Financial theory states that an asset with high expected risk would, on average, pay a higher return (Xekalaki and Degiannakis 2010). This relationship between investors’ expected return and risk was measured by Engle et al. (1987) using auto-regressive conditional heteroscedasticity (ARCH) and is given as
$$ \left.\ \begin{array}{c}{y}_t={x}_t^{\prime}\beta +\varphi \left({\sigma}_t^2\right)+{\varepsilon}_t\ \\ {}\left.{\varepsilon}_t\right|{I}_{t-1}\sim f\left(0,{\sigma}_t^2\right),\\ {}{\sigma}_t^2=g\left({\sigma}_{t-1},{\sigma}_{t-2},\cdots, {\varepsilon}_{t-1},{\varepsilon}_{t-2},\cdots, {v}_{t-1},{v}_{t-2},\cdots \right),\end{array}\right\} $$
(1)
where xt is a k × 1 vector of endogenous and exogenous explanatory variables included in the set It − 1, and \( \varphi \left({\sigma}_t^2\right) \) represents the risk premium, which means the increase in the rate of return due to an increase in the variables of the returns.
Financial econometrics and financial time series analysis provides better understanding of how prices behave and how knowledge of price behavior can reduce risk or enhance better decision-making (Aas and Dimakos 2004). This is done using time series models for forecasting, option pricing and risk management. The remainder of this section focuses on some GARCH models and their extensions.
Autoregressive conditional Heteroskedasticity (ARCH) family model
Every ARCH or GARCH family model requires two distinct specifications, namely: the mean and the variance equations (Atoi 2014). The mean equation for a conditional heteroskedasticity in a return series, yt, is given as
$$ {y}_t={E}_{t-1}\left({y}_t\right)+{\varepsilon}_t $$
(2)
where εt = ϕtσt
The mean equation in eq. (2) also applies to other ARCH family models. Et − 1(.) is the expected value conditional on information available at time t − 1, while εt is the error generated from the mean equation at time t, and ϕt is the sequence of independent and identically distributed random variables with zero mean and unit variance. The variance equation for an ARCH(p) model is given as
$$ {\sigma}_t^2=\omega +{\alpha}_1{a}_{t-1}^2+\cdots +{\alpha}_p{a}_{t-p}^2 $$
(3)
It can be seen in the equation that large values of the innovation of asset returns exert a bigger impact on the conditional variance because they are squared, which means that a large shock tends to follow another large shock, similar to how clusters of the volatility behave. So, the ARCH(p) model becomes:
$$ {a}_t={\sigma}_t{\varepsilon}_t,\kern1.5em {\sigma}_t^2=\omega +{\alpha}_1{a}_{t-1}^2+\cdots +{\alpha}_p{a}_{t-p}^2 $$
(4)
where εt~N(0, 1) iid, ω > 0, and αi ≥ 0 for i > 0. In practice, εt is assumed to follow the standard normal or a standard student-t distribution or a generalized error distribution (Tsay 2005).
Asymmetric power ARCH
According to Rossi (2004), the asymmetric power ARCH model proposed by Ding et al. (1993) given below forms the basis for deriving the GARCH family models. Given that
$$ {\varepsilon}_t={\sigma}_t{\varepsilon}_t, $$
$$ {\varepsilon}_t\sim N\left(0,1\right) $$
$$ {\sigma}_t^{\delta }=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(\left|{a}_{t-i}\right|-{\gamma}_i{a}_{t-i}\right)}^{\delta }+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^{\delta } $$
(5)
where:
$$ \omega >0,\delta \ge 0. $$
$$ {\alpha}_i\ge 0\kern0.5em i=1,2,\cdots, p $$
$$ -1<{\gamma}_i<1\kern0.5em i=1,2,\cdots, p $$
$$ {\beta}_j>0\kern0.5em j=1,2,\cdots, q $$
This model imposes a Box-Cox transformation of the conditional standard deviation process and the asymmetric absolute residuals. The leverage effect is the asymmetric response of volatility to positive and negative “shocks.”
Standard GARCH(p,q) model
The mathematical model for the sGARCH(p,q) model is obtained from eq. (5) by letting δ = 2 and γi = 0, i = 1, 2, ⋯, p to be:
$$ {a}_t={\sigma}_t{\varepsilon}_t,\kern1em {\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{a}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2 $$
(6)
where at = rt − μt (rt is the continuous compounding log return series), and εt~N(0, 1) iid, the parameter αi is the ARCH parameter and βj is the GARCH parameter, and ω > 0, αi ≥ 0, βj ≥ 0, and \( {\sum}_{i=1}^{\max \left(p,q\right)}\left({\alpha}_i+{\beta}_j\right)<1 \), (Rossi 2004; Tsay 2005; Jiang 2012).
The restriction on ARCH and GARCH parameters (αi, βj) suggests that the volatility (ai) is finite and that the conditional standard deviation (σi) increases. It can be observed that if q = 0, then the model GARCH parameter (βj) becomes extinct and what is left is an ARCH(p) model.
To expatiate on the properties of GARCH models, the following representation is necessary,
Let \( {\eta}_t={a}_t^2-{\sigma}_t^2 \) so that \( {\sigma}_t^2={a}_t^2-{\eta}_t \). By substituting \( {\sigma}_{t-i}^2={a}_{t-i}^2-{\eta}_{t-i},\left(i=0,\dots, q\right) \) into Eq. (4), the GARCH model can be rewritten as
$$ {a}_t={\alpha}_0+\sum \limits_{i=1}^{\max \left(p,q\right)}\left({\alpha}_i+{\beta}_j\right){a}_{t-i}^2+{\eta}_t-\sum \limits_{j=1}^q{\beta}_j{\eta}_{t-j} $$
(7)
It can be seen that {ηt} is a martingale difference series (i.e., E(ηt) = 0 and cov(ηt, ηt − j) = 0, for j ≥ 1). However, {ηt} in general is not an iid sequence.
A GARCH model can be regarded as an application of the ARMA idea to the squared series \( {a}_t^2 \). Using the unconditional mean of an ARMA model results in
$$ \mathrm{E}\left({a}_t^2\right)=\frac{\alpha_0}{1-\sum \limits_{i=1}^{\max \left(p,q\right)}\left({\alpha}_i+{\beta}_j\right)} $$
provided that the denominator of the prior fraction is positive, (Tsay 2005). When p = 1 and q = 1, we obtain the GARCH(1,1) model given by
$$ {a}_t={\sigma}_t{\varepsilon}_t,\kern1.5em {\sigma}_t^2=\omega +{\alpha}_1{a}_{t-1}^2+{\beta}_1{\sigma}_{t-1}^2 $$
(8)
GJR-GARCH(p,q) model
The Glosten-Jagannathan-Runkle GARCH (GJRGARCH) model, which is a model that attempts to address volatility clustering in an innovation process, is obtained by letting δ = 2.
When δ = 2 and 0 ≤ γi < 1,
$$ {\displaystyle \begin{array}{c}{\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(\left|{\varepsilon}_{t-i}\right|-{\gamma}_i{\varepsilon}_{t-i}\right)}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2\\ {}=\omega +\sum \limits_{i=1}^p{\alpha}_i\left({\left|{\varepsilon}_{t-i}\right|}^2+{\gamma}_i^2{\varepsilon}_{t-i}^2-2{\gamma}_i\left|{\varepsilon}_{t-i}\right|{\varepsilon}_{t-i}\right)+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2\end{array}} $$
(9)
$$ {\sigma}_t^2=\left\{\begin{array}{c}\omega +\sum \limits_{i=1}^p{\alpha}_i^2{\left(1+{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2,{\varepsilon}_{t-i}<0\ \\ {}\ \\ {}\omega +\sum \limits_{i=1}^p{\alpha}_i^2{\left(1-{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2,{\varepsilon}_{t-i}>0\end{array}\right. $$
i.e.:
$$ {\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(1-{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{i=1}^p{\alpha}_i\left\{{\left(1+{\gamma}_i\right)}^2-{\left(1-{\gamma}_i\right)}^2\right\}{S}_i^{-}{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2 $$
$$ {\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(1-{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2+\sum \limits_{i=1}^p4{\alpha}_i{\gamma}_i{S}_i^{-}{\varepsilon}_{t-i}^2 $$
where: \( {S}_i^{-}=\left\{\begin{array}{c}1\ if\ {\varepsilon}_{t-i}<0\\ {}0\ if\ {\varepsilon}_{t-i}\ge 0\end{array}\right. \).
Now, we define
$$ {\alpha}_i^{\ast }={\alpha}_i{\left(1-{\gamma}_i\right)}^2\kern0.24em and\kern0.24em {\gamma}_i^{\ast }=4{\alpha}_i{\gamma}_i $$
then
$$ {\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(1-{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2+\sum \limits_{i=1}^p{\gamma}_i^{\ast }{S}_i^{-}{\varepsilon}_{t-i}^2 $$
(10)
which is the GJRGARCH model (Rossi 2004).
However, when −1 ≤ γi < 0, then recall Eq. (9)
$$ {\displaystyle \begin{array}{c}{\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(\left|{\varepsilon}_{t-i}\right|-{\gamma}_i{\varepsilon}_{t-i}\right)}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2\\ {}=\omega +\sum \limits_{i=1}^p{\alpha}_i\left({\left|{\varepsilon}_{t-i}\right|}^2+{\gamma}_i^2{\varepsilon}_{t-i}^2-2{\gamma}_i\left|{\varepsilon}_{t-i}\right|{\varepsilon}_{t-i}\right)+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2\end{array}} $$
$$ {\sigma}_t^2=\left\{\begin{array}{c}\omega +\sum \limits_{i=1}^p{\alpha}_i^2{\left(1-{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2,{\varepsilon}_{t-i}>0\ \\ {}\ \\ {}\omega +\sum \limits_{i=1}^p{\alpha}_i^2{\left(1+{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2,{\varepsilon}_{t-i}<0\end{array}\right. $$
$$ {\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(1+{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2+\sum \limits_{i=1}^p{\alpha}_i\left\{{\left(1+{\gamma}_i\right)}^2-{\left(1-{\gamma}_i\right)}^2\right\}{S}_i^{+}{\varepsilon}_{t-i}^2 $$
$$ =\omega +\sum \limits_{i=1}^p{\alpha}_i{\left(1+{\gamma}_i\right)}^2{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2+\sum \limits_{i=1}^p{\alpha}_i\left\{1+{\gamma}_i^2-2{\gamma}_i-1-{\gamma}_i^2-2{\gamma}_i\right\}{S}_i^{+}{\varepsilon}_{t-i}^2 $$
where: \( {S}_i^{+}=\left\{\begin{array}{c}1\ if\ {\varepsilon}_{t-i}>0\\ {}0\ if\ {\varepsilon}_{t-i}\le 0\end{array}\right. \).
Also define
$$ {\alpha}_i^{\ast }={\alpha}_i{\left(1+{\gamma}_i\right)}^2\kern0.24em and\kern0.24em {\gamma}_i^{\ast }=-4{\alpha}_i{\gamma}_i $$
then
$$ {\sigma}_t^2=\omega +\sum \limits_{i=1}^p{\alpha}_i^{\ast }{\varepsilon}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2+\sum \limits_{i=1}^p{\gamma}_i^{\ast }{S}_i^{+}{\varepsilon}_{t-i}^2 $$
(11)
which allows positive shocks to have a stronger effect on volatility than negative shocks (Rossi 2004). However, when p = q = 1, the GJRGARCH(1,1) model will be written as
$$ {\sigma}_t^2=\omega +\alpha {\varepsilon}_t^2+\gamma {S}_i{\varepsilon}_{t-1}^2+\beta {\sigma}_{t-1}^2 $$
(12)
IGARCH(1,1) model
Integrated GARCH (IGARCH) models are unit-root GARCH models. The IGARCH(1,1) model is specified in Tsay (2005) as
$$ {a}_t={\sigma}_t{\varepsilon}_t; $$
$$ {\sigma}_t^2={\alpha}_0+{\beta}_1{\sigma}_{t-1}^2+\left(1-{\beta}_1\right){a}_{t-1}^2 $$
(13)
where
εt~N(0, 1) iid, and 0 < β1 < 1. Ali (2013) used αi to denote 1 − βi. The model is also an exponential smoothing model for the \( \left\{{a}_t^2\right\} \) series. To see this, we rewrite the model as
$$ {\displaystyle \begin{array}{c}{\sigma}_t^2=\left(1-{\beta}_1\right){a}_{t-1}^2+{\beta}_1{\sigma}_{t-1}^2\\ {}=\left(1-{\beta}_1\right){a}_{t-1}^2+{\beta}_1\left[\left(1-{\beta}_1\right){a}_{t-2}^2+{\beta}_1{\sigma}_{t-1}^2\right]\\ {}=\left(1-{\beta}_1\right){a}_{t-1}^2+\left(1-{\beta}_1\right){\beta}_1{a}_{t-2}^2+{\beta}_1^2{\sigma}_{t-2}^2\end{array}} $$
(14)
By repeated substitution, we obtain
$$ {\sigma}_t^2=\left(1-{\beta}_1\right)\left({a}_{t-1}^2+{\beta}_1{a}_{t-1}^2+{\beta}_1^2{a}_{t-3}^3+\cdots \right) $$
(15)
which is a well-known exponential smoothing formation in which β1 is the discounting factor (Tsay 2005).
TGARCH(p,q) model
The threshold GARCH model is another model used to handle leverage effects, and a TGARCH(p,q) model is given by the following:
$$ {\sigma}_t^2={\alpha}_0+\sum \limits_{i=1}^p\left({\alpha}_1+{\gamma}_i{N}_{t-i}\right){a}_{t-i}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2 $$
(16)
where Nt − i is an indicator for negative at − i; that is,
$$ {N}_{t-i}=\left\{\begin{array}{c}1\ \mathrm{if}\ {a}_{t-i}<0,\\ {}0\ \mathrm{if}\ {a}_{t-i}\ge 0,\end{array}\right. $$
and αi, γi and βj are nonnegative parameters satisfying conditions similar to those of GARCH models, (Tsay 2005). When p = 1, q = 1, the TGARCH(1,1) model becomes:
$$ {\sigma}_t^2=\omega +\left(\alpha +\gamma {N}_{t-1}\right){a}_{t-1}^2+\beta {\sigma}_{t-1}^2 $$
(17)
NGARCH(p,q) model
The nonlinear GARCH model has been presented variously in the literature by the following scholars: Hsieh and Ritchken (2005), Lanne and Saikkonen (2005), Malecka (2014) and Kononovicius and Ruseckas (2015). The following model can be shown to represent all representations:
$$ {h}_t=\omega +\sum \limits_{i=1}^q{\alpha}_i{\varepsilon}_{t-i}^2+\sum \limits_{i=1}^q{\gamma}_i{\varepsilon}_{t-i}+\sum \limits_{j=1}^p{\beta}_j{h}_{t-j} $$
(18)
where ht is the conditional variance, and ω, β and α satisfy ω > 0, β ≥ 0 and α ≥ 0.
This can also be written as
$$ {\sigma}_t=\omega +\sum \limits_{i=1}^q{\alpha}_i{\varepsilon}_{t-i}^2+\sum \limits_{i=1}^q{\gamma}_i{\varepsilon}_{t-i}+\sum \limits_{j=1}^p{\beta}_j{\sigma}_{t-j} $$
(19)
The EGARCH model
The exponential GARCH (EGARCH) model was proposed by Nelson (1991) to overcome some weaknesses of the GARCH model in handling financial time series, as pointed out by Enocksson and Skoog (2012). In particular, to allow for asymmetric effects between positive and negative asset returns, he considered the weighted innovation:
$$ g\left({\varepsilon}_t\right)=\theta {\varepsilon}_t+\gamma \left[\left|{\varepsilon}_t\right|-\mathrm{E}\left(\left|{\varepsilon}_t\right|\right)\right] $$
(20)
where θ and γ are real constants. Both εt and |εt| − E(|εt|) are zero-mean iid sequences with continuous distributions. Therefore, E[g(εt)] = 0. The asymmetry of g(εt) can easily be seen by rewriting it as
$$ g\left({\varepsilon}_t\right)=\left\{\begin{array}{c}\left(\theta +\gamma \right){\varepsilon}_t-\gamma E\left(\left|{\varepsilon}_t\right|\right)\ if\ {\varepsilon}_t\ge 0,\\ {}\left(\theta -\gamma \right){\varepsilon}_t-\gamma E\left(\left|{\varepsilon}_t\right|\right)\ if\ {\varepsilon}_t<0.\end{array}\right. $$
(21)
An EGARCH(m,s) model, according to Tsay (2005),Dhamija and Bhalla (2010), Jiang (2012), Ali (2013) and Grek (2014), can be written as
$$ {a}_t={\sigma}_t{\varepsilon}_t, $$
$$ \ln \left({\sigma}_t^2\right)=\omega +\sum \limits_{i=1}^s{\alpha}_i\frac{\left|{a}_{t-i}\right|+{\theta}_i{a}_{t-i}}{\sigma_{t-i}}+\sum \limits_{j=1}^m{\beta}_j\ln \left({\sigma}_{t-i}^2\right) $$
(22)
which specifically results in EGARCH(1,1) being written as
$$ {a}_t={\sigma}_t{\varepsilon}_t $$
$$ \ln \left({\sigma}_t^2\right)=\omega +\alpha \left(\left[\left|{a}_{t-1}\right|-\mathrm{E}\left(\left|{a}_{t-1}\right|\right)\right]\right)+\theta {a}_{t-1}+\beta \ln \left({\sigma}_{t-1}^2\right) $$
(23)
where |at − 1| − E(|at − 1|) are iid and mean zero. When the EGARCH model has a Gaussian distribution of error term, then \( \mathrm{E}\left(\left|{\varepsilon}_t\right|\right)=\sqrt{2/\pi } \), which gives:
$$ \ln \left({\sigma}_t^2\right)=\omega +\alpha \left(\left[\left|{a}_{t-1}\right|-\sqrt{2/\pi}\right]\right)+\theta {a}_{t-1}+\beta \ln \left({\sigma}_{t-1}^2\right) $$
(24)
The AVGARCH model
An asymmetric GARCH (AGARCH), according to Ali (2013), is simply
$$ {a}_t={\sigma}_t{\varepsilon}_t; $$
$$ {\sigma}^2=\omega +\sum \limits_{t=1}^p{\alpha}_i{\left|{\varepsilon}_{t-i}-b\right|}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2 $$
(25)
while the absolute value GARCH (AVGARCH) model is specified as
$$ {a}_t={\sigma}_t{\varepsilon}_t; $$
$$ {\sigma}^2=\omega +\sum \limits_{t=1}^p{\alpha}_i{\left(\left|{\varepsilon}_{t-i}+b\right|-c\left({\varepsilon}_{t-i}+b\right)\right)}^2+\sum \limits_{j=1}^q{\beta}_j{\sigma}_{t-j}^2 $$
(26)
The N(a)GARCH or NAGARCH model
The nonlinear (Asymmetric) GARCH (NAGARCH or N(A)GARCH) model plays key role in option pricing with stochastic volatility because, as we shall see later on, NAGARCH allows one to derive closed-form expressions for European option prices in spite of rich volatility dynamics. Because a NAGARCH may be written as
$$ {\sigma}_{t+1}^2=\omega +\alpha {\sigma}_t^2{\left({z}_t-\delta \right)}^2+\beta {\sigma}_t^2 $$
(27)
If zt~IIDN(0, 1), zt is independent of \( {\sigma}_t^2 \) as \( {\sigma}_t^2 \) is only a function of an infinite number of past-squared returns, it is possible to easily derive the long run, unconditional variance under NGARCH and the assumption of stationarity:
$$ {\displaystyle \begin{array}{c}E\left[{\sigma}_{t+1}^2\right]={\overline{\sigma}}^2=\omega +\alpha E\left[{\sigma}_t^2{\left({z}_t-\delta \right)}^2\right]+\beta E\left[{\sigma}_t^2\right]E\left[{\sigma}_{t\_+1}^2\right]={\overline{\sigma}}^2\\ {}=\omega +\alpha E\left[{\sigma}_t^2{\left({z}_t-\delta \right)}^2\right]+\beta E\left[{\sigma}_t^2\right]\\ {}\begin{array}{c}=\omega +\alpha E\left[{\sigma}_t^2\right]E\left({z}_t^2+{\delta}^2-2\delta {z}_t\right)+\beta E\left[{\sigma}_t^2\right]\\ {}=\omega +\alpha {\overline{\sigma}}^2\left(1+{\delta}^2\right)+\beta {\overline{\sigma}}^2\end{array}\end{array}} $$
(28)
where \( {\overline{\sigma}}^2=E\left[{\sigma}_t^2\right] \) and \( E\left[{\sigma}_t^2\right]=E\left[{\sigma}_{t+1}^2\right] \) because of stationarity. Therefore,
$$ {\overline{\sigma}}^2\left[1-\alpha \left(1+{\delta}^2\right)+\beta \right]=\omega \Rightarrow {\overline{\sigma}}^2=\frac{\omega }{1-\alpha \left(1+{\delta}^2\right)+\beta } $$
(29)
which exists and is positive if, and only if, α(1 + δ2) + β < 1. This has two implications
- (i)
the persistence index of a NAGARCH(1,1) is α(1 + δ2) + β and not simply α + β; n and
- (ii)
an NAGARCH(1,1) model is stationary if, and only if, α(1 + δ2) + β < 1.
Further details about these implications can be found in Nelson (1991), Hall and Yao (2003), Enders (2004), Christoffersen et al. (2008) and Engle and Rangel (2008).
