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Amsterdam School of Economics
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The dynamic relationship between
the Euro Stoxx 50 returns and the
percentage changes in the VSTOXX
N ikki W esselius
Master’s Thesis to obtain the degree in Econometrics
Specialization in Financial Econometrics University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics Author: N ikki W esselius Student number: 10338942
Date: July 14, 2018
Email: nikkiwesselius@gmail.com Supervisor UvA: Prof. Dr. H.P. Boswijk Supervisor KPMG: E. Blom
Statement of Originality
This document is written by Student N ikki W esselius who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Acknowledgement
I would like to thank the people from Financial Risk Managment at KPMG, the Netherlands, especially Erwin Blom, my thesis supervisor, who was always there for a peptalk when I needed one. A special thanks to professor Peter Boswijk who gave me great advice during thesis meet-ings. I also want to thank my mother for rereading this entire thesis over and over again to give critical notes. Last and most of all I want to thank Beer, who helped me during this research with his infinite patient and creativity to improve my programming skills.
Abstract
This paper uses the time-varying correlated jump intensity MGARCH model and correlated bivariate Poisson MGARCH model to investigate the dynamics between the log returns of the Euro STOXX 50 and the log changes of the VSTOXX. We find that the diffusion correlation coefficient of the indexes is negative as well as the jump correlation coefficient. Even though the Euro STOXX 50 has some individual jumps, the jumps in the indexes mostly occur at the same time. In the time-varying model, we show that the intensity of correlated jumps is influenced by the past residuals and the jump intensity of the last period. We find that the expected jump size of the Euro STOXX 50 returns is negative while the expected size of the jumps of the changes in the VSTOXX is positive and of bigger magnitude. The changes in the returns of the VSTOXX lead to a higher risk of jumps than the changes in the Euro STOXX 50 returns. We conclude that adding a jump component to the model adds extra information to better un-derstand the relation between the index returns, without affecting the MGARCH part of the model.
Contents
1 Introduction 1
2 Literature Review 4 2.1 Vector Autoregressive model . . . 4 2.2 Multivariate Generalized Autoregressive Conditional Heteroscedasticity
model . . . 5 2.3 Poisson Correlation function . . . 8 2.4 Relationship between the indexes . . . 8 3 Model & Method 10 3.1 Model Setup . . . 10 3.2 Variations of the model . . . 14
4 Data 16 5 Results 19 6 Discussion 26 7 Conclusion 28 Bibliography 29 A Proofs 32
A.1 Poisson density function . . . 32 A.2 Covariance . . . 33
1. Introduction
On February 26, 1998 the Euro STOXX 50 Index was introduced by STOXX, an index provider owned by the Deutsche B¨orse Group (STOXX, 2018a). The price data of the Euro STOXX 50 was calculated retroactively to the year 1986. The Euro STOXX 50 Index contains the 50 largest and most liquid stocks in Europe. It represents Europe’s 50 biggest companies among 19 super sectors out of 11 European countries: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Luxembourg, the Netherlands, Por-tugal, and Spain. It is considered the leading benchmark of the performance of the Eu-ropean blue-chip stock market. The index serves as the underlying of financial products as exchange-traded funds (ETFs), futures, options and structured products worldwide. The Euro STOXX 50 captures around 60 per cent of the free-float market capitalization of the EURO STOXX Total Market Index, which covers about 95 per cent of the free-floated market capitalization of the represented countries. The joint venture of SWX, Deutsche B¨orse and Dow Jones & Company decide if a stock should be included or excluded from the index (STOXX, 2018a).
The Euro STOXX 50 volatility (VSTOXX) aims to measure the volatility of the EURO STOXX 50 Index. It measures the implied variance across all STOXX option contracts of thirty days to expiry available on the Eurex Exchange. The model used for the VSTOXX is developed by Goldman Sachs and Deutsche B¨orse and is based on no-arbitrage pricing of variance swaps (STOXX, 2018b). The goal of the index is to make it possible to trade solely on volatility and not on price fluctuations. The VSTOXX is often referred to the ”fear index” of the European stock market.
It is interesting to look at the relationship between the Euro STOXX 50 returns and the log changes in the VSTOXX. There has not been done much research on this subject. Nevertheless, the relationship between Euro STOXX 50 and the VSTOXX can be compared with the American version of the two indexes: the S&P 500 and the VIX. The S&P 500 contains 505 stocks issued by 500 big American companies. The VIX is the Market Volatility Index for the upcoming thirty days based on the S&P 500.
Much research has been done on the relationship between these two indexes. For example, Hibbert, Daigler, and Dupoyet (2008) and Dennis, Mayhew, and Stivers (2006) found evidence for a negative, asymmetric relation between an S&P 500 related stock and the VIX. These articles explain the, sometimes causal, relationship between the
indexes extensively. The main issue these studies miss is the fact that the models do not take jumps (in volatility) into consideration, which makes the models vulnerable for mispricing. Other studies like Lin and Lee (2010) and Becker, Clements, and McClelland (2009) focus on building a model that does contain jumps. These studies extend the bi-variate models for different kind of indexes to models with discontinuous jump-diffusion processes. Unfortunately, they do not tell much about the relationship between the VIX log changes and S&P returns.
The Euro STOXX 50 index has quite different dynamics than the S&P 500, so carrying out research on these European data can give a different outcome than the studies described above. Here we will use models with jump characteristics to look for evidence for the causal relationships in two directions between the Euro STOXX 50 returns and the percentage changes in the VSTOXX. This leads us to the following research question: ”What is the dynamic relationship between the Euro STOXX 50 returns and the VSTOXX log changes and what is the role of jump dynamics in this relationship?” The purpose of this research is to develop a bi-variate jump model to study the relationship between both indexes, with focus on the jump dynamic relation. An appropriate model to see how the log changes in the Euro STOXX 50 and VSTOXX affect one another is a bivariate extension of the Generalized Autoregressive Conditional Heteroscedasticity (hereafter, GARCH) model. The GARCH model takes into account heteroscedasticity and is able to capture clustered volatility movements. In addition to the GARCH model, the Bivariate Generalized Autoregressive Conditional Heteroscedasticity (hereafter, BGARCH) model does not only look at the aspects men-tioned above but also looks at the way the lagged volatilities affect one another, like the spillover effects and the dynamic covariance.
The BGARCH model might be extended since it does not perfectly fit the underlying data. The data might show leptokurtosis in the conditional distribution. The normal density in the BGARCH model can be changed to a Student t distribution or Generalized Error distribution since these distributions have fatter tails and more skewed conditional distribution.
The other possibility is to extend the BGARCH by including a correlated jump process to add infrequent large movements to the smooth volatility. Correlated Bivariate Poisson (hereafter, CBP) jumps will allow jumps in the Euro STOXX 50 and VSTOXX separately, but also jumps that occur in both at the same time. This CBP-GARCH model, which Chan (2003) uses, can be expanded further by making the jump intensities time-varying. This shows the jump correlation between the indexes and ensures a better understandable relationship between the jump dynamics and volatilities.
The data chosen for this research is the daily data of the Euro STOXX 50 and VSTOXX over the period of 1st of January 2008 up till the 31st of December 2017. Sev-eral tests will be done on the model to check for, among other things, misspecification, autocorrelation and robustness.
jumps and time-varying correlated jump intensities will be analyzed. With this model, there will be taken a closer look at the relationships between the indexes.
This research has been structured as follows: chapter 2 contains a literature review, in which the origin of the used models is being described. Also, relevant studies are exposed and compared with each other. Subsequently, chapter 3 describes the model set up and the way the model is estimated. This is followed by a description of the data in chapter 4. In chapter 5 the results are being scrutinized. The discussion, held in chapter 6, gives advice for further studies on this subject. The conclusion can be found in the final chapter 7.
2. Literature Review
This section provides a background for the chosen model and method and will give a review of the relevant previous studies. It gives an opportunity to compare the re-sults of this study with former ones. First, the Vector Autoregressive (VAR) model is reviewed. Next, we give an overview of the different kinds of MGARCH models and their (dis)advantages. Then the bivariate correlated Poisson distribution is discussed and maximum likelihood estimation is described. Subsequently, the leverage effect and the effect of jumps are discussed. Lastly, the relevant researches on the discussed models will be compared with each other
2.1
Vector Autoregressive model
Due to globalization in the financial world, movements in returns in one market easily expand to another market. These dynamic relationships should be structured in financial econometric models for multiple return series. The multivariate time series analyze the returns of multiple assets or indexes. Modelling can show dynamic relationships between sets of variables. The VAR model is a way of modelling index returns. The VAR model in matrix notation looks like this:
xt= φ0+ p
X
l=1
Φlxt−l+ at (2.1)
where xt= (x1t, ..., xnt)0is the n-dimensional vector time series at time t, xt−lis the same
kind of time series but than at time t − l, Φ0 = (φ10, ..., φn0)0 is the vector of intercepts,
φl=
h φij(l)
i
are n × n coefficient matrices and at= (a1t, ..., ant)0 is a multivariate white
noise with V(at) = Σ.
In the bivariate variant of the VAR(1) model, the log returns depend on their own lagged log returns and on the one of the other index. The VAR(1) model will be the starting point of our model, to capture the relationship between the changes in Euro STOXX 50 and VSTOXX returns and their lag. Different tests, discussed in chapter 5, check whether the VAR model fits well and if it is adequate. In multivariate time series the use of Vector Autoregressivemoving-average (VARMA) models is unusual, because of the difficulties when identifying parameters and for that reason they will not be used in our modelling
2.2
Multivariate Generalized Autoregressive Conditional
Heteroscedasticity model
An obstruction in classical econometric models forecasting price indexes is that they assume that the one-period forecast variance is constant. In times of economic agitation, a high variance today implies a high variance tomorrow. This means that the volatility of indexes is clustered. Therefore Engle (1982) introduces the Autoregressive conditional heteroskedastic (ARCH) process, in which variances are non-constant. In this model, the conditional variance is influenced by the past squared errors. This is a simple way to capture the dependence of volatility. Bollerslev (1986) introduced an extension of the ARCH model: the GARCH model. The benefit of the GARCH model compared to the ARCH model is the more flexible lag structure, since it allows for past conditional variances in the model for the current conditional variance. GARCH models can be used when data exhibits heteroskedasticity and volatility clustering. The GARCH(m,s) model looks like this:
σt2= α0+ m X i=1 αia2t−i+ s X j=1 βjσt−j2 (2.2)
where α0 > 0, αi≥ 0, βj ≥ 0 to get σt2> 0 and where σt2 is the conditional variance of
at.
In the research of Bollerslev, Engle, and Wooldridge in 1988, this univariate GARCH model is extended to a multivariate one. Their study aims to estimate returns to T-bills, bond and stocks by using the MGARCH model since they foresee the expected returns to be proportional to the conditional covariance. The MGARCH model complements the univariate GARCH model because of the way markets influence each other becomes visible. It shows how the volatility of one market follows the volatility of the other. It’s also possible to see if a shock in one market increases the volatility of another market. Next to that it can also take into account spillover effects, to see the information transmission between the different stocks. When using an MGARCH model we will be able to see whether the conditional correlations change over time, for example, if they have a higher value during financial crises.
An example of a research that uses an MGARCH model is the one of Baillie and Bollerslev (1990), where spot and forward exchange rates of different currencies are modelled. The MGARCH model is used to test if the risk premium is a linear function of the conditional variances and covariances.
The MGARCH model can take different forms, that will be discussed, based on assumptions, restrictions and effects. We will start with a vector stochastic process xt
of dimension N×1 and θ is a finite vector of parameters. This gives: xt= µt(θ) + at
where µt(θ) is the conditional mean vector and
with Ht(θ)1/2 is an n × n positive definite matrix. Also assumed is that zt is an n × 1
random vector with:
E(zt) = 0
V(zt) = In
with In an n × n identity matrix. Because V(xt|It−1) = Ht, Ht(θ) must be a positive
definite matrix.
From this starting point, we go to the generalizations of the univariate standard GARCH model: the VEC and BEKK. In the VEC model Ht is a linear function of the
lagged squared residuals, cross-products of residuals and lagged values of the elements of Ht (Bauwens, Laurent, and Rombouts, 2006). This model is very flexible in terms of
persistence in volatility and covariation, time variation in correlation and volatility spill-over effects. Nevertheless, the huge disadvantage of the VEC model is the complicated parameter restrictions to keep H
1 2
t (θ) positive definite. Another disadvantage is that the
number of parameters drastically increases when the number of time series increases. These criteria mentioned make the VEC model in practice only useful for bivariate models.
Because the VEC model has such hard restrictions, Engle and Kroner came up with a new model in 1995, the so-called BEKK (Baba, Engle, Kraft, Kroner) model. The advantage of this model is that the number of parameters is significantly less than in the VEC model. The BEKK model is a restricted version of the VEC model but is still very flexible. The parameter restrictions to keep the model positive definite are less strict. Even though the BEKK model looks like an easier model to estimate, in practice the estimation of the parameters of both the VEC and BEKK model is very difficult. That’s why these models are rarely used for multiple time series of more than three elements. Since we are working with a bivariate model, we can make use of the BEKK model. Both the models are good for presenting dynamics of variances and covariances. If the volatility transmission is more important in the research, these two models might not be the best to use. Despite the loss of finding volatility transmission, these models indicate the dynamics of variances and covariances very well. The cross-dynamics between the variance of the Euro STOXX 50 returns and the variance of the VSTOXX log changes in this research are important to be indicated, so a BEKK model could be a right model to use.
There are also nonlinear combinations of univariate GARCH models like the CCC and DCC model, which have a big reduction in terms of computational complexity in comparison with the models mentioned above. The flip side is that stationarity and the moments are not easy to obtain. The advantage of the nonlinear combinations is that both the individual conditional variances and the conditional correlation matrix are estimated between series. The Constant Conditional Correlation (CCC) Model by Bollerslev (1990), as the name reveals, assums that the conditional correlations are constant, but the MGARCH model still has time-varying conditional variances and co-variances. Ht(θ) is positive definite if all conditional variances are positive and constant
conditional correlations are between -1 and 1.
Because constant conditional correlations are too restrictive in many situations, the Dynamic Conditional Correlation (DCC) model is formed, in which the conditional correlation matrix is time-dependent (Engle, 2002). DCC models allow for dynamically varying conditional correlations. The way the models are estimated is with the two-step method. In the first step for every individual return, a univariate GARCH model is estimated. The second step is to involve a correlation matrix and estimate this. DCC models do not allow for volatility spill-overs. Engle (2002) finds that bivariate DCC models give a good estimation to a variety of time-varying correlation processes.
Financial data tend to be very sensitive to shocks. A large negative shock, like a stock market crash, is expected to affect the volatility more than a large positive shock. This phenomenon is known as the leverage effect. In MGARCH models the variances and covariances also tend to be more sensitive for negative shocks. The leverage effect causes asymmetry, this asymmetry can be included in for example, the BEKK, Factor or DCC model.
To estimate the models of volatility mentioned above, maximum likelihood is used. MGARCH models tackle heteroskedasticity. A disadvantage is that financial data, for which these extended GARCH models are frequently used, usually do not have a normal distribution. Financial data are often more fat-tailed, so with a large skewness and/or kurtosis. The MGARCH model does not nicely capture this leptokurtosis in the un-conditional distribution. If a un-conditional normal distribution is used a mismatch will potentially arise. Many solutions are proposed to solve the described problem.
The first was presented by the study of Bollerslev and Wooldridge in 1992. In their research, they claim that the consistent estimator can be obtained by maximizing the likelihood, even if the data generation process (DGP) is not conditionally Gaussian distributed. They show that if the conditional mean and variance are correctly spec-ified, the quasi-maximum likelihood (QML) or pseudo-maximum likelihood (PML) is applied and will give a consistent estimator, even if conditional normality is violated. The following years different researches, like Jeantheau (1998), proved that Gaussian QML gives strongly consistent estimators under certain conditions.
Although QML is justified for non-Gaussian MGARCH models, it is also possible to replace the normal density by the Power Exponential distribution or Student t distribu-tion, like in the study of Fiorentini, Sentana, and Calzolari (2003). The estimators are often more efficient using the Student t distribution, assuming this is indeed the correct conditional distribution for the DGP. A disadvantage is that the estimators might lose consistency if Student t distribution is not the correct conditional distribution. An im-portant aspect for financial data is that the multivariate normal distribution tends to underestimate extreme events compared with the Student t distribution. But these ex-treme events can be implemented in another way, for example with a Poisson correlated function.
2.3
Poisson Correlation function
An extreme event, like a large movement over a short time interval, can be seen as a jump. Jumps can occur, because of new announcements, unexpected disasters with economic, natural or terroristic reasons. These events can affect on macroeconomic level multiple stocks or can be specific for one or both of the returns. The effect of these jumps is not modeled in the above described MGARCH models since these models capture smooth and persistent changes in volatility and not large, discrete changes. Nevertheless, these large return differences are important to estimate since they have a large influence on expected values of the returns. It is important to notice that if the modelled returns are leptokurtic and asymmetric with respect to zero, a jump diffusion model might fit better. A combination of an MGARCH model with a Correlated Bivariate Poisson part will describe time-variation in volatility, fat-tails and skewness of financial series. The jumps of the CBP part will occur when the information arriving is irregular.
The importance of jumps in market returns is investigated by, among others, Ander-sen, Benzoni, and Lund (2002). In their study, they claim that every continuous-time model for index returns should allow for both discrete jumps and stochastic volatility. The stochastic volatility includes a strong negative relationship between the index re-turns and the volatility changes. The jumps are Poisson distributed with time-varying intensity. Pan (2002) mentions three kinds of risks that influence the S&P 500 index returns; the diffusive price shocks, the diffusive volatility shocks and the price jumps. The latter is added to the model by jump arrivals that are Poisson distributed with a time-varying intensity. There are also studies that use other distributions than the Poisson for the jump arrivals. In their study, Vlaar and Palm (1993) compare the use of Poisson and Bernoulli distributed jump processes. They do not have time variation included in the jump intensity. Their conclusion is that both distributions fit very well. It is also possible to let the parameter λ change over time. Vlaar and Palm (1993) find that the parameters for jump intensity of several different European currencies depend on the inflation in Germany. Chan (2003) lets the λ depend on the returns at time t − 1. Other possibilities to let the λ change over time are discussed in chapter 3.
2.4
Relationship between the indexes
Several researchers have looked at the relationship between the S&P 500 returns and the log changes in the VIX. As we mentioned above we find these two indexes comparable with the indexes of the Euro STOXX 50 and the VSTOXX. Lin and Lee (2010) find that the once lagged S&P 500 returns has a negative impact on the current S&P 500 returns, which is in line with the literature of Hibbert, Daigler, and Dupoyet (2008) and Dennis, Mayhew, and Stivers (2006). Next to that the lag VIX changes also have a negative impact on the current S&P 500 returns, which is in line with the negative return-volatility relationship described in the studies above.
negatively impacted by the lag VIX percentage changes. Hibbert, Daigler, and Dupoyet (2008) argue that investors expect volatility changes to maintain a trend in the near future, so the negative relationship between the one-lagged VIX log changes and the current one is consistent with the behavioural bias theory. Meanwhile in Lin and Lee (2010) the VIX log changes are positively impacted by the once and twice lags of the S&P 500 returns, which indicates a positive risk-return relationship.
Chan (2003) and Lin and Lee (2010) both find that the conditional variance equa-tions (α + β) are close but less than, one, which indicates a strong GARCH effect. Becker, Clements, and McClelland (2009) investigate the jump component of the S&P 500 volatility and the VIX index and find that the jump component cannot be rejected. The jump processes used in Lin and Lee (2010) are also significant. The relationship between the jump component and the S&P 500 returns and the VIX percentage changes are negative and highly correlated. They found that the log S&P 500 returns and the VIX percentage changes only have joint jumps and that the individual jumps are in-significant. They argue that the jump behaviour is not time-varying. Finally, they find that the relationship between the S&P 500 returns and the VIX log changes is causal in two directions.
Chiu and Lee (2007) looked at the relationship between the stock and exchange rates in Taiwan. They found that both the stock and exchange rate have a fat tail and JB sig-nificant statistic. For the financial time series they use the jump MGARCH model. The jump component part of the model has all significant parameters, for both the size and the intensity. With a likelihood ratio test, they test if the diffusion volatility spillover effects do not exist. This test rejects significantly, which indicates that there is informa-tion transmission between the stock market and the exchange rate. The jump intensity of both the stock and exchange returns in the model of Chiu and Lee (2007) depend on the returns of both the stock and exchange index of the last period. When testing this with a likelihood ratio test, this jump intensity spillover effect is not significant.
Another research of Chiu and Hung (2007) looks at the interaction between the share markets of Shanghai and Shenzen. They find that there is significant bidirectional normal information transmission between the two markets. Both the researches of Chiu and Lee (2007) and Chiu and Hung (2007) look at the models before and after an important event at a certain period in time. The model has a dummy variable which is equal to zero before the event and one at the time and after the event happened. In both the researches the coefficient for the dummy is significant.
We can conclude that in all researches found, the spillover effects are significant. Also, the diffusion volatility is found significant, in almost all papers. In these papers, the GARCH effects are in general strong.
3. Model & Method
3.1
Model Setup
The basis of the model is the CBP-GARCH model of Chan (2003). The CBP-GARCH model is a combination of the BEKK model of Engle and Kroner (1995) and the Pois-son Bivariate Correlation distribution of Campbell (1934), described in section 2.2 and section 2.3, respectively. The aim is to investigate the relationship between the Euro STOXX 50 returns and the log changes of the VSTOXX. When combining these two models both diffusion volatility and jump intensity spillovers will be captured. Con-structing the model, we will start with defining the 2×1 vector that presents the Euro STOXX 50 returns and log changes in the VSTOXX:
r1,t r2,t ! = (ln STt− ln STt−1) × 100 (ln V STt− ln V STt−1) × 100 ! (3.1) where STt and STt−1 are the prices of the Euro STOXX 50 stock index at trading day
t and t − 1 respectively. V STt and V STt−1 are, respectively, the VSTOXX index at
trading day t and t − 1.
The CBP-GARCH model is described as follows: Rt= r1,t r2,t ! (3.2) r1,t = c1+ β11r1,t−1+ β12r2,t−1+ 1,t+ J1,t (3.3) r2,t = c2+ β21r1,t−1+ β22r2,t−1+ 2,t+ J2,t (3.4)
where r1,t and r2,t are defined as above. These returns on day t depend on the returns
of day t − 1. Further, the 1,t and 2,t are the error terms on trading day t of the Euro
STOXX 50 returns and the log changes in the VSTOXX, respectively. The J1,t and J2,t
are the jump components on trading day t of r1,t and r2,t.
The random disturbances 1,t and 2,t follow a bivariate normal distribution with a
expectation of zero and conditional variance-covariance matrix ˜Ht:
1,t
2,t
!
∼ N0, ˜Ht
The variance-covariance matrix for the normal random disturbance ˜Ht, is assumed
to have the form of a bivariate CCC model. This is defined as follows: ˜ Ht= σ21,t σ12,t σ12,t σ2,t2 ! (3.5) With σ21,t= ω1+ α1˜21,t−1+ β1σ1,t−12 σ22,t= ω2+ α2˜22,t−1+ β2σ2,t−12 σ12,t= ω12 q σ21,tσ2,t2
The ˜t is the sum of the disturbance term and the jump component. The shocks the
variance-covariance matrix ˜Ht gets are from this ˜t, which is influenced by the returns.
It is also possible to use the BEKK model, then the model looks as follows: ˜Ht is
assumed to have the form of a bivariate symmetric BEKK model. ˜
Ht= C0C + A0˜t−1˜0t−1A + B 0˜
Ht−1B (3.6)
A,B and C are all N × N matrixes. A0A and B0B are symmetric and non-singular and C is a lower triangular matrix. The ˜0t−1 is the sum of the disturbance term and the jump component. The variance-covariance matrix is always positive definite, because of the quadratic form of the matrix.
Next, the jump components J1,t and J2,t will be discussed. During every time period
t, which is one trading day, many jumps can occur. For example, several news items that are reported during a trading day can result in multiple jumps in the indexes. This means that an index can have n number of jumps during a trading day t. The jump component can be divided into two parts, the size of the jumps and the number of jumps during one period. For r1,t and r2,t respectively
n1t P i=1 Y1t,i and n2t P j=1
Y2t,j are the sums of
series of random jump variables Y1i and Y2j respectively. The stochastic variables Y1t,i
and Y2t,i can be seen as the jump size. These jump sizes are assumed to have a normal
distribution with a constant mean and variance for each different index. The jump sizes are look like this:
Y1t,i∼ N (θ1, δ12) and Y2t,j∼ N (θ2, δ22) (3.7)
The jump component conditional on the number of jumps also follows a bivariate normal distribution with a expected mean of zero and conditional variance-covariance matrix ∆t:
(Jt|n1,t, n2,t) ∼ N
0, ∆t
The mean equation of the jump component has expected value zero. This is done by subtracting the expected values from the series of random jumps. The jump component is constructed as follows: Jt= n1t P i=1 Y1t,i− Et−1( n1t P i=1 Y1t,i) n2t P j=1 Y2t,j− Et−1( n2t P j=1 Y2t,j) (3.8)
We will assume that the error term and the jump component are independent of each other; E(1,t, J1,t) = 0, E(2,t, J2,t) = 0, E(1,t, J2,t) = 0 and E(2,t, J1,t) = 0.
In equation (3.8) there are two counting variables; n1t and n2t, which are discrete
variables managing the arrival of jumps. Both jumps counting variables are characterized by three independent Poisson variables; n∗1t, n∗2tand n∗3t. These stochastic variables count the number of jumps that occurred. The counting variable n∗1t generates independent jumps of the EURO STOXX 50 returns in time period t. The counting variable n∗2t generates independent jumps of the log changes in the VSTOXX in time period t. The third counting variable n∗3t generates jumps that occur in both indexes in time period t. The counting variables are Poisson distributed, which gives the following density function for all three separately:
P(n∗it = j|Φt−1) =
λji j!e
−λi (3.9)
The jump intensity is λi, which is the expected value and the variance of n∗it, because
of the Poisson distribution.
The individual counting variables of the expected number of jumps n1t and n2t are
the linear combination of the following jump counting variables above:
n1t = n∗1t+ n∗3t and n2t= n∗2t+ n∗3t (3.10)
By changing the variables and integrating out n∗3t, the following bivariate Poisson dis-tribution of n1t and n2t can be found:
P(n1t= i, n2t= j|Φt−1) = min(i,j) X k=0 e−(λ1+λ2+λ3) λ i−k 1 λ j−k 2 λk3 (i − k)!(j − k)!k! (3.11) The full elaboration of this can be seen in the Appendix A.1.
The expected number of jumps is equal to:
E(nit) = λi+ λ3, with i = 1, 2 (3.12)
The covariance of n1tand n2tis λ3, which is always a non-negative value. The correlation
of n1t and n2t is as follows:
ρ(n1t, n2t) =
λ3
p(λ1+ λ3)(λ2+ λ3)
, (see Appendix A.2 for the proof)
which is also always non-negative. This positive correlation between n1t and n2t means
that the model assumes that the number of jumps in the series is always positively related to each other. This can be removed, when using another distribution. But even when the correlation between the number of jumps is always positive, it is still possible that the means of the jump sizes are different. This implies that the number of jumps of one index influences the number of jumps of the other index positively, but that the jump has a positive effect on one of the series and a negative effect on the other. In the special case of λ3 = 0, there is no relation between the series, which makes it
then n1t ≤ n2t. This is also the case for n2t, when λ2 = 0. When both λ1 = λ2 = 0,
then n1t = n2t are the only possible jumps. This case will be further elaborated in the
variations section 3.2.
∆ij,tis the variance-covariance matrix of the jump component. This matrix is based
on the assumptions that the correlation between the jump sizes is constant between the different indexes and that there is no variation across time:
ρ(Y1t, Y2t) = ρ12 and ρ(Y1t, Y2s) = 0, with t 6= s (3.13)
This means that the variances of the jump components conditional on i and j are the following: V P iY1t|n1t = i, n2t= j = iδ12 VP jY2t|n1t= i, n2t = j = jδ22 The covariance will be defined as follows:
covP iY1t, P jY2t = ρ12 p ijδ1δ2
When merging this, the following variance-covariance matrix for the jump component can be defined: ∆ij,t= iδ12 ρ12 √ ijδ1δ2 ρ12 √ ijδ1δ2 jδ22 ! (3.14) The ∆ij,t is, like ˜Ht, a positive definite matrix.
The probability density function for Rt, given i jumps in the returns of the Euro
STOXX 50 and j jumps in the log changes of the VSTOXX consists constructed by a combination of the GARCH model and the CBP function, defined by:
f (Rt|n1t= i, n2t = j, Φt−1) = 1 2πN2 |Hij,t|− 1 2exp h
−u0ij,tHij,t−1uij,t
i
(3.15) where uij is the error term including the jump component Jij,t, that presents the effect
of i and j jumps: uij,t= Rt− µ − Jij,t= " r1,t− µ1− iθ1+ (λ1+ λ3)θ1 r2,t− µ2− jθ2+ (λ2+ λ3)θ2 # (3.16) Hij,t is the variance-covariance matrix of the returns given i jumps in the Euro
STOXX 50 log returns and j jumps in the log changes in the VSTOXX. We assume that the jump component is independent of the normal disturbance t. With this assumption,
Hij,t is divided into the two parts calculated above. The first part is the
variance-covariance matrix of the CCC model and the second part is the variance-variance-covariance matrix of the jump component: Hij,t= ˜Hij,t+ ∆ij. Hij,tis a sum of two positive definite
matrices, which makes the matrix itself also positive definite. The combination of the variance-covariance matrix of both the disturbance term and the jump component makes that the covariances are driven by both terms.
The jumps are unobserved, but with help of the Bayes’ rule we are able to identify the jumps in the series:
P(n1t = i, n2t= j|Φt) =
f (Rt|n1t= i, n2t= j, Φt−1)
P(Rt|Φt−1)
× P(n1t= i, n2t = j|Φt−1) (3.17)
From the probability density of 3.15 we further develop to the conditional density func-tion of Rt: P(Rt|Φt−1) = ∞ X i=0 ∞ X j=0 f (Rt|n1t = i, n2t= j, Φt−1)P(n1t= i, n2t= j|Φt−1) (3.18)
Because the function has no maximum value, a truncation point must be chosen. This truncation point is chosen for the summation in equation 3.18 and must be large enough to converge and stabilize the likelihood function and the parameter estimations. This truncation point indicates the maximum number of jumps that will occur during one period, which is one trading day.
From here on it is easy to make the log-likelihood functions which is the sum of the log conditional densities:
ln L =
N
X
t=1
ln P(Rt|Φt−1) (3.19)
This function will be used in the maximum likelihood estimation method to estimate the unknown parameters, indicated in the model.
3.2
Variations of the model
The model described above will be compared with the multivariate GARCH model with-out a jump component, so with the following parameters equal to zero: θ1, θ2, δ1, δ2, ρ12,
λ1, λ2, λ3, η1, η2, η3.
Next, it will also be compared with the model where there are only jump moments that happen at the same time in both the indexes. Possibly the indexes are in such a way correlated that there will only happen jumps in both indexes at the same time. The size of the jumps can differ in this case, but the jump happens at the same moment. When there are only jumps at both indexes that happen at the same time, the intensity looks as follows: n1t= n2t= n∗3t. The density function will look as in 3.9, with n∗it= n
∗ 3t
(i = 1, 2). It is very likely to have a λ3 that can change over time so that the likelihood
of jumps intensities are time-varying. The following structure is created to make the jump intensities time-varying:
λ3t = λ3+ γ121,t−1+ γ222,t−1+ η1λ3,t−1 (3.20)
This function gives additional jump dynamics to the model. This time-varying jump intensity can be seen as a form of cross- and self-excitation. The λ3,t depends on the
square of the 21,t−1 and 22,t−1. These squared residuals of period t − 1 are chosen to mimic the market conditions, which is an approximation of the last period volatility of respectively the returns of the Euro STOXX 50 and percentage changes in the VSTOXX.
Another dynamic added is the number of jumps at period t − 1, λ3,t−1, which makes the
expected number of jumps in period t be depending on the jump intensity of the period before, which is a form of self-excitation. It is also possible to change λ3,t−1 to n∗3,t−1.
In the case of the new time-varying λ3,t, the term uij changes to:
uij,t= Rt− µ − Jij,t= " r1,t− µ1− (i − λ3,t)θ1 r2,t− µ2− (i − λ3,t)θ2 # (3.21) Also the variance-covariance matrix ∆ij will look different:
∆ij,t= iδ2 1 iρ12δ1δ2 iρ12δ1δ2 iδ22 ! (3.22) Further this model can use the same formulas and follows the same assumptions as the model described in the Model Setup 3.1.
4. Data
The data that is used for this research are the daily Euro STOXX 50 index and VSTOXX. The data of the Euro STOXX 50 are obtained from Yahoo Finance, the data of the VSTOXX from the website of STOXX. It is daily data over a nine year period. This period starts on the first of January 2008 and ends the 31st of December 2017. This period is chosen because it is interesting to look at the indexes in times of financial crisis and recovery from that period. Based on the number of observations we need, the length of the period is chosen. Ng and Lam (2006) took a closer look at how sample size affects the GARCH model. They concluded that to estimate the parame-ters by maximum likelihood there must be at least 1000 observations in the sample. A trading year has around 252 days a year. The number of observations in our sample is 2546. The dataset is incomplete, since the Euro STOXX 50 index contains stocks from different European countries. This means that there are some trading holidays in certain countries in which there is no daily return visible of stocks on that day. We delete the days on which not both the Euro STOXX 50 and the VSTOXX are available, in order to make the dataset complete. This leads to a sample of 2501 observations.
First, we take a look at the graphs of the Euro STOXX 50 index prices. The first of the four graphs in the Figure 4.1 shows a fluctuating function of the value of the Euro STOXX 50 in time. The prices show time-varying trends. The variation is proportional to the price level. Because of this, it is preferred to take the natural logarithm (log) of the Euro STOXX 50. The second graph shows that the log of the Euro STOXX 50 is both decreasing and increasing in different periods of time. We are not interested in the price level of the Euro STOXX 50 index, but only in the changes in the prices. The percentage changes in the stock index are visible in graph III of Figure 4.1. The log returns of the Euro STOXX 50 shows clear volatility clustering with an expectation that is around zero and slow mean-reversion. In the fourth plot, we make a histogram of the log returns. We see a histogram steeper than the normal distribution, which indicates a positive excess kurtosis. For the VSTOXX, we also take the logarithm of the first differences, since in this way it is easier to compare the two indexes with each other.
For VSTOXX the same four plots are made, these are visible in the Appendix B.1. The outliers of VSTOXX are proportionally bigger than those of the Euro STOXX 50. The log of the VSTOXX shows clearly that the function has a decreasing trend.
The log-differences of the VSTOXX also fluctuates around zero. Nevertheless, both the outliers and the volatility of the log changes of the VSTOXX are significantly bigger than those of the Euro STOXX 50.
Figure 4.1: Euro STOXX 50 graphs
Next we take a closer look at the descriptive statistics of the logarithm of the first differences of the Euro STOXX 50 and the VSTOXX. Figure 4.2 presents the mean return of the Euro STOXX 50 and the mean changes of the VSTOXX over a period of 9 years, the values are slightly negative but close to zero. The volatility of the returns of the Euro STOXX 50 is significantly smaller than the one of VSTOXX. The lowest value of the Euro STOXX 50 happened on the 24th of June 2016, the day the United Kingdom voted to leave the European Union. The maximum values of the Euro STOXX 50 and VSTOXX occur at respectively the 10th and the 11th of October 2008. This is the beginning of the global financial crisis. The market is extremely volatile at this moment and because of this the percentage changes in the VSTOXX are big during these days. The returns in the Euro STOXX 50 are also very volatile at the beginning of October. The second biggest daily return of the Euro STOXX 50 is on the 10th of May 2010 and has the value of 9.8466. This is the day after the EU-ministers of finance together with the IMF agree upon a deal that makes 720 billion euro available for the EU countries
that are in big financial trouble. This way the collapse of the euro is prevented.
The skewness, a measure of symmetry, of the Euro STOXX 50 returns is slightly negative, which indicates that the distribution of the sample set is left-skewed. The mean of the returns is less than the median. The log changes in the VSTOXX are in contrast, somewhat right-skewed. The kurtosis of the returns of the Euro STOXX 50 (VSTOXX) is 8.3865 (6.1700), which is substantially bigger than the kurtosis of a normal distribution. The distributions are clearly fat-tailed, or so-called leptokurtic. The high kurtosis signals that the probability of obtaining extreme values is higher than in a normal distribution. Next to that the points are clustering. The Jarque-Bera (J-B) test tests if the sample data matches the normal distribution. The J-B test statistics are highly significant, which indicates that it is rejected with a p-value of < 0.01 that the distributions of the log changes in the Euro STOXX 50 and VSTOXX are normally distributed. The Ljung-Box statistic tests whether the series of observations over time are random and independent. The Ljung-Box statistic has for both sample sets 5 lags and is highly significant with a p-value < 0.01. This rejects that both the returns of the Euro STOXX 50 and VSTOXX show no serial autocorrelation. The Ljung-Box Q test for the squared returns also shows that the squared returns exhibit autocorrelation. These features indicate that a GARCH model can be used to analyze the data.
Figure 4.2: Descriptive statistics
Index STOXX Returns VSTOXX Changes Sample Mean -0.0085 -0.0165 Standard Deviation 1.5381 6.4242 Minimum Value -9.0110 -43.4716 Maximum Value 10.4377 32.7680 Range 19.4486 76.2396 Skewness -0.0502 0.4310 Kurtosis 8.3865 6.1700 Jarque-Bera*** 3023.3 1124.2 Ljung-Box*** 15.678 22.268 Ljung-Box2*** 605.95 114.49
Note: *** denotes significance at the 1% level. The Jarque-Bera statistic tests whether the data is normally distributed. The Ljung-Box Q test statistics with lag 5 is used to find serial correlation in the standardized residuals.The Ljung-Box Q test statistics with lag 5 is used to find serial correlation in the standardized residuals. The Ljung-Box2 Q test statistics with lag 5 is used to find serial correlation in the squared standardized residuals.
5. Results
In this chapter we look at outcomes of applying the model, we discussed in chapter 3, to our data. We look at the results and the significance of them. We do some tests and substantiate everything with tables and figures. The model and tests used in this research are programmed in R.
We start with checking if the VAR(p) model is the right model to use. The right number of lags in the model will be looked at with the Akaike Information Criterion (AIC). The PACF plots in the appendix B.3 shows some significant partial autocorre-lations, which indicates that some lags might have to be added to the model. The AIC asymptotically selects the model with the lowest mean squared error. Based on the cri-terion, we start by looking at the VAR model with 16 lags. Almost all of the coefficients of the lags of the VAR(16) model are not significant. That’s why we choose to look at Bayesian information criterion (BIC) value. The BIC selects the model with only one lag. Figure 5.1 shows the coefficients of the VAR(1) model of the Euro STOXX 50 and VSTOXX written in equation 3.3 and 3.4. For Euro STOXX 50 are the coefficient of the lagged Euro STOXX 50 returns, the coefficient of the lagged VSTOXX and value of the constant close to zero. If we look at the p-values we see that the coefficients of all three parameters are not significant. Looking at the returns of the VSTOXX, the coefficients of the parameters are again close to zero. Likewise, these values are insignificant. The ACF plots in the appendix B.2, also show that there is no significant sign for correlation between the returns and the lagged returns. That is why we choose to further work with the following formula for the returns:
Rt= µt+ t+ Jt (5.1)
with Rtthe returns as in equation 3.2, µt a 2 × 1 vector of the mean that is calculated
at time t, t the 2 × 1 random disturbance and Jt the 2 × 1 jump component.
After we reformulated the returns we go to the MGARCH models with jump com-ponents. Table 5.1 shows all the parameter estimates, confidence intervals and standard errors of the correlated jump intensity-CCC models. We will start by looking at the least complicated model, which is the model in which the number jumps only depends on the correlated jump intensity λ3. Remarkable is that the α1+ β1 ≈ 1 but not more than
Figure 5.1: VAR(1) model
(a) Euro STOXX 50
r1,t Coefficient P-value c1 -0.00774 0.801 β11 0.00305 0.921 β12 0.00952 0.195 (b) VSTOXX r2,t Coefficient P-value c2 -0.0198 0.878 β21 -0.0310 0.809 β22 -0.0214 0.486
that the variance is stationary. The GARCH effect at the VSTOXX is also strong and stationary. The sum of αi and βi, with i = 1, 2 must be less than one in order to stay
stationary, which is the case here. The variance-covariance matrix has to be positive definite. In order to have a positive definite variance-covariance matrix, the variance elements σ1,t and σ2,t must be positive, which means that ωi > 0, αi ≥ 0 and βi ≥ 0,
i = 1, 2. This is the case for the λ3-CCC models.
The effect of the lagged variance on the Euro STOXX 50 returns (0.8950) is bigger than the effect of the lagged estimated residuals (0.1050). This is also the case for the changes in the VSTOXX. All the coefficients of the GARCH part are very significant, except for ω1, which is not significant at all but close to zero. The diffusion correlation
coefficient ω12is very significant and has the value -0.7995. It shows that the correlation
between the Euro STOXX 50 and the VSTOXX is strongly negative. This corresponds with our expectations and the theory. When there are periods of high returns at the Euro STOXX 50, the VSTOXX changes will be low. This makes sense because the VSTOXX can be seen as the fear index of the Euro STOXX 50. When the returns are high, people get less afraid, volatility decreases and the VSTOXX changes decline. The other way around is that when the changes in the VSTOXX increase, this indicates that the market of the Euro STOXX 50 might be unstable and the returns of the Euro STOXX 50 decrease.
The jump part of the CCC model contains the correlated jump intensity coefficient λ3, which has the value 0.1758 and is significant. The average size of the Euro STOXX
50 jumps, θ1, is -0.0236, with only a significance level of 10 percent. The variance of
the size of the Euro STOXX 50 is significant and has a value of 0.8910. The size of the changes in the VSTOXX, θ2, is 0.1431 with a variance of 4.7100, both significant.
We expected the Euro STOXX 50 returns to be more sensitive to negative shocks than positive ones, as described in the Literature Review section 2.2, so we expected the mean of the jumps size of the Euro STOXX 50 to be negative. The ρ12has value −0.7590 and
is significant. Because of this negative relationship between the jumps, we expect the jump size of the VSTOXX to be positive. The VSTOXX has a bigger variance, because the VSTOXX values are more sensitive to changes. The implied volatility of the Euro STOXX 50 increases more than the changes in value in the index itself. From the values and significance of both the size and intensity of the jumps mentioned here, we can conclude that the returns certainly show jump behaviour.
Now we look at the model in which the correlated jump intensity has expanded. The model in which λ3 is time-varying. The MGARCH part of the time-varying correlated
jump intensity model does not differ that much from the not time-varying model. The value of the non-time-varying component λ3 of the jump intensity is significant and
slightly higher than the λ3 in the non-time-varying CCC model. The parameter for
the squared residuals of the Euro STOXX 50 is close to zero and is insignificant. The parameter for the squared residuals of the VSTOXX is, in contrast, very significant and has a value of 0.1000. This means that if the absolute value of the residuals in the past period is bigger, the jump intensity of the correlated jumps will be larger the next period. This makes sense because this means that when the volatility in the last period is higher than expected, there will be a bigger chance of having more correlated jumps in the following period. Agitation in the last period could be a sign of unexpected jumps in the next period. The increase of the amount of jointly jumps is driven by the VSTOXX. The parameter for the lagged correlated jump intensity has value 0.4000. This shows that when the number jumps in the past period is high then this has a positive influence on the number jumps in the following period.
The jump sizes of both of the indexes are smaller than the ones in the non-time-varying λ3-CCC model. The variances of the jump sizes are larger than the variances
of the non-time-varying model. The higher variance and jump intensity are connected to lower expected jump sizes.
We can compare the two models with each other with the Likelihood-Ratio test (lr test) and see which model fits better. Note that the log-likelihood values are negative since we minimized when doing MLE. We take the λ3-CCC model as the null model
against the λ3t-CCC model as the alternative, who have respectively the values -10641.49
and -12689.03. The time-varying correlated jump intensity-CCC model has three extra parameters and will be seen as the alternative model. The lr test statistic of no time-varying CCC model has value 4095.08. The critical value of the chi-square distribution of three degrees of freedom is used. This rejects the null hypothesis with a p-value of 0.01, which means that the not time-varying correlated jump intensity-CCC model is rejected. From this, we can conclude that adding time variation is very important for estimation of the model.
The correlated jump intensity model with only λ3is expanded to a Correlated
Bivari-ate Poisson CCC model. From now on the jumps cannot only occur at both the indexes but also independently at one of them, with λ1and λ2. Table 5.2 shows the parameters,
coefficients and significance levels of the CBP-CCC and the CBP-BEKK model. We will start by discussing the CBP-CCC-model. We find that also at this model there is a strong GARCH effect and that the conditional variance of the r1 and r2 persists.
CBP-CCC model is stationary and the variance-covariance matrix is positive definite. The diffusion correlation coefficient is very significant and is comparable with the value calculated at the correlated jump intensity-CCC models described above.
Table 5.1: correlated jump intensity-CCC models λ3-CCC λ3,t-CCC
model model
Parameter Coefficient Standard Parameter Coefficient Standard
error error GARCH GARCH ω1 0.0038 0.0095 ω1 0.0407 *** 0.0152 α1 0.1050 *** 0.0132 α1 0.1066 *** 0.0120 β1 0.8950 *** 0.0114 β1 0.8933 *** 0.0093 ω2 1.2983 ** 0.5605 ω2 3.0010 *** 0.9518 α2 0.0751 *** 0.0169 α2 0.0808 *** 0.0184 β2 0.8737 *** 0.0201 β2 0.8577 *** 0.0021 ω12 -0.7995 *** 0.0171 ω12 -0.7902 *** 0.0106 Jump Jump λ3 0.1758 *** 0.0164 λ3 0.2004 *** 0.0034 γ1 0.0327 0.0265 γ2 0.1000 *** 0.0510 η1 0.4000 ** 0.1563 θ1 -0.0236 * 0.0135 θ1 -0.0051 0.0048 δ1 0.8910 *** 0.0506 δ1 2.4801 *** 0.2056 θ2 0.1431 ** 0.0654 θ2 0.0852 0.0538 δ2 4.7100 *** 0.1781 δ2 12.620 *** 0.9558 ρ12 -0.7590 *** 0.0218 ρ12 -0.7523 *** 0.0295 Log-likelihood -10641.49 Log-likelihood -12689.03
The expected intensities of the jumps in the CCC-model, which are λ1, λ2 and
λ3, are all significant. The three coefficients of the jumps intensities are non-negative,
which confirms with the condition, there cannot be a negative amount of jumps. λ3 is
notably bigger than the other two jump intensities, 0.2216 against 0.0473 and 0.0230 of λ1 and λ2 respectively. This means that there are mostly jumps that occur at both
returns jointly. The jump intensity of Euro STOXX 50 (VSTOXXX) is λ1+ λ3 = 0.2689
(λ2+ λ3 = 0.2446). The ρ(n1t, n2t) is the correlation between the jumps n1t and n2t,
this parameter is per definition positive, but the high value shows the strong positive relationship between the number jumps of the Euro STOXX 50 and the VSTOXX.
The average size of the jump of the returns of the Euro STOXX 50 is negative and insignificant. On the other hand the jump size of the changes in VSTOXX is very significant, with a big positive value. The variance of the jump size of VSTOXX, with a value of 11.9896, is enormous. This can explain that there will be big negative shocks in the returns of the VSTOXX as well as big positive shocks. The bigger absolute value of the VSTOXX jump size coefficient compared to the Euro STOXX 50 jump size coefficient, shows that the impact of shocks on the VSTOXX changes is bigger than the impact of shocks on the Euro STOXX 50. The high variance of the jump size of the VSTOXX indicates that changes in the VSTOXX will be likely to cause jump risk.
The parameter ρ12is the jump correlation coefficient. This parameter has a value of
-0.899. This shows a negative interaction between the jump sizes of both returns. The correlation coefficient has around the same value as the diffusion correlation coefficient and the jump correlation coefficients as the other model. The estimated values are in line with the theory and the MGARCH part of the three models described lookalike and show no important differences. Because the jump intensities of both λ1 and λ2
are significant, we can conclude that the returns show bivariate jump behaviour. The CBP part of the model is indispensable for estimating daily returns behaviour. The conditional variance of the CCC-model is not altered by adding the independent jump components to the model. The jump dynamics will be enriched by this expansion of the model.
The CBP-CCC model is compared with the time-varying correlated jump intensity-CCC model. The lr statistic tests the goodness of fit. The null model is the time-varying model, the CBP-CCC model is the alternative one. The values of them are -12698.03 and -12700.34 respectively. Again we use the lr-test from which we obtain the value 4.62. The chi-square degrees of freedom is one. With a p-value of 0.05 we can reject that the time-varying model has a better fit.
The CBP-CCC model is compared with the CBP-BEKK model. The CBP-CCC model is very strict, because of the constant correlation part of the MGARCH model. The CBP-BEKK model adds time variation in the correlations and it allows for spillover effects. The elements of the BEKK part of the model are mostly very significant. If we look at the expected jump intensities of the CBP-BEKK model, we see that the jump intensity of only Euro STOXX 50 is around two times less than the jump intensity of the
Euro STOXX 50 at the CBP-CCC model. The jump intensity of only the VSTOXX is close to zero and is insignificant. The jump intensity for both indexes (λ3) is
unexpect-edly smaller than the λ3 in all the three models above. We see that that the CBP-CCC
model has in total around 1.7 times more jumps than the CBP-BEKK model. The total amount of jumps intensities of the CBP-BEKK model is comparable with the jump in-tensity in the λ3-CCC model. Although the jumps intensities at the CCC-BEKK model
are smaller, the sizes of the jumps in the CCC-BEKK model are in absolute value bigger. The Ljung-Box test statistics are robust for heteroskedasticity and check for serial correlation. The Euro STOXX 50 standardized residuals and squared standardized resid-uals show no serial correlation. The squared standardized residresid-uals of the VSTOXX show also no serial correlation, which means that there is no remaining volatility clustering. Nevertheless, the normal standardized residuals of the VSTOXX are highly correlated, which indicates that the model might be miss specified. This miss specification can be caused by the use of equation 5.1, instead of a VAR(1) or VAR(16) model.
Summarized, we find no significant correlation between the returns of the Euro STOXX 50 and the VSTOXX and their lags. Therefore we chose to not work with a VAR(p) model. Nevertheless the Ljung-Box test indicates that the VSTOXX residuals do contain serial correlation. If we focus on the MGARCH model, we see that we need to add time-variation to the model. We find that the changes in the VSTOXX increase the jump intensity. The jointly jumps are also driven by the jumps in the past. Further we see that the volatilities of the changes in Euro STOXX 50 and the VSTOXX strongly negatively influence each other. This is also the case for the correlation between the jump sizes. When the VSTOXX has a highly positive jump, the returns of the Euro STOXX 50 will be influenced by a negative shock. Nonetheless is the jump intensity relationship highly positive, indicating that the amount of jumps of the indexes drives up each other and itself. We can conclude that adding jumps and making the model time-varying are necessary expansions to the model to explain the relationship between the indexes well.
Table 5.2: CBP-GARCH models
CBP-CCC CBP-BEKK
model model
Parameter Coefficient Standard Parameter Coefficient Standard
error error GARCH GARCH ω1 0.0103 ** 0.0040 c1 0.5624 *** 0.0916 α1 0.0629 *** 0.0001 c12 0.2413 0.2582 β1 0.9351 *** 0.0747 c2 1.0497 *** 0.1064 ω2 5.0353 *** 0.0026 a1 -0.6255 *** 0.0333 α2 0.0613 0.8819 a12 0.7366 *** 0.1210 β2 0.8136 *** 0.0242 a21 -0.0239 *** 0.0072 ω12 -0.7824 *** 0.0077 a2 -0.2654 *** 0.0301 b1 0.5502 *** 0.0616 b12 0.7470 *** 0.1532 b21 -0.0687 *** 0.0094 b2 1.0385 *** 0.0235 Jump Jump λ1 0.0473 *** 0.0143 λ1 0.0264 ** 0.0105 λ2 0.0230 ** 0.0107 λ2 0.0080 0.0063 λ3 0.2216 *** 0.0233 λ3 0.1395 *** 0.0164 θ1 -0.1112 0.0739 θ1 -0.2806 ** 0.1311 δ1 2.3648 *** 0.1476 δ1 3.1220 *** 0.2047 θ2 1.8483 *** 0.4187 θ2 2.9654 *** 0.6297 δ2 11.9896 *** 0.7042 δ2 14.0779 *** 0.8940 ρ12 -0.8990 *** 0.0203 ρ12 -0.8711 *** 0.0390 Log-likelihood -12700.34 Log-likelihood -12859.39 Q1(5) 3.3578 (0.6450) Q1(5) 5.3529 (0.3743) Q2 1(5) 5.2733 (0.3834) Q21(5) 0.6213 (0.9870) Q2(5) 18.0342 *** (0.0029) Q2(5) 17.0470 *** (0.0044) Q22(5) 2.9441 (0.7086) Q22(5) 0.7041 (0.9827) Note: ***, **, * denotes significance at the 1% level, 5% and 10% respectively. Qi is the
Ljung-Box portmanteau test if the standardized residuals are independently distributed . Q2
i is the Ljung-Box test for serial correlation in the squared standardized residuals.
6. Discussion
In this discussion, we look back at the model and results that have been obtained and argue if these are correct or if they could be improved. We look at the model structure, the maximization process and other literature.
We start with the variation part of the model setup. In the section 3.2 we looked at the variation in which the jump intensities only happened at the same time. We made the correlated jump intensity parameter, λ3, dynamic in time in equation 3.20.
Following studies can expand the jump intensity part of the model even more. We can change the original λ1, λ2 and λ3 and let all these jump intensities be dynamic in time.
It is very likely to have jump intensities that can change over time so that the likelihood of jumps changes over time. After all the market conditions will influence the correlation between the counting variables. An example is the following structure is created to make the jump intensities time-varying:
λ1t= λ1+ η21r21,t−1+ γ1λ1,t−1 (6.1)
λ2t= λ2+ η22r22,t−1+ γ2λ2,t−1 (6.2)
λ3t= λ3+ η23r21,t−1+ η42r2,t−12 + γ3λ3,t−1 (6.3)
The square of these r1,t−1and r2,t−1 are chosen to mimic the market conditions, which
is an approximation of the last period volatility of respectively the returns of the Euro STOXX 50 and log changes in the VSTOXX. This adds additional jump dynamics to the model. The coefficient in front of the λi,t−1, with i = 1, 2, 3 shows the way the jump
intensity in this period is depending on the jump intensity of the period before. This part of the formula can also be changed to ni,t. In that case, the jump intensity depends
on the number of jumps that happened the period before.
Another variation to the model is to let the size of the jumps vary asymmetrically in time. In the model used in this report, the size of the jumps are normally distributed with an expected mean that is time independent, see equation 3.7. The expected mean is θi, with i = 1, 2. An extension could be the following:
θ1t θ2t ! = θ1+ θ − 1r1,t−1Ir1,t−1+ θ + 1r1,t−1(1 − Ir1,t−1) θ2+ θ−2r2,t−1Ir2,t−1+ θ + 2r2,t−1(1 − Ir2,t−1) ! (6.4)
with Ir1,t−1 = 1 if r1,t−1> 0 0 if r1,t−1< 0
and this is also the case for r2,t−1.
This makes the returns of the period before influence the size of the jumps at period t. We choose to not use all the model extensions above since they are very time consuming when running the script. Also the reliability of the parameter values might be considered, since the more dimensions the likelihood function gets, the more likely it will find a local instead of the global maximum.
Frankel et al. (2008) presented a new model that fits their options dataset better than the CBP-GARCH model used in this report. In their model, the volatility of the returns is also divided into two components: a long and a short run component. The long run part of the model is fully persistent. The short run is not and has an expected mean of zero. Their model scores better on the RMSE fit when making use of the MLE parameters. Nevertheless, the CBP-GARCH model scores better on the log-likelihood criterion. They claim that CBP-GARCH model only performs better in periods where volatility is low. In following studies these models can be compared for the data that is used here and see if this is also the case in these circumstances.
In this section, it became clear that there are a lot of variations and extensions possible within the used CBP-GARCH model. Also some researchers found comparable models that seemed to work better for some index datasets.
7. Conclusion
In the conclusion a brief summary of the results found in this research is given. The aim of this research is to look at the dynamic relationship between index returns and the expected implied volatility of the index and to see what the role of jump dynamics in this relationship is. The data used for this research are the daily returns of the Euro STOXX 50 and VSTOXX in the period of 2008 up till 2017.
First, we found no significant correlation between the changes in the Euro STOXX 50, the VSTOXX and their lags. Next, we started looking at different models used to find the best fitting MGARCH model. We started with the non-time-varying correlated jump intensity CCC model and soon we found that extending this model gives some advantages. Expanding the correlated jump intensity model with individual jump inten-sities is also increasing the goodness of fit of the model by allowing for both individuals as well as correlated jumps at both of the indexes. When expanding the non-time-varying correlated jump intensity CCC model to a time-varying correlated jump intensity CCC model or to a CBP-CCC model, the MGARCH part of the model stays more or less equal to the MGARCH part of the not extended model. This indicates that the expansions are a good supplement to the model that otherwise would not have been found.
Both the CCC- and BEKK-model are capable of modelling the conditional variance-covariance of the Euro STOXX 50 and VSTOXX. The MGARCH model that must be used for the optimal model is debatable. A model like the BEKK model is easier to calculate. This makes the parameter estimation easier and reliable. Nevertheless, a more expanded model also has its advantages, since the assumption of constant correlation is very restrictive. The BEKK model shows dynamics in variances and covariances and might be a better model to use, even though the parameter estimations is difficult.
The time-varying correlated jump intensity CCC model, CBP-CCC model and the CBP-BEKK model are all capable of improving the goodness of fit. Even though the pa-rameter estimations in the different models vary, they all substantiate economic theories without contradicting each other. The time-varying CCC model shows that the number jumps gets influenced by what happened the period before. If the squared residuals of the changes in the VSTOXX in the last period are big, it indicates that the returns in the same period differ a lot from the expected returns in that period. These VSTOXX squared residuals of last period have a positive influence on the jump intensity of the
next period. This means that the joint jump intensity gets driven up by the VSTOXX changes. Furthermore, it seems that jumps occur in a way comparable to volatility clus-tering. If the number of jumps in the period before is high, then it is likely to have more jumps in the following period, a sort of jump clustering. In this restless climate the chance for jumps increases. Nonetheless this does not indicate that the jumps sizes at all these jumps are the same. The opposite can be true; in a period of high jump intensity it is very likely that one jump has positive peak when the following one has a negative one. The strongly positive correlation coefficient of the number of jumps complements this conclusion.
The high level of volatility clustering in the financial return data is clearly visible in the CCC part of the models. We found a diffusion correlation coefficient in the models that is strongly negative. This corresponds with the theory; if the Euro STOXX 50 returns decrease, the VSTOXX changes increase. The CCC model and the CBP-BEKK model both have correlated and individual jumps. The number of correlated jumps is higher than the number individual jumps, which in line with theory since the indexes are highly correlated. Note that this does not indicate that the jumps that occur move the same way. The jump correlation coefficient at the models is about the same value as the diffusion correlation coefficient. This means that the relationship between the size of the jumps is also strongly negative. The size of the VSTOXX jumps is in absolute value bigger than the size of the jumps of the Euro STOXX 50. This is in line with the theory since changes of the implied volatility of an index are bigger than the changes of the index itself. The jumps size of the Euro STOXX 50 is negative. News flashes mostly have a negative jump impact on the index, since people are risk-averse, negative news has more impact on their trading activities than positive news. We found that the variance of the changes in the VSTOXX is bigger than the variance of the Euro STOXX 50. This means that the changes in the VSTOXX lead to more risk of jumps than the Euro STOXX 50 return changes.
The bivariate jump component in all the models is significant and a necessary ex-pansion of the MGARCH models to completely capture the indexes and the dynamic relationship between them. Since the time-varying part of the model is also indispensable for looking at the dynamic relationship between the Euro STOXX 50 and the VSTOXX returns, the best expansion in the future will be to make a model that captures both the time-variation in the jump intensities and the individual and correlated jumps.
