1. Introduction
The stochastic behavior of stock prices is a critical concern in the area of financial economics. An appropriate specification of price dynamics plays a crucial
role in determining the equilibrium price of the stock itself capital asset pricing model, as well as in pricing its derivatives option pricing model. A classical and
widely used assumption for stock price dynamics is that stock prices follow a diffusion process, in particular, geometric Brownian motion, in which the continu-
ous compound stock return log-return for a certain period is normally distributed. Despite its attractive statistical properties and computation convenience which lead
to it being unanimously accepted for theoretical derivation, the independent Gaussian process does not adequately describe real-world stock price fluctuations.
Beginning with the seminal studies of Mandelbrot 1963, 1967 and Fama 1965, extensive empirical evidence has shown that the statistical characteristics of stock
return distributions, such as a high level of kurtosis, a non-zero value of skewness, and volatility clustering, are inconsistent with the independent Gaussian process.
Concerning the non-zero skewness and high-level kurtosis of stock return distri- butions, there are three classes of models which can provide possible explanations.
First, the stock return might follow a Gaussian process with time-varying parame- ters, such as the autoregressive conditional heteroscedasticity ARCH model
proposed by Engle 1982, or the generalized autoregressive conditional het- eroscedasticity GARCH model proposed by Bollerslev 1987. In the ARCH and
GARCH models, according to Milhoj 1985, the unconditional distribution of stock returns is non-Gaussian with heavier tails than a Gaussian distribution. Some
other extended GARCH models were also proposed, such as the exponential GARCH proposed by Nelson 1991, the non-linear asymmetric GARCH model
proposed by Engle and Ng 1993, and the leveraged GARCH model proposed by Glosten, Jagannathan, and Runkle Glosten et al., 1993. These extended GARCH
models are able to specify the characteristic that stock volatility and return are negatively correlated, i.e. the leverage effect. More recently, Duan 1997 proposed
an augmented GARCH model that can contain all the above GARCH models.
Second, the stock return could be a mixture of stationary distributions, such as a mixture of two or more normal distributions with different means and variances
called the mixed – normal distribution proposed by Kon 1984, or a mixture of a normal and a discontinuous Poisson jump process called the Poisson – normal
distribution discussed by Akgiray and Booth 1986. In a mixed – normal distribu- tion model, each stock return is an independent observation drawn from one of a
finite number of normal distributions. The mixed – normal model can accommodate both structural and cyclical parameter shifts, thus can explain the observed skew-
ness and leptokurtosis in stock returns. Similarly, Akgiray and Booth 1986 pointed out that the stock return distribution under the mixture of a normal
diffusion and a Poisson jump process is leptokurtic as the jump intensity parameter is greater than zero, and is skewed when the expected jump size is non-zero. The
Poisson – normal model is essentially an infinite mixture of normal distributions with parameter restrictions.
Third, the stock return might be a stationary process, such as a stable Paretian distribution proposed by Mandelbrot 1963, 1967 and Fama 1965, Student’s t
distribution proposed by Blattberg and Gonedes 1974 and Bollerslev 1987, or a generalized exponential distribution proposed by Nelson 1991. The stock return
distributions under the above stationary processes all have thicker tails than that of the normal distribution. However, these symmetric distributions are unable to
account for the skewness observed in the empirical data.
The Student-t distribution is actually a continuous mixture of normal distribu- tions where the variance is a random variable. Blattberg and Gonedes 1974 prove
that if the variance of the normal follows an inverted gamma distribution then the resulting posterior distribution is the Student-t. Alternatively, the discrete mixture
of normal distributions model has more economic meanings. In fact, financial theory predicts that changes in the investment and financial decision variables, such
as financial and operating leverage of firms will result in adjustment to the expected return and standard deviation parameters of the distribution of a stock’s return.
Moreover, there are also information signal scenarios concerning the disclosure of a firm’s earnings that lead to parameter shifts. Seasonal announcements result in
return observations with higher variance during the disclosure period than during the non-announcement periods. In addition, other firm-specific information or
market-wide information will also change the parameters of the distribution of the stock’s return. After examining daily returns of 30 Dow Jones stocks, Kon 1984
concluded that the discrete mixture of normal distributions model has substantially more descriptive validity than the symmetric Student-t model.
Ball and Torous 1983 also derive and provide evidence consistent with a mixture of two normal distributions model resulting from a Bernoulli jump process
to describe information arrivals which will result in changes in parameters of the distribution of a stock’s return. If stochastic jumps are modeled by means of a
Poisson distribution, the resulting distribution is a discrete mixture of an infinite number of normal distributions. Ball and Torous 1985, Akgiray and Booth
1988, and Jorion 1988 all provide evidence for the Poisson – normal distribution model.
Obviously, both the GARCH specification and the jump process either the Bernoulli jump or the Poisson jumps can explain the leptokurtic behavior of the
series. Since the statistical and economic motivations for GARCH effects and jumps are quite different, we choose a model specification that can account for the
two components simultaneously. In the return process under the mixed model, after a jump has taken place, volatility will be high, but gradually it will return to normal
values when a new equilibrium is reached. If we did not include the GARCH model, the large volatility following a jump would mistakenly be taken for
additional jumps causing the jump intensity parameter to be overestimated.
In asset pricing, apart from the classical assumption of stock price dynamics used in financial models, alternative assumptions of the diffusion Gaussian processes
with time-varying parameters and discontinuous jump processes are also widely used to capture the observed skewness and leptokurtosis in stock price distribu-
tions. For example, Hull and White 1987 and Heston 1993 priced options under
the assumption of stochastic volatility. Duan 1995 developed a GARCH option pricing model. Cox and Ross 1976, Merton 1976 and Ahn and Thompson 1988
priced options assuming discontinuous jump processes for underlying assets. It is intuitively clear that, if jumps are present, the classical model may severely
overestimate the effectiveness of short-term hedging strategies based on dynamic portfolio adjustment. Similarly, the prices of out-of-the-money options that are
close to maturity will be underestimated if jumps are present that are not taken into account. Not only option prices are sensitive to jumps but also, as noted by Amin
1993, the optimal early-exercise decision for American options alters significantly when jumps are present.
In empirical studies, Pagan and Schwert 1990, Day and Lewis 1992, and Kim and Kon 1994 studied various models with conditional heteroscedasticity for
stock price dynamics. Bollerslev 1987, Nelson 1991, and Kim and Kon 1994 also examined GARCH models without assumption of conditional normality for
stock returns. Jarrow and Rosenfeld 1984, Ball and Torous 1985, and Akgiray and Booth 1986 all found evidence indicating the presence of jumps in the stock
price process. While Jorion 1988 combined the ARCH 1 model and the jump-diffusion process, conducted tests for the nested hypotheses, and found
systematic discontinuous jumps for stock prices even after allowing for conditional heteroscedasticity in the diffusion process. The problem is more important in
Taiwan, since as found by De Santis and Imrohoroglu 1997, emerging markets exhibit higher conditional volatility and conditional probability of large price
changes than mature markets. Thus incorporating discontinuous jumps and condi- tional heterscedasticity in the stock return process may be more meaningful.
Nieuwland et al. 1991 also followed Jorion’s methodology to analyze weekly DM rates. Since the ARCH 1 model is not appropriate to model most financial prices,
Vlaar and Palm 1993 and Drost et al. 1998 extended Jorion’s model by combining GARCH 1,1 process and the Poisson – normal distribution. Although
these Poisson – normal – GARCH models can incorporate both Poisson jumps and conditional heteroscedasticity in the process, they only allow the GARCH process
in the diffusion component. It is unfortunately inconsistent with actual observations in the structure of a GARCH model. In this study we will allow the GARCH
model in the whole stock return process rather than in the diffusion part only.
The purpose of this study is to study the distribution and conditional het- eroscedasticity in stock returns on the Taiwan stock market. Apart from the normal
distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson – normal, and
the mixed – normal distributions, which are essentially a mixture of normal distribu- tions, as conditional distributions in the stock return process. We also use the
ARMA 1,1 model to adjust the serial correlation, and adopt the GJR – GARCH 1,1 model to account for the conditional heterscedasticity that exists in the return
process. Extensive evidence was obtained by including a large number of individual stocks and the value-weighted stock index as research samples. Weekly returns from
January 1985 to May 1997 were examined. Maximum likelihood estimation MLE is used to estimate parameters in various models, and the likelihood ratio test is
used to test nested hypotheses. One of the testing hypotheses is that stock prices exhibit GARCH phenomena and leverage effects under whatever conditional
distribution is assumed. Another hypothesis is that normality is not an appropriate description for conditional distributions of stock prices.
The contribution of this study to the literature is the new GARCH models we use in this study, in particular the Poisson – normal – GARCH model which is modified
from Jorion 1988 and other extended models, and the mixed – normal – GARCH model which has never been found in previous literature. For practical purposes,
the estimation also provides information for understanding the stock behavior of the Taiwan stock market. This may have important implications for stock warrants,
and stock index futures markets in pricing, hedging, as well as trading.
The rest of the paper is organized as follows: Section 2 describes the models and methodology used in this paper. Section 3 describes the data and the statistical
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