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Modeling share returns -an empirical study on the Variance Gamma model

A. Rathgebera, J. Stadlera, S. Stocklb,∗

aFIM Research Center Finance & Information Management, Department of Information Systems Engineering & FinancialManagement, University of Augsburg, Universitatsstrasse 12, 86159 Augsburg, Germany

bDepartment of Economics, Chair for Corporate Finance (Visiting Professor), University of Constance, Box 141,78457 Constance, Germany

Abstract

Due to the fact that there has been only little research on some essential issues of the Variance Gamma(VG) process, we have recognized a gap in literature as to the performance of the various estimationmethods for modeling empirical share returns. While some papers present only few estimated parame-ters for a very small, selected empirical database, Finlay and Seneta (2008) compare most of the possibleestimation methods using simulated data. In contrast to Finlay and Seneta (2008) we utilize a broad,daily, and empirical data set consisting of the stocks of each company listed on the DOW JONES overthe period from 1991 to 2011. We also apply a regime switching model in order to identify normal andturbulent times within our data set and fit the VG process to the data in the respective period. We find outthat the VG process parameters vary over time, and in accordance with the regime switching model, werecognize significantly increasing fitting rates which are due to the chosen periods.

Keywords: Variance Gamma Model; Parameter Estimation Methods; Regime Switching Model

JEL classification: G15

1. Introduction

Distribution assumptions and jump discontinuities play an important role in modeling share returns.Jumps can generate fat tails, and in doing so they can influence the excess skewness and kurtosis. Fama(1965) studies the distribution of daily returns on the Dow Jones Industrial Average Index (DOW JONES)over the period from 1957 to 1972. In his analysis, he includes the stocks of the thirty companies listed onthe DOW JONES, rejects the normality assumption and suggests that non-normal distribution assump-tions would probably fit more accurately. However, he is not able to find a Paretian stable distribution for

∗Corresponding author: Tel: +49 (0)7531 88-3425 Fax: +49 (0)7531 88-3440Email address: [email protected] (S. Stockl)

his non-normal distribution assumptions. According to Mandelbrot (1963) this means that the distribu-tions have means but their variances are infinite and therefore, many commonly used statistical methodscannot be applied any longer. Based on these results Officer (1972) or Hsu et al. (1974) - amongst others -demand better modeling approaches for non-stationary share returns as their attempts to estimate Paretianstable distributions (at least one finite mean and one variance) failed. Praetz (1972) was the first tryingto improve share return modeling by means of an inverse gamma distribution which implies a rescaledStudent t-distribution for returns. His results are superior to Fama’s (1965) and are regarded as the basisfor all future stochastic variance models.

Since Madan and Seneta (1987) took the results of Praetz (1972) and published the first symmetricversion of the Variance Gamma process (VG process) with mean zero, there has been some progressin developing alternatives to the common Brownian Motion model for stock market returns, which hadbeen developed by Osborne (1959). The Brownian Motion is an essential part of the Black-Scholes op-tion pricing model devised by Black and Scholes (1973) although it does not include the third and fourthmoment. This is in contrast to the VG process which contains all four moments. By integrating the jumpdiffusion model into the traditional Black-Scholes option pricing model, Merton (1976) was the first tosucceed in taking the market’s reactions on incoming information into account. Cox and Ross (1976),Jones (1984), Hull and White (1987), Heston (1993a) and Heston (1993b) - amongst others - followthe trend of modeling market fluctuations with Levy processes or other stochastic processes with jumps.With regard to that they build on a diffusion concept which has a martingale component with samplepaths being continuous functions of calendar time. Madan and Seneta (1990) extend the Black-Scholesmodel by applying the VG process within the pricing framework to the Variance Gamma option pricingmodel. Madan et al. (1998) conclude that Variance Gamma option pricing reduces the pricing bias - incontrast to the Black-Scholes model - as the VG process covers the excess kurtosis, which is a result ofjumps. Daal and Madan (2005) use this new idea for an empirical examination of the Variance Gammaoption pricing model, the traditional Black-Scholes model and Merton’s (1976) jump diffusion model forforeign currency options. They reaffirm Madan and Seneta’s (1990) findings that the Variance Gammaoption pricing model performs better than the others do. In contrast to the two-parametric BrownianMotion the VG process mentioned above is a four-parametric stochastic process. Therefore, these twomethods used for stock market modeling differ widely as the VG process captures the skewness andkurtosis in addition to the mean and standard deviation. The VG process considers both, the symmetricincrease in the left and right tail probabilities of the return distribution (kurtosis) and the asymmetry ofthe left and right tails of the return density (skewness). These properties allow for a more accurate rep-resentation of stock returns. The VG process parameters can be obtained by the application of severalmethods, such as the simplified method of moments, the method of moments, the maximum likelihoodestimation, the empirical characteristic function, the Bayesian inference and Markov chain Monte Carlomethod, or the minimum χ2 method. For an overview see Madan and Seneta (1987), Madan and Seneta(1990), Seneta (2004) and particularly Finlay and Seneta (2008).

However, there has been only little research on some essential issues of the VG process, so far. Wehave recognized a gap in literature as to the performance of the various estimation methods for model-ing empirical share returns. While some papers present only few estimated parameters for a very small,selected empirical database, Finlay and Seneta (2008) compare most of the possible estimation methodsusing simulated data. In contrast to Finlay and Seneta (2008), we utilize a broad, daily, and empiricaldata set consisting of the stocks of each of the companies listed on the DOW JONES over the period from1991 to 2011. Additionally, the calibration quality as well as the parameters’ range of the VG process are

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dependent on time. This means that parameters vary in different market phases. As market participantsare exposed to varying situations, the selection of the correctly fitted model is an essential element oftheir work. This leads us to apply a regime switching model in order to identify normal and turbulentregimes within our data set and to fit the VG process to the data in the respective period. This approachhas two major advantages. First, it results in a more accurate parameter estimation, which avoids over-or underestimation of the real VG process. Second, the fitting rate - the fact that the returns follow a VGprocess - increases significantly. Thus, the use of the regime switching model adds new knowledge to theframework of the VG process.

The remainder of this paper is structured as follows. The next section provides an overview of the theoret-ical background of the VG process itself, the several estimation methods for the VG process parameters,and our hypotheses. Section 3 introduces the research design including the data set and the methodologyof differentiating between normal and turbulent times in financial markets. Section 4 presents the resultsof the VG process parameter estimation. Subsequent to the discussion of the results in section 5, section6 concludes the paper.

2. Theoretical Background

This section presents the framework for modeling risky assets by means of a VG process and providesa brief overview of the estimation methods for the VG process parameters. We define the price of a riskyasset St at point in time t (t ≥ 0) with the following traditional model

St = S0ert , (1)

where rt = ln StS0is the compounded log return from t = 0 to t. rt follows the stochastic process

rt = ct + θTt + σW (Tt), where c, θ and σ(> 0) are real constants as defined by Seneta (2004). W (·)represents the traditional Brownian Motion. Therefore, this approach is also known as the VG processas a modified Brownian Motion. Besides, Schoutens (2003) and Cont and Tankov (2004) describe theVG process as the difference between two independent gamma processes. Luciano and Schoutens (2006)take these results and define a Levy Triplet by means of the difference between two gamma processes.Seneta and Tjetjep (2006) model the VG process as a normal-variance-mean-mixture-model. Seneta(2004) models the market activity time (Tt)t≥0 as a positive, monotonically non-decreasing randomprocess with stationary increments τt = Tt − Tt−1 (t ≥ 1) and T0 = 0 (almost surely). It is assumedthat the expected duration of an increment is E(τt) = 1 ∀ t ≥ 0. This simplification allows normalizingthe expected economically relevant measure to 1. Seneta (2004), Finlay and Seneta (2008) - amongstothers - use the corresponding increments Xt of St instead of the compounded return rt for fitting theVG process. This means that Xt = c+ θ(Tt − Tt−1) + σ(W (Tt)−W (Tt−1)) and from the distributionproperties of W (·), σ(W (Tt) −W (Tt−1))

D= σ

√Tt − Tt−1W (1), the relevant formula for Xt can be

derived

Xt = c+ θτt + σ√τtW (1). (2)

3

Both, Seneta (2004) and Finlay and Seneta (2006) model τt with a gamma distribution γ(

1ν ,

1ν

)1 with

ν > 0 and a gamma function Γ(

1ν

)embedded in the gamma probability density function (PDF)

fτ (w) =

1

ν1ν

w1ν

−1e−wν

Γ( 1ν )

, w, ν > 0

0, else,(3)

which fulfills the requirements that E(τt) = 1 and V ar(τt) = ν. Before Madan and Seneta (1990)introduced the VG process in 1990, there had not existed a closed form for the PDF of the VG process.They only identified a closed form for the characteristic function. By means of a Bessel function K it ispossible to close this research gap (see Cont and Tankov (2004)) and to define fX(x) as the PDF of theVG process

fX(x) =2

σ√

2πν1ν Γ(

1ν

)eθ x−cσ2

| x− c |√2σ2

ν + θ2

1ν−

12

K 1ν−

12

| x− c |√

2σ2

ν + θ2

σ2

, (4)

with θ, σ, ν and c representing the VG process parameters, according to Seneta (2004). As suggested byboth, Madan and Seneta (1990) and Cont and Tankov (2004), we apply the Bessel function of the secondkind. Nevertheless, there is another concept developed by Seneta (2004), where K is modeled as theBessel function of the third kind. The characteristic function φX(x) = E(eiux) for the VG process canbe generated by means of the PDF (see formula 4) and Madan et al. (1998) define it as

φX(u) = eicu(

1− iθνu+σ2νu2

2

)− 1ν

, (5)

with −∞ < u < ∞ and i =√−1. On the basis of these properties the four moments of the VG

process can be calculated. By using the moment generating function mτ (x) = E(eux) for the timeincrements τt, mτ (u) = (1 − νu)−

1ν , the i-th moment E[(τt − 0)i], with i = 1, . . . , 4, about zero,

the i-th central moment Mi = E[(τt − 1)i], with i = 1, . . . , 4 , we get E[(Xt − E(Xt))2] = c + θ,

E[(Xt − E(Xt))2] = σ2 + θ2M2, E[(Xt − E(Xt))

3] = 3θσ2M2 + θ3 and E[(Xt − E(Xt))4] =

3σ4(1 +M2) + 6σ2θ2(M2 +M3) + θM4. Finally, we present the four moments Mi, with i = 1, . . . , 4,of the VG process mean (µ), variance (σ2), skewness (β) and kurtosis (κ) depending on the parameterset η = (σ, θ, ν, c)

M1 : = µ = c+ θ, (6)

M2 : = σ2 = σ2 + θ2ν, (7)

M3 : = β =2θ3ν2 + 3σ2θν

(σ2 + θ2ν)32

and (8)

M4 : = κ = 3 +3σ4ν + 12σ2θ2ν2 + 6θ4ν2

(σ2 + θ2ν)2. (9)

1In general, a γ(b, p) distribution has f(x) = bp

Γ(p)xp−1e−bx ∀ b, p, x > 0 as PDF.

4

Seneta and Tjetjep (2006) calculate some upper and lower bounds for the skewness and the kurtosis. Allthe bounds depend on the subordinated VG process with its parameter ν. The skewness is embedded inthe range −3

√ν < β < 3

√ν and the kurtosis is embedded in the range 3ν + 3 < κ < 6ν + 3. For a

more detailed derivation of the respective formulas see, for example, Madan and Seneta (1987), Madanand Seneta (1990) and Seneta (2004).

For fitting the PDF of the VG process and the normalized moments mentioned above, the related liter-ature suggests several approaches. In the subsequent paragraphs we provide a short overview of the sixdifferent VG process estimation methods.

Simplified method of moments

The simplified method of moments (SMOM) approach is the simplest way of determining the param-eter set η of a VG process. Seneta (2004) and Finlay and Seneta (2006) assume that we face a symmetriccase of the empirical distribution of the log returns and we can approximate θ ≈ 0. This restrictionimplies θ2 = θ3 = θ4 = 0 and by means of the empirical moments ME

i , with i = 1, . . . , 4, we get thefollowing four estimators for

σ∗ =√ME

2 , ν∗ =ME

4

3− 1, θ∗ =

ME3 σ∗

3ν∗and c∗ = ME

1 − θ∗. (10)

The results of the SMOM will be used as the starting parameters for the following optimization methods.

Method of Moments

The method of moments (MOM) is a numerical algorithm trying to identify a parameter set for themoments which is very close to the empirical one. As Finlay and Seneta (2008) suggest, we use theleast-square method and minimize the relative deviation between the i-th sample momentME

i calculatedfrom the empirical data set and the i-th moment Mi, with i = 1, . . . , 4 (see formulas 6 - 9), derived fromthe parameter set. From this framework we get the optimization model

(σ∗, θ∗, ν∗, c∗)MOM = argminσ,θ,ν,c

4∑i=1

(MEi − Mi(σ, θ, ν, c)

MEi

)2

, (11)

with σ∗ > 0 and ν∗ > 0. Seneta and Tjetjep (2006) modify the optimization to

(σ∗, θ∗, ν∗)MOM = argminσ,θ,ν

4∑i=2

(MEi − Mi(σ, θ, ν)

MEi

)2

(12)

and c∗ = ME1 − θ∗, as the parameter c, only has an effect on M1.

Maximum Likelihood Estimation

The maximum likelihood estimation (MLE) is a method for evaluating the density of an N -dimensional

5

random process. Based on the ideas of Efron (1982) and Aldrich (1997), as suggested by Seneta(2004), we use the MLE for fitting the VG process to the parameter set η = (σ, θ, ν, c) at Xi, withX = X1, . . . , XN representing the random log-return process. Finally, we optimize

fX(x | σ∗, θ∗, ν∗, c∗)MLE = argmaxσ,θ,ν,c

N∏i=1

fX(x | σ, θ, ν, c), (13)

where∏Ni=1 fX(x | σ, θ, ν, c) is defined as the so-called likelihood function L(η). As we face i.i.d

samples of X , Aldrich (1997) concludes that we can use the practical log-likelihood function LL(η) =ln(L(η)) for the maximization

fX(x | σ∗, θ∗, ν∗, c∗)MLE = argmaxσ,θ,ν,c

N∑i=1

ln (fX(x | σ, θ, ν, c)) . (14)

With respect to the PDF of the VG process (see formula 4) and the parameter ν used for defining theBessel function of the second kind, we have to use a numerical approach as the partial derivation is notknown. Due to the fact that the surface of the vicinity of the moments is very likely to be flat, previousoptimization approaches are unsuccessful. Therefore Seneta (2004) modifies the log-likelihood approachas follows. First, he suggests that all the observations of the random VG process are adjusted by the firstempirical moment ME

1 in order to get a VG process of the three parameters σ, θ and ν with the PDF

fX(z) =2

σ√

2πν1ν Γ( 1

ν )eθ

z−θσ2

| z − θ |√2σ2

ν + θ2

1ν−

12

K 1ν−

12

| z − θ |√

2σ2

ν + θ2

σ2

, (15)

with z = Xt−ME1 . After this optimization, c∗ has to fulfill the requirement that c∗+ θ∗ = 1

N

∑Ni=1Xi.

Empirical characteristic function

The empirical characteristic function (ECF) method tries to match the characteristic function derivedfrom the VG process model with the empirical characteristic function obtained from empirical data.Seneta (2004) states that this approach has some advantages compared to the MLE as this method cansometimes cause maximization problems as mentioned in the previous paragraph. In accordance withYu (2004), we first calculate an estimator for the empirical characteristic function ΦECF (ω) using the Nobserved log-returns X = (X1, . . . , XN )

ΦECF (ω) =1

N

N∑j=1

eiωXj , (16)

where i =√−1 and ω are the evaluation points. Then we take the characteristic function of the VG

process ΦX,η(ω) (for a detailed overview see function 5) and set up the optimization problem

η∗ = argminη

∫ +∞

−∞| ΦX,η(ω)− ΦECF (ω) |2 e−ω

2

dω (17)

6

for estimating the parameter set η∗, this is in accordance with Yu (2004).

Bayesian inference and Markov chain Monte Carlo

Finlay and Seneta (2008) suggest the so-called Bayesian inference and Markov chain Monte Carlo (BI)method. Roberts (1965) already recommends the approach to look at the posterior distribution of theparameters of a data distribution which is usually the major objective of a Bayesian statistical analysis.This means that the posterior probability is the probability of the parameter set η which is based on theevidence of X : fX(η | x), where X = (X1, . . . , XN ) represents the N observed log returns. Further-more, the likelihood function L(η) describes the probability of the evidence given fX(x | η). Finally,assuming a prior probability fX(η), we can use the Bayes theorem and calculate the posterior probabilityfX(η | x) as

fX(η | x) =L(η)fX(η)∫L(η)fX(η)dη

. (18)

On the basis of function 18 the mean of the posterior probability η∗ is defined as

η∗ = µ(fX(η | x)). (19)

As already mentioned in the context of equations 13 and 14, there are some difficulties in handlingthe Bessel function. In this case it is hardly possible to find a solution to the integral

∫L(η)fX(η)dη.

Roberts and Rosenthal (2004) propose a Markov chain Monte Carlo (MCMC) simulation as a possi-ble solution, because Markov’s solution is similar to the original problem. In accordance with Finlayand Seneta (2008) we use WinBUGS for the MCMC simulation as it was designed for the Bayesian es-timation of parameters. We run the MCMC simulation as described by Finlay and Seneta (2008) in detail.

χ2 method

Another estimation approach is the minimum χ2 (χ2) method. Berkson (1980) defines the chi-squarefunction as a function of the observed frequencies and their expectations or estimates of their expecta-tions which are asymptotically distributed in the tabular chi-square distribution. Finlay and Seneta (2008)confirm that this method is a very good approximation of the empirical PDF by means of the estimatedPDF. While Berkson (1980) mentions five different versions of the χ2 method we use the approach pro-posed by Finlay and Seneta (2008).

The basis of this method are N observed log returns X = (X1, . . . , XN ) and the determination of Iintervals with a vector B := (Bi)i=0,...,I as right borders of the intervals. The number of the expectedobservations Oi(η) in any interval is dependent on the parameter set η and can be approximated with

Oi(η) = N

∫ Bi

Bi−1

fX,η(x)dx, (20)

where fX,η(x) is the density function of the VG process (see formula 4). Finlay and Seneta (2008)suggest that the log returns are divided into 1% sample quantile bands. This means that Oi is a constantover all intervals and has a value of N

100 . We set the integration bounds to B0 = −10 and BI =

7

10, because a numerical integration to infinity is not possible due to the properties of fX,η(x). Theoptimization problem

η∗ = argminη

M∑i=1

(Oi − Oi(η)

)2

Oi(η)(21)

minimizes the relative difference between the observed and expected numbers of log returns in the deter-mined intervals and in this way identifies the optimal parameter set η∗.

For testing the quality of the parameter estimation methods we use the Kolomogorov-Smirnov (KS)test as well as the χ2 test. An overview of these two distribution tests can be found in appendix A1.

Having described the theoretical background, we can develop some hypotheses. All hypotheses de-rive from the characteristic behavior of the stock markets and the estimation methods of the VG processparameters. So far, Finlay and Seneta (2008) have been the only authors who conducted a broad com-parison of the estimation methods of the VG process. Their findings are mainly based on a simulatedVG process data set and could be summarized, that in most instances, the MLE method is the superiormethod and that the χ2 estimation is - on average - the next best method. In accordance with Madanand Seneta (1987), Madan and Seneta (1990) and Daal and Madan (2005), who use only one or twoestimation methods, there is no evidence for a large empirical data set. For the purpose of closing thisgap, we assimilate the properties of the estimation methods and the typical features of empirical assetreturns. The SMOM can be obtained from the empirical data set more or less directly. The empirical logreturns often include outliers due to extreme events at the stock markets. These events affect the kurtosisdirectly and thus, produce some bias. Therefore, the minimization problem of the MOM leads to an over-estimation of the parameter ν and thus, to a higher market activity as to the gamma function itself. TheECF method is closely related to the MOM. We have derived the four moments of the VG process fromthe moment generating function, which is similar to the characteristic function. For that reason, the ECFmethod is frequently confronted with the same overestimation of ν. The BI method is independent of aparticular optimization approach and relies on a simulation. As it is true for all simulations, the resultsare affected by the assumptions made within the simulation framework, such as the prior distribution ofthe parameter set η or the intervals for the parameter set η. Nevertheless, this approach should providemore stable and better results compared to the SMOM, MOM and ECF method. The MLE and the χ2

methods are two approaches, which directly use the PDF for fitting the VG process. While the MLEmethod modifies the PDF to a three-parametric PDF for numerical reasons, the χ2 method utilizes theoriginal PDF. The use of the PDF enables a close fitting between the empirical PDF and the PDF of theVG process. Particularly, the problem of extreme events can be captured more easily as the χ2 methodemploys intervals being able to handle these effects. Therefore, it can be stated that the χ2 method couldbe superior to that of the MLE. We build the first hypothesis (H1) on these fundamental aspects:

H1: The χ2 method provides the best approximation of the VG process for anempirical data set.

According to Hamilton (1989), Jeanne and Masson (2000) or Cerra and Saxena (2003) - amongst others -

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the stock markets’ behavior can change from normal (= non-volatile) times to turbulent (=volatile) timesover a longer period as a result of structural breaks. This market attitude can also affect the VG process’fitting procedure and the quality of the estimated parameter set η. Therefore, the fact whether the VGprocess is fitted to a data set taken from a normal time, or a turbulent time, or a mixture of both canbe important. For example, the use of a badly selected data set can imply that the variance influencingparameter σ under- or overestimates the real VG process or that the asymmetry component θ is biased.With this knowledge, we apply a regime switching model (for an overview of this model see Hamilton(1989) or Hamilton (2005)) for the purpose of identifying normal and turbulent times at the stock marketsand calibrate the VG process using this selected database. Therefore, we formulate the next hypothesis(H2):

H2: If a stock faces different market phases, the parameter set η of the VGprocess must vary over time.

Previous literature on the VG process’ fitting models - e.g. Seneta (2004), Seneta and Tjetjep (2006) andFinlay and Seneta (2008) - use simulated data or only an empirical data set taken from a short period oftime for the whole sample. Due to insufficient empirical testing of the above mentioned six fitting meth-ods for large data sets, we use the DOW JONES as well as all index stocks over the period from 1991to 2011. In accordance with hypothesis H2 and the assumption that the inclusion of a regime switchingmodel contributes to a parameter improvement we state our last hypothesis (H3):

H3: The inclusion of a regime switching model in a VG process’ frameworkincreases the fitting rate of each parameter estimation method.

We can state that H1 provides a hint for the results of the other two hypotheses. Furthermore, H2 isclosely connected to H3. Therefore, the tests of these hypotheses will also be joined in section 4.

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3. Research Design

3.1. Data

In general, we must distinguish between the data set needed for fitting the VG process and the dataset needed for the regime switching model. For analyzing the VG process we use daily returns of DOWJONES stocks and of the index itself over the period from 01.01.1991 to 31.12.2011. The stocks repre-sent the actual composition of the DOW JONES at the end of the year 2011. With this framework wecalibrate the VG process for 29 stocks and the index itself. We only exclude Kraft Foods Inc. from ourresearch, because they went public only in June 2001. Therefore only limited daily returns are available.

Furthermore, we calibrate the regime switching model with weekly closing prices of the DOW JONES.For back-testing the results of this model we take the yields of United States of America (US) Treasurieswith a maturity of 6 months (m), 12 months, 5 years (y) and 10 years as well as the closing prices ofgold. Using the daily data set we conduct a correlation analysis based on the theoretical background ofthe safe haven theory.

Our data are provided by Thomson Reuters Datastream. Table 1 summarizes the data set and table12 in appendix A1 provides some descriptive information - such as mean and standard deviation - aboutthe data set. Our data set includes the different phases of the past 21 years. It covers bullish markets

Underlying Timeframe Interval Observations

Dow Jones Industrial Average Index 01.01.1991 to 31.12.2011 daily 5298Dow Jones Industrial Average Index 01.01.1991 to 31.12.2011 weekly 1094All Dow Jones Industrial Average Index stocks 01.01.1991 to 31.12.2011 daily 5298US Treasury 6 Months Middle Rate 01.01.1991 to 31.12.2011 daily 5298US Treasury 12 Months Middle Rate 01.01.1991 to 31.12.2011 daily 5298US Treasury 5 Years Middle 01.01.1991 to 31.12.2011 daily 5298US Treasury10 Years Middle 01.01.1991 to 31.12.2011 daily 5298gold 01.01.1991 to 31.12.2011 daily 5298

Table 1: Empirical data overview

which could be observed in the periods from 1991 to 1998 and from 2003 to 2007, the dot-com hype andthe crisis subsequent to it, the events of 9/11, the financial crisis as well as the effects of the Europeansovereign debt crisis on US stocks. This broad range of data allows us to provide a profound analysis ofthe VG process in the next sections.

3.2. Sector classification

In some cases, it can be useful to group shares into sectors in order to identify common behavior andtrends in different market phases. We use the North American Industry Classification System (NAICS)for classifying the companies listed on the DOW JONES into sectors. As to that, the classificationsystem of the NAICS employs a six-digit code. While the first five digits are of general nature, the lastdigit indicates the national industries. The first two digits indicate the largest business sector, the thirddigit the sub-sector, the fourth digit the industry group, and the fifth digit the particular industry. For adetailed overview of the NAICS, see http://www.census.gov/cgi-bin/sssd/naics/naicsrch. For instance, itoften suffices to rely on the first two digits. In some cases, however, such as in the manufacturing sector,

10

which a lot of companies listed on the DOW JONES operate in, the third digit has to be employed. Weidentify the following six sectors, which sometimes cover more than one industry. For an overview seetable 2.

Sector Coverageinformation & entertainment information sector

entertainment sectorfinance & insurance financial sector

insurance sectorengineering transportation equipment manufacturing

machinery manufacturingprimary metal manufacturing

trade & food retail tradewholesalefood industry

chemistry & oil chemical manufacturingpetroleum and coal products manufacturing

electrical & component electrical sectorelectrical component sector

Table 2: Sectors covered by the DOW JONES according to NAICS

This structure allows us to assign the DOW JONES stocks to sectors as presented in table 3.

information finance engineering trade chemistry electrical& entertainment & insurance & food & oil & component

ATT AMEX ALCOA COCA COLA CHEVRON 3MMICROSOFT BOA CAT HOME DEPOT DU PONT CISCOVERIZON JP MORGAN UTC MCDONALDS EXXON GEWALT DISNEY TRAVELERS WAL MART MERCK HP

JNJ PFIZER INTELPG IBM

Table 3: Classification of DOW JONES into sectors according to NAICS

3.3. Identification of normal and turbulent timeframes

For identifying normal and turbulent timeframes in our economic time series we use a regime switch-ing model. The change from a normal to a turbulent timeframe is often connected with a break in thetime series caused by a financial crisis or by political changes (for an overview see Hamilton (1988),Jeanne and Masson (2000) and Cerra and Saxena (2003)). For an overview of the main idea of a regimeswitching model see Hamilton (1988), Hamilton (1989) or Hamilton (2005). We follow Hamilton (1989)

11

and Hamilton (1994) by using the maximum likelihood estimation method2for estimating the transi-tion matrix P ∗, which indicates the time series switches from state i to state j. Our objective is todetermine periods when the DOW JONES was in a normal market period or in a turbulent period respec-tively (resp.). For calibrating the regime switching model we use a database of weekly closing pricesfrom January 1991 to December 2011. We prefer the weekly database to the daily database as the resultsare more stable. Figure 1 shows the development of the DOW JONES as well as the smoothed proba-bility that the share index is in a turbulent regime. By means of these smoothed probabilities, we divide

Jan91 Feb92 Mar93 May94 Jun95 Jul96 Aug97 Oct98 Nov99 Dec00 Jan02 Mar03 Apr04 May05 Jun06 Jul07 Sep08 Oct09 Nov10 Dec110

0.5

1

Pro

babi

lity

Tur

bule

nt R

egim

e

Jan91 Feb92 Mar93 May94 Jun95 Jul96 Aug97 Oct98 Nov99 Dec00 Jan02 Mar03 Apr04 May05 Jun06 Jul07 Sep08 Oct09 Nov10 Dec11

2.500

5.000

7.500

10.000

12500

15000

DO

W J

ON

ES

Timeframe 101.01.1991 to 08.06.1998

Timeframe 215.06.1998 to 07.04.2003

Timeframe 314.04.2003 to 15.10.2007

Timeframe 422.10.2007 to 31.12.2011

Figure 1: Regime switching model - DOW JONES

the total timeframe into four sub-periods - two normal states (from 01.01.1991 to 08.06.1998 and from14.04.2003 to 15.10.2007) and two turbulent states (from 15.06.1998 to 07.04.2003 and from 22.10.2007to 31.12.2011). Hamilton (1989) concludes that his model makes very clear decisions about the proba-bility of being in a certain state. In Hamilton’s context this means that only few smoothed probabilitiesshould lie between 0.3 and 0.7 and the algorithm usually identifies fairly clear decisions about the states.With this knowledge he suggests using the decision criterion smoothed probability ≥ 0.5 in a two statemodel. In the first sub-period we treat the two short periods with a high smoothed probability as outliersand therefore, consider them to be within a normal time. In the second and fourth sub-period we facetimes switching from turbulent to normal and back to turbulent with high frequency. We apply Hamilton’s(1989) decision criteria and demand an expected smoothed probability µsm ≥ 0.5 for a turbulent time.Therefore, we perform an approximate Gauss test with a 95% significance level and the null hypothesisH0 : µ ≥ 0.5 and H1 : µ < 0.5 as the alternative hypothesis. For more information about the regime

2As proposed by Alexander (2008) we use the package ”MS regress”, which was provided by Perlin (2007) for the implementationin MATLAB.

12

switching model and the applied approximate Gauss-test see appendix A2.

To sum up, we classify our four sub-periods as shown in table 4 and use this division in the subse-

Timeframe Regime switching state

01.01.1991 to 08.06.1998 normal15.06.1998 to 07.04.2003 turbulent14.04.2003 to 15.10.2007 normal22.10.2007 to 31.12.2011 turbulent

Table 4: Classification sub-periods into normal and turbulent times

quent sections. We also do a robustness test by using the safe haven theory. The results are also shownin appendix A2.

4. Empirical results

At first, we try to fit a VG process over the total period of 21 years by means of the six presented esti-mation methods. Table 5 demonstrates that the quality of the fitting rates (=percentages of shares whichfollow a VG process) tested by the KS test is not acceptable except for the χ2 method. In contrast to thesimulated VG process data, we even notice a strong decrease of the fitting rates with increasing levels ofsignificance. Furthermore, µd indicates the mean maximum deviation of the 30 empirical samples. As isthe case for the fitting rates, µd is too large and therefore does not reveal a high-quality approximationfor all methods. Only the χ2 method positively stands out from these methods. For a robustness test ofthe fitting rates, have a look at the χ2 test’s results in appendix A3 (see table 15). These results make

SMOM MOM MLE BI ECF χ2

µd 7.96 8.37 6.21 8.75 8.79 2.431% 0 0 3.33 6.67 0 505% 0 0 0 6.67 0 26.6710% 0 0 0 3.33 0 26.6715% 0 0 0 3.33 0 2020% 0 0 0 3.33 0 20

All fitting rates are indicated in %.

Table 5: KS test: fitting rates of all estimation methods - total timeframe

clear that it is hardly possible to estimate a VG process over a very long period. Problems result fromdata covering too many different market phases in which the stocks change their behavior. This meansthat the increments’ or the log returns’ PDF, resp., has various symmetries, fat tails on the left or rightend of the distribution or a time varying volatility as most stocks behave differently during the differentperiods. Table 10 presents - among other things - the parameter estimation results via the χ2 methodand the corresponding significance levels over the period from 1991 to 2011. We observe that it is quitedifficult to get significant results for the finance & insurance sector (for classification of sectors see sec-tion 3.2). Taking the descriptive statistics (see table 12 in appendix A1) into account it becomes apparent

13

that the finance & insurance sector has had a very high volatility, which has also led to a high kurtosis,particularly since the beginning of the financial crisis. The VG process tries to cover this stock behaviorwith σ and ν. The comparison of the finance & insurance sector’s ν with other sectors’ features reveals alarge upward deviation. It seems that the change of the market behavior of financial institutions’ stocks inthe last two decades has also affected the fitting characteristics of the stochastic processes and thereforeemphasizes our hypotheses H2 and H3. In contrast to the highly volatile financial & insurance sector,the results for the more defensive sub-sectors electrical & component, trade & food and oil & chem-istry or the typical cyclical engineering sector are generally more significant. From the results we learnthat a randomly selected timeframe used for fitting a stochastic process does not ensure significant results.

With this background knowledge we run the same parameter estimations for the four timeframes identi-fied by the regime switching model and the safe haven theory. Tables 6, 7, 8 and 9 clearly show increasingfitting rates for all estimation methods compared to the total timeframe estimation (see table 5). For arobustness check see tables 16, 17, 18 and 19 in appendix A3. While, for example, the fitting rate forthe χ2 method is 50% (1% significance level) over the total timeframe, the fitting rates vary within theinterval [70%, 100%] over the four identified timeframes. This fact highlights that the data used to fita model plays an important role and a well-defined period of the data set increases the probability ofobtaining high-quality estimations. Besides, we also consider the mean maximum absolute deviation µdbetween the estimated and the empirical CDF. In turbulent timeframes this indicator is smaller (2.45 and2.01, resp.) than in normal timeframes (3.55 and 2.57, resp.). Further, there is a trend that the deviationsthemselves decrease in general in normal and turbulent timeframes (as to the χ2 method from 3.55 to2.01). Again, these trends reaffirm that stock markets are more likely to follow a VG process. We cannotreject our hypothesis H2 as we obtain strongly improved fitting rates from the application of the regimeswitching model. We find an outperformance of the χ2 method for all sub-periods and the total period.The MLE method is the second best approach and particularly outperforms the moment based estimationmethods in the µd figures. The BI method only seems to work well in a highly volatile market phase(timeframe 4) and does not justify the efforts undertaken. These results confirm the simulated VG pro-cess’ results and our first hypothesis H1.


µd 5.52 5.5 4.63 13.54 5.59 3.551% 30 30 30 3.33 30 705% 20 20 20 0 20 53.3310% 6.67 6.67 6.67 0 6.67 4015% 0 0 0 0 0 26.6720% 0 0 0 0 0 16.67


Table 6: KS test: fitting rates of all estimation methods - timeframe 1

So far we have focused on the fitting rates and the quality of the estimations. Now we start a more detailedanalysis of the stocks’ parameters of the VG process (see table 10). At first, we take the parameters of thetotal timeframe and the four sub-periods into account. We see that the total timeframe tends to average

14


µd 5.17 5.14 4.06 7.93 5.26 2.451% 63.33 63.33 80 3.33 63.33 905% 50 50 60 3.33 50 9010% 40 40 43.33 3.33 33.33 9015% 30 30 33.33 0 26.67 9020% 23.33 23.33 26.67 0 23.33 90




µd 7.11 6.98 4.59 40.81∗ 7.29 2.571% 53.33 53.33 56.67 6.67 53.33 905% 46.67 46.67 50 6.67 46.67 9010% 46.67 46.67 46.67 6.67 46.67 9015% 43.33 43.33 43.33 6.67 40 9020% 43.33 43.33 43.33 6.67 40 90

∗ This figure is biased as some shares could not be fitted correctly.All fitting rates are indicated in %.



µd 7.96 7.9 6.4 3.51 8.5 2.011% 23.33 23.33 36.67 93.33 23.33 1005% 20 20 36.67 80 16.67 10010% 16.67 16.67 23.33 73.33 13.33 10015% 13.33 13.33 10 70 10 96.6720% 6.67 6.67 6.67 70 3.33 93.33



15

over the four sub-timeframes. This means, for example, that we overestimate σ, θ or ν in a normal periodand underestimate them in a turbulent period. With respect to simulation and forecasting applications ofasset prices by means of a VG process model this can lead to enormous biases. While σ and ν errorsparticularly lead to a mismatch of the actual market behavior, a θ error results in a wrong assumption ofthe log returns’ symmetry. In normal timeframes there is a trend towards a right symmetric stochasticprocess, whereas in turbulent timeframes there is a trend towards a left symmetric VG process’ behavior.

Having looked at the VG process parameters in a more general manner, we continue with some de-tailed information about the sectors:

Information & entertainment and electrical & component:

We notice different behavior in the two turbulent periods. During the dot-com crisis, these two sec-tors have a higher σ and ν in relation to the other sectors, while these sectors are less volatile duringthe financial crisis (see table 12 in appendix A1). Particularly, enterprises in the telecommunication andtechnology sector reveal a stable performance and offer a good opportunity for diversification.

Finance & insurance:

This sector has an extremely high σ and ν during the financial crisis. The VG process is able to cap-ture this behavior according to the KS test and the respective significance levels. Therefore, the VGprocess and the jumps generated by a gamma function using the parameter ν are helpful approaches tocover this difficult market behavior. These findings are in contrast to the first timeframe where the finance& insurance stocks do not directly follow a VG process. This change in market behavior highlights thetransition from a stable to a volatile and nervous sector as a consequence of a financial crisis and politicaland economic changes.

Engineering:

This sector seems to be strongly dependent on the current economic situation. In turbulent phases σ andν increase. For that reason, the VG process is a good procedure for modeling stocks of cyclical industries.

Chemistry & oil and trade & food:

The VG process parameters σ and ν show that these sectors decouple from the rest of the market inturbulent phases. The parameters indicating nervous markets are lower than those of other sectors. Froma risk perspective, we can - again - find some opportunities for diversification.

To sum up, the parameters analysis emphasizes the importance of a clearly defined separation of nor-mal and turbulent market phases and verifies our second hypothesis H2. Additionally, the examination ofthe four parameters allows a more detailed analysis of stocks from a risk perspective. In some cases, thereis only a concentration on the mean and the variance. The VG process allows an extended considerationof risk by also taking the symmetry of stocks’ log return PDF, the kurtosis or the market activity with theparameter ν into account.

16

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17

5. Discussion

After presenting the empirical results, we put them in relation to each other. The quality of a VGprocess depends on the estimation method used. Unlike comparable cases, where there often exists atrade off between estimation quality and estimation time, the χ2 method allows a high quality estimationby keeping the estimation effort acceptable. Furthermore, we notice that we even get high fitting ratesfor the χ2 method when we apply the very restrictive KS test. This fact verifies the good VG processapproximation by means of the χ2 method.

We contribute to existing literature, such as Seneta (2004) or Finlay and Seneta (2008), by compar-ing all estimation methods with a large empirical data set of the DOW JONES for a period of more than21 years. In contrast to Seneta (2004) and Finlay and Seneta (2008), we do not choose the timeframes forfitting the VG process randomly, we apply a regime switching model for identifying normal and turbulenttimeframes. This additional aspect offers a good opportunity for comparing the estimated VG processparameters over time for an underlying. We observe that the parameters vary over time and this modelingapproach is a progress in avoiding the over- or underestimation of the actual parameters. However, weare aware that the differentiation between normal and turbulent timeframes is only based on the DOWJONES and not on a single asset decision. Therefore, our results and the behavior of an asset are alwayscompared to the market behavior of the DOW JONES. Nevertheless, the regime switching model in-creases the fitting rates of the six estimation methods significantly. We also notice that markets are morelikely to follow jump processes, such as the VG process, as we identified increasing fitting rates over thefour timeframes, particularly for the χ2 method. Besides improved fitting rates, the regime switchingmodel highlights that VG process parameters vary over time and the sensitivity of the VG process itselfbecomes apparent. As a consequence, we state that a correct estimation method selection is essential fora proper application of the VG process in the context of modeling share returns.

6. Conclusion

We have taken the common knowledge about the VG process (see Madan and Seneta (1987), Madanand Seneta (1990), Seneta (2004), Finlay and Seneta (2008), amongst others) and the knowledge aboutthe estimation methods of the VG process estimation methods by means of a simulated VG process dataset (see Finlay and Seneta (2008)) and applied it on a large empirical data set, the DOW JONES. Basedon the theoretical background we develop three hypotheses, each of which includes an aspect of practicalusage of the VG process. We reach our aim to improve the quality of a VG process by integrating aregime switching model developed by Hamilton (1989) and Hamilton (2005) into the VG process frame-work, and in this way, identify VG process parameters which depend on timeframes (hypothesis H2).This approach results in the improvement of fitting rates (hypothesis H3), which are determined by theconservatively adjusted KS test for CDF. Finlay and Seneta (2008) suggest using a MLE method or χ2

method in order to fit a VG process by means of a simulated data set. We can verify this knowledge forempirical data, but in contrast to Finlay and Seneta (2008), we find a superiority of the χ2 method forempirical data (hypothesis H1).

The regime switching model allows us to have a closer look at the parameters of the VG process over thevarious normal and turbulent regimes over our daily data set comprising 21 years. Furthermore, we canlearn that such a differentiation helps to avoid an over- or underestimation of the real VG process stockmarket behavior.

18

However, we state that our results only cover the DOW JONES and the new knowledge about the VGprocess is limited to this data set. The scope of the analysis could be broadened to cover other marketssuch as the commodities, currency or bond markets by extending the data set. Furthermore, this paperfocuses on the identification of the estimation method fitting the VG process best. On the basis of theseresults we could compare the VG process with other stochastic models.

To sum up, our findings should ensure a better understanding of the VG process in connection withan empirical dataset. This empirical study on the VG process should provide a useful basis for furtherresearch on the permanent consideration of the combination of the regime switching model and the VGprocess.

19

References

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21

Appendix A1

There are various ways to measure the fitting quality between the real model and the estimated model.A common approach is to compare the estimated PDF or the cumulative distribution function (CDF)with the respective empirical PDF or CDF. Literature mainly proposes the χ2 test, which is very sim-ilar to the χ2 method. The close connection to the χ2 method can lead to less objectivity and resultin a bias. Bearing our aim of identifying the best fitting method in mind, we choose the Kolmogorov-Smirnov (KS) test as suggested by Massey (1951). This test is based on the maximum difference be-tween an empirical and a hypothetical CDF. For the reasons mentioned, we first run a KS test, and thenback-test the results with the χ2 test. The KS test assumes that the log returns follow a VG process asnull hypothesis. The alternative hypothesis simply rejects this assumption. The main framework con-sists of the comparison between the maximum absolute deviation of the estimated CDF and empiricalCDF , FEX (xi) = 1

N

∑Nj=1 1{xj≤xi}, where Xi = (X1, . . . , XN ) are the N observed log returns of

the VG process and 1{xj≤xi} is 1 if xj ≤ xi, else 0. By means of the estimated parameter set η andthe PDF of the VG process (see formula 4), the estimated CDF of the VG process FX(x) is defined asFX,η(xi) =

∫ xi−∞ fX,η(x)dx. In the last step, the maximum absolute deviation d between FEX (xi) and

FX(xi), d = maxi=1...N

| FEX (xi) − FX,η(xi) |, is calculated and compared with the critical values (see

table 11) as defined by Massey (1951). α stands for the significance level. If d exceeds dα, we reject thenull hypothesis. The KS test is sometimes criticized as the decision is based on one maximum deviation

α 1% 5% 10% 15% 20%

dα 1.63√N 1.36

√N 1.22

√N 1.14

√N 1.07

√N

Table 11: Significance levels KS test

only. For that reason, an estimated CDF that fits badly to the empirical CDF at one point in time only isregarded as non-convenient, although the other points in time are estimated correctly. We treat this aspectby taking it as a more conservative method.

The χ2 test is similar to the χ2 method. It is a common approach for testing the quality of an esti-mated CDF. Let η, r,B,Oi and Oi be defined as described in the section dealing with the χ2 methodpart (see section 2) and Q be defined as the intervals. By means of these parameters, the χ2 test value

χ2(η) =∑Qi=1

(Oi−Oi(η))2

Oi(η)is calculated. Subsequently, χ2(η) is compared to the quantile (1 − α) of

the χ2 distribution with Q − 5 degrees of freedom for various significance levels α according to Finlayand Seneta (2008). If χ2(η) exceeds the quantile, we reject the null hypothesis that the observationsXt follow a VG process. The degrees of freedom are calculated as the number of intervals minus thenumber of estimated parameters minus one. Instead of using I intervals, as in the χ2 method, we applyQ(Q = 10 which is similar to 10% quantile bands). We change this technical aspect in order to reduce theadvantage of the χ2 method over the other estimation methods and, in this case, differ from Finlay andSeneta (2008). They apply wider intervals on the right and left hand side of the distribution and smallerintervals in between.

22

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23

Appendix A2

As we have no idea about the distribution of the smoothed probabilities, we do not assume any distributionfor calculating the test statistic value V . Table 13 demonstrates that we can rejectHo in the first and thirdsub-period, but not in other periods.

Timeframe Observations µsm σsm V Rejection interval

01.01.1991 to 08.06.1998 387 0.067651 0.16408 -51.8376 (-∞,-1.6449)15.06.1998 to 07.04.2003 252 0.48732 0.40017 -0.50313 (-∞,-1.6449)14.04.2003 to 15.10.2007 236 0.013478 0.036818 -202.9983 (-∞,-1.6449)22.10.2007 to 31.12.2011 219 0.60208 0.44601 3.3869 (-∞,-1.6449)

Table 13: Descriptive statistics: approximate Gauss test for smoothed probabilities

We also present the estimated transition matrix P ∗ for the regime switching model for the DOW JONES

P ∗ =

0.99 0.01

(0.00∗∗∗) (0.01∗∗∗)0.04 0.96

(0.00∗∗∗) (0.00∗∗∗)

, (22)

which provides the probabilities of switching from one state to another. The figures in brackets denotedare the test values and the significances levels, with ∗∗∗ indicating a 1% significance level.

For running a robustness test which is in accordance with the safe haven theory, we have a look at thecorrelations between the DOW JONES, US T-Bills yields and gold. In turbulent times, the correlationbetween the DOW JONES and the US T-Bills yields increases. This means that while stock marketsdecrease, US T-Bills prices increase (yields =decrease). Furthermore, gold decouples from the DOWJONES in turbulent times.

TimeframeYield US

T-Bills 6mYield US

T-Bills 12mYield UST-Bills 5y

Yield UST-Bills 10y

gold

01.01.1991 to 08.06.1998 0.32654 0.29208 -0.25329 -0.51722 -0.4598615.06.1998 to 07.04.2003 0.59068 0.61947 0.76689 0.80246 -0.6785214.04.2003 to 15.10.2007 0.761 0.75848 0.73629 0.63986 0.9028222.10.2007 to 31.12.2011 0.45577 0.41158 0.24329 0.23187 0.31944

Table 14: Correlation between DOW JONES and US T-Bills and gold

24

Appendix A3

The following six tables show the fitting rates based on the χ2 test. As mentioned above we use theseresults as a robustness check for the KS test. Each time a robustness test is needed, a reference can befound in the main part of this paper. The results of the χ2 test are in accordance with the KS test andverify the χ2 method as the best estimation method of the VG process. All in all, the results only differin that they reveal lower fitting rates.


1% 0 0 0 0 0 6.675% 0 0 0 0 0 3.3310% 0 0 0 0 0 015% 0 0 0 0 0 020% 0 0 0 0 0 0


Table 15: χ2 test: fitting rates of all estimation methods - total timeframe


1% 16.67 16.67 16.67 0 13.33 33.335% 3.33 3.33 3.33 0 3.33 13.3310% 0 0 0 0 0 1015% 0 0 0 0 0 1020% 0 0 0 0 0 6.67


Table 16: χ2 test: fitting rates of all estimation methods - timeframe 1

25


1% 16.67 20 16.67 0 16.67 86.675% 13.33 13.33 13.33 0 13.33 66.6710% 6.67 6.67 6.67 0 6.67 56.6715% 6.67 6.67 6.67 0 6.67 4020% 6.67 6.67 6.67 0 6.67 40




1% 43.33 40 43.33 13.33 40 86.675% 30 26.67 30 13.33 26.67 76.6710% 26.67 23.33 26.67 10 23.33 63.3315% 23.33 20 23.33 10 23.33 56.6720% 20 16.67 20 10 20 43.33




1% 6.67 6.67 10 46.67 6.67 905% 3.33 3.33 3.33 26.67 3.33 83.3310% 3.33 3.33 3.33 20 3.33 7015% 3.33 3.33 3.33 16.67 3.33 53.3320% 0 0 0 16.67 0 43.33

All fitting rates are presented in %.


26

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