CONDITIONAL VOLATILITY OF MOST ACTIVE SHARES
OF CASABLANCA STOCK EXCHANGE

ABDELHAMID EL BOUHADI*
89, rue Al Azhar, Quartier Amerchich
Marrakech, Maroc

email address : [email protected]

Summary

Volatility plays an important role in the explanation of prices of securities and their derivatives as well as risk which are relative to them. The stock exchange of Casablanca constitutes a market meadow emergent of MEA zone for which the problem of volatility should not be underestimated for the causes of lack of the making in the order book. The introduction of the electronic system in 1998 allowed continuous quotation of a number restricted by securities liquidity of which does not seem even so important. We test in this paper the conditional volatility of a certain number of securities considered as active and which to be result from the MADEX index. Results show a certain asymmetric volatility of the major securities. The use of the asymmetric GARCH allowed us better to describe rough variations of prices provoked by large quantities traded in block market. These models throw back quadratic specification of the conditional variance lauded by GARCH standard models. Indeed, with regard to these last ones, exponential model GARCH and threshold GARCH have two advantages. While standard model GARCH puts that only the amplitude of shock and not the sign of the past residuals has an impact on the conditional variance, EGARCH and TGARCH models allows an asymmetric answer to shocks. Second advantage is that not conditional variance is discrete.

Keywords

Volatility ; Asymmetric GARCH ; None Linearity ; MADEX ; The Stock Exchange of Casablanca.

JEL Classification: G14

VOLATILITÉ CONDITIONNELLE DES VALEURS LES PLUS
ACTIVES DE LA BOURSE DE CASABLANCA

Résumé

La volatilité joue un rôle important dans l’explication des prix des titres et leurs dérivés ainsi que le risque qui leur sont afférents. La bourse de Casablanca constitue une bourse pré-émergente de la zone MEA pour laquelle le problème de volatilité n’est pas à sous-estimer pour les causes de manque de contreparties dans le carnet d’ordres. L’introduction du système électronique en 1998 a permis une cotation au continu d’un nombre restreint de titres dont la liquidité ne semble pas encore si importante. Nous testons dans ce papier la volatilité conditionnelle d’un certain nombre de valeurs considérées comme actives et qui sont issues de l’indice MADEX. Les résultats montrent une certaine volatilité de type asymétrique. L’utilisation de la modélisation GARCH asymétrique nous a permis de mieux décrire les variations brutales dans les cours provoquées par les cessions de marché de blocs. Ces modèles rejettent la spécification quadratique de la variance conditionnelle prônée par les modèles GARCH standard. En effet, relativement à ces derniers, les modèles GARCH exponentiels et à seuils ont deux avantages. Alors que le modèle GARCH standard pose que seule l’amplitude de choc et non le signe des résidus passés a un impact sur la variance conditionnelle, les modèles EGARCH et TGARCH permettent une réponse asymétrique aux chocs. Le second avantage est que la variance non conditionnelle est finie.

Mots-clés

Volatilité ; Asymétries ; Non Linéarité ; EGARCH ; MADEX ; La Bourse de Casablanca.

Classification JEL : G14

* Assistant Professor of Finance at the University of Marrakech, Morocco.
1

INTRODUCTION

Volatility can be simply defined as the variability of security price during a given period. In the models of asset valuation such the CAPM either APM, volatility is likened to risk measured by the standard deviation or its square ( its variance) under normal probability. One distinguishes three types of volatility : none conditional or historic volatility which is observation ex post past variations concerning stock exchange prices; the conditional volatility as ARCH and its extensions which allows to extract the early part of the historic volatility describing the behavior of the conditional variance at the time and finally the implied volatility which supplies the element of forecast. This present article is dedicated to describe and to test the conditional volatility of the most active shares of stock market of Casablanca. Our study distributed so in two points : the first treats models ARCH and its extensions and the second tests the various classes of these models as for the volatility of the most liquid shares.

1. CONDITIONAL VOLATILITY

The measure of the conditional volatility resulting from econometric models of ARCH types allows to extract the anticipated volatility by pushing aside the influence of unpredictable shocks. This measure allows so to arrest volatility such as it is anticipated by the market ex ante on the basis of the available relevant information. Also, it allows to measure the persistent effect which can have the behavior of stock exchange in the time : at the periods of strong fluctuations, in increase or in decrease, behavior of which is repetitive at the duration and the amplitude succeed one another quiet periods in which prevail weak fluctuations in stock exchange.

1.1. Model ARCH

ARCH Process (AutoRegressive Conditional Heteroskedasticity) introduced by Engle¹
[1982] is a part of a set of process treating the conditional Heteroskedasticity. Unlike the linear models which are interested only at the first order of the moment (expected value), ARCH model introduces the study of second order the moments (conditionals and not

¹ Engle R., « Autoregressive Conditionnal Heteroskedasticity whih Estimates of The Variance of UK inflation », Econometrica, vol. 50, n°1, 1982.

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conditionals) of the time series. This type of models allows the variance of a series to depend on the available of information set, and notably of the time. The object of these representations is to mitigate the insufficient of linear ARMA models not capable to describe the behavior of financial series. These last ones are indeed characterized by a variable volatility on the time and by phenomena of asymmetry which can not be taken into account with the ARMA models. Besides, if process followed by the volatility is correctly specified, this one can supply information useful for the determination of the process generating of returns and leaving to be used in the forecast. ARCH model is based on a endogenous parametrisation of the conditional variance. These models were the object of several extensions of which the most famous is GARCH. Let us consider a model with correlated errors of class AR (1) in which :

Y = X_t β + ε_t

(1) (2)

t

ε_t = aε_t−1 + u_t

With,
Y : endogenous variable (scalar of dimension n) ;

t

X_t : vector of the exogenous variables (matrix of dimension n × (k +1)); β : vector of coefficients which the dimension is k +1 ; ε_t :innovation or error of specification (which is an unpredictable term) ; u_t : Process i.i.d. of average zero and of finished varianceσ_u².

(1) Can rewrite :

∞

ε = aⁿu

(3)

∑

• t
• t−n

n=0

As E[ε_t ] = 0, by hypothesis, we obtains then :

V[ε_t ] = E[ε_t − E[ε_t ]]² = E[ε_t² ] = E[aε_t−1 + u_t ]² = a²E[ε_t²₋₁] +σ_u²

If process is stationary, we should have :

E[ε_t² ] = E[ε_t²₋₁] = V[ε_t ]

3

Hence :

σ_u²
V[ε_t ] = a²V[ε_t ] +σ_u² =

1− a²

In conditional terms we have :

E[ε_t ε_t−1] = E[(aε_t−1 + u_t )ε_{t −1}] = aε_{t −1} V[ε_t ε_t−1] = E[ε_t ε_{t −1} − aε_t−1]² = E[aε_{t −1} + u_t − aε_t−1]² = E[u_t² ] = σ_u²

These moments seem so constant and independent of time. To characterize in a correct way the behavior of variable financial, it was necessary to introduce models of regression where the variance of the noise depends on time : these models are heteroskedastic by construction,. It is in this optical that Granger and Andersen¹ puts a model such as the conditional variance of ε_t depends on its past realizations ε_t−1
:

ε_t = u_tε_t−1

(4)
Where u_t is white noise with zero average and finished varianceσ_u² , et ε_t−1 is a unpredictable variable which stochastic variations are independent from the u_t. The calculation of the moments of the unpredictable variable ε_t gives us the following thing :

ƒ

In not conditional terms, we have :

E[ε_t ] = 0 V[ε_t ] = σ_u²ε_t²₋₁

ƒ

In conditional terms, we have :

E[ε_t ε_t−1] = 0

1

Granger C. et Andersen A., An Introduction to Bilinear Time-Series Models, Göttingen : Vandenhoeck and Ruprecht, 1978.

4

V[ε_t ε_t−1] = σ_u²ε_t²₋₁

One notices that the stochastic properties of the conditional and unconditional estimators are the same, what limits their use within the framework of the forecast, notably in the case of financial series, because shock or effect of a large-scale operation on the shares¹ can not be taken into account. This is of as really as this effect is persisting. Engle's ARCH model [1982] appears to supply the means to take into account these phenomena and a frame sharply more adapted for the treatment and the description of the characteristics of financial series. The work of Engle consists obviously in generalizing bilinear model of Granger and Andersen. For that purpose, he poses :

ε_t = u_th

(5)

t

Where u_t follows a normal distribution N(0,1) , and h replaces ε_t−1 in the equation (5).

t

Consequently :

p

h² = a + a ε ² = a₀ + a(B)ε_t²

(6)

∑

t

0

i t−i i=0

p

• 
• 

h = a + a ε ²

∑

t

0

i t −1

• 
• 

i=0

• 
• 

Such as :

a₁ > 0, a_i ≥ 0 ∀i a(B) = a₁B + a₂B² + ...+ a_pB^p

And h Process is called ARCH of order p and it is noted ARCH(P). If we adds the hypothesis of

t

normality, ε_t , defined in the equation (5), can be directly expressed according to the available relevant information at the moment t :

(ε_t I_t−1) → N(0,h² )

t

¹ All the operations on a share that is a takeover bid (TOB) entailing a limit crossing or a speculative intervention entailing a momentary gap of the price are not taken into account in the calculation of the hoped earning. The change of the variance (which can have for origin a speculative intervention) can not be taken into account because the stochastic properties of conditional and marginal moments are the same.

5

With :

p

h² = a + a ε²

∑

t

0

i t−i i=1

The conditional distribution of ε_t is standard normal (centered). The variance h² depends

t

linearly on the last values p of the process ; so :

E[ε_t I_{t −1}] = 0

p

V[ε I ] = h² = a + aε ²

∑

t t −1 t

0

i t−i i=1

It is possible to present ε_t² under the process of AR( p).For this, we set :

v_t = ε_t² − h²

(7)
(8)

t

hence :

p

ε² = a + a ε² + v_t

∑

t

0

i t−i i=1

v_t has an average and a covariance equal to zero, but its variance is not constant. According to this formulation of AR(P), the model of regression with ARCH is obtained supposing that the average of ε_t is a linear combination of exogenous and lagged dependent variables (X_t ),

multiplied by a vector of unknown parameters (β) :

(ε_t I_t−1) → N(X_t β,h² )

t

h = h(u_t−1,u_t−2 ,...,u_{t− p} ,a)

(9)

t

u_t = ε_t − X_t β

The last expression possesses very interesting properties for econometric applications. McNees¹ showed in 1980 that uncertainty closely linked to the forecast varied according to periods and this by report not only on the horizon of forecast but as well on the fact as errors often regrouped in high errors followed by weak errors. The ARCH model, which the

¹ McNees S. K., « A Critique of Alternative Methods of Comparing Macroeconomic Models », in Ramsey J. and

Kmenta ed. Methodology of Macroeconomic Models, North-Holland, 1980.

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variance depends on the time and past errors, allows to realize this phenomenon. Indeed, as show it the expressions bellow, the conditional and marginal moments of order 2 are not identical. For ARCH(1) for example, we have the following thing :

ƒ

In marginal terms, we have :

E[ε_t ] = 0

a₀

V[ε_t ] =

1− a₁

ƒ

In conditional terms, we have :

E[ε_t I_{t −1}] = 0
(1− a₁^h )

V[ε_t I_{t −1}] = a₀
1− a₁

+ a₁^hε_{t −h2}

One notices that marginal and conditional variances are not identical; especially since conditional variance is not a constant because it depends on past errors. However, another remark deserves to be indicated : when h aims towards the infinity, conditional variance aims towards the marginal variance :

a₀

lim V[ε_t I_{t −1}] =

(10)

h→∞

1− a₁

If coefficients a_i are all positive and rather big, there is a certain persistence of alternated levels by volatility : periods of strong volatility followed by periods of weak volatility and periods of weak volatility followed by strong volatility. Besides, the asymmetrical character of ARCH processes is one a reason of more its interest in the field of the finance because the big majority of stock exchange series present this character¹ : they possess tails of distribution more thick than those of the normal distribution and they are all more important as the kurtosis is big.

¹ It is for Fama that falls this discovery in 1963.
7

In the same spirit of definition of the normal distribution, the kurtosis of ARCH(1) process defines as the report of standard moment (centered) with order 4 on the square of standard moment centered with order two :

3a₀ (1+ a₁)
E[ε_t⁴ ] =
(1− a₁)(1− 3a₁² )



²

a₀

• 2
• 2




E[ε_t ] =

1− a₁

• 
• 

hence :

3(1− a₁² )
K =

(11)

1− 3a₁²

If 3a₁² is lower than 1, this expression appears value of K always superior to 3 (value of the kurtosis of the normal distribution). It is notably case of financial series. Being this, model ARCH is not without insufficiencies ; the main difficulty appears when the order of delay of the function of conditional variance is very high. In that case the number of parameters to be estimated is very important. To reduce the number of parameters to calculate, Engle suggested considering the equation of following conditional variance: