 2005. Estimating the tail-dependence

coefficient: properties and pitfalls. Insurance: mathematics and Economics, 37(1),

pp.80-100.

22

[12] Glosten, L.R., Jagannathan, R. and Runkle, D.E., 1993. On the relation between

the expected value and the volatility of the nominal excess return on stocks. The

journal of finance, 48(5), pp.1779-1801.

[13] Joe, H. and Xu, J.J., 1996. The estimation method of inference functions for

margins for multivariate models.

[14] Joe, H., 1997. Multivariate models and multivariate dependence concepts. CRC

Press.

[15] Juri, A. and Wüthrich, M.V., 2002. Copula convergence theorems for tail

events. Insurance: Mathematics and Economics, 30(3), pp.405-420.

[16] King, M. A., & Wadhwani, S., 1990. Transmission of volatility between stock

markets. The Review of Financial Studies, 3(1), pp.5-33.

[17] Li, D.X., 2000. On default correlation: A copula function approach. The Journal

of Fixed Income, 9(4), pp.43-54.

[18] Nelson, D.B., 1991. Conditional heteroskedasticity in asset returns: A new

approach. Econometrica: Journal of the Econometric Society, pp.347-370.

[19] Rodriguez, J.C., 2007. Measuring financial contagion: A copula approach. Journal

of empirical finance, 14(3), pp.401-423.

[20] Patton, A. J., 2004. On the out-of-sample importance of skewness and asymmetric

dependence for asset allocation. Journal of Financial Econometrics, 2(1), pp.130-168.

[21] Patton, A.J., 2006. Modelling asymmetric exchange rate dependence.

International economic review, 47(2), pp.527-556.

[22] Salmon, F., 2009. A formula for disaster. Wired, March, pp.74-79.

[23] Sklar, M., 1959. Fonctions de repartition an dimensions et leurs marges. Publ.

inst. statist. univ. Paris, 8, pp.229-231.