1. Introduction
It is well-known that many financial time series such as stock returns exhibit leptokurtosis and time-varying volatility [1]. The generalized autoregressive conditional heteroscedasticity (GARCH) and the random coefficient autoregressive (RCA) models have been extensively used to capture the time-varying behaviour of the volatility. Studies using GARCH models commonly assume that the time series is conditionally normally distributed; however, the kurtosis implied by the normal GARCH tends to be lower than the sample kurtosis observed in many time series Bollerslev [1]. Thavaneswaran et al. [2] use an ARMA representation to derive the kurtosis of various classes of GARCH models such as power GARCH, nonGaussian GARCH, non-stationary and random coefficient GARCH. Recently, Thavaneswaran et al. [3], Appadoo et al. [4] have extended the results to stationary RCA processes with GARCH errors and Paseka et al. [5] further extended the results to RCA processes with stochastic volatility (SV) errors.
Leptokurtosis is commonly observed in financial time series, as well as in currency and commodity markets. The opening and closure of the markets, time-of-the-day and day-of-the-week effects, weekends and vacation periods cause changes in the trading volume that translates into regular changes in price variability. Financial, currency, and commodity data also respond to new information entering into the market, which usually have large kurtosis. Recently, there has been growing interest in using volatility models [3,4]. Most of the studies use GARCH models with dummy variables in the volatility equation, and a few of them have been extended to a more flexible form such as the RCA GARCH. However, even though much research has been performed on volatility models applied to market data such as stock returns, more general specifications accounting for RCA with GARCH errors have been little explored. First we derive the kurtosis of a simple time series model with behaviour in the mean. Then we introduce various classes of RCA GARCH models and study the moments and discuss applications in option pricing. We extend the results for RCA GARCH volatility models to RCA quadratic GARCH models. The RCA GARCH model is appropriate for time series where significant autocorrelation exists. Option pricing with RCA model with quadratic GARCH errors is also discussed in some detail. The moments derived for the RCA GARCH volatility models provide more accurate estimates of market data behaviour and help investors, decision makers, and other market participants develop improved trading strategies. The rest of the paper is organized as follows. In rest of Section 1, we present results on standard GARCH models. These results are interesting for their own sake. In Section 2, we derive the higher order moments of some RCA models with GARCH errors, and in Section 3 we discuss some option pricing applications with RCA models with GARCH errors.
GARCH Models
Consider the general class of GARCH (P,Q) model for the time series yt, where
(1.1)
(1.2)
where Zt is a sequence of independent, normally distributed random variables with zero mean, unit variance. Let
be the martingale difference and let
be the variance of
, (2.12) and (1.2) could be written as:
(1.3)
(1.4)
(1.5)
where
![](https://www.scirp.org/html/2-1490056\1dd7f0f1-f87b-41d2-98f9-744be810c51e.jpg)
and R = max(P,Q).We shall make the following stationarity assumptions for
which has an ARMA(R,Q) representation.All the zeroes of the polynomial Φ(B) lie outside of the unit circle.
where the ![](https://www.scirp.org/html/2-1490056\a909a626-32c8-4eb1-a88d-dd56a4af2ce7.jpg)
are obtained from the relation
with
. The assumption sensure that the
are uncorrelated with zero mean and finite variance and that the
process is weakly stationary. In this case, the autocorrelation function of
will be exactly the same as that for astationary ARMA(R,Q) model. For any random variable ε with finite fourth moments, the kurtosis defined by
and if the process {Zt}
is normal then the process {εt} defined by equations (1.3) and (1.4) is called a normal GARCH (p, q) process. The kurtosis ofthe GARCHprocess is denoted by
when it exists.
2. Random Coefficient Volatility Models
Consider the class of random coefficient autoregressive (RCA) models defined by allowing random additive perturbations of the autoregressive (AR) coefficients of ordinary AR models. That is, we assume that the process yt is given by,
(2.1)
where the parameters θi, i = 2, ···, p, are assumed to be known,
and
are zero mean square integrable independent processes and the variances are denoted by
and
.
are independent of
and
and may be thought of as incorporating structural changes. In order to motivate nonlinear forecasts for nonlinear models, we consider a class of estimating functions of the form
[6], where
and
is a function of
and possibly the known parameters
(i.e. We assume that the fitted model is available). If we restrict ourselves to a class of estimating functions of the above form then we can forecast the future value of
based on the observed values
as
. That iswhether we have an AR(p) model or RCA(p) model we will get the same linear predictor of
. However, for the RCA model under consideration, we have
and
.
Thus, the conditional variance is a nonlinear function and hence the RCA model may be viewed as a non-linear time series model. Nicholls and Quinn [7] studied linear as well as some nonlinear (proposed) forecast by fitting a nonlinear (RCA) model for the classical lynx cycle data. Using heuristic reasoning they proposed a nonlinear forecast and
theyshowed empirically that the forecast
is a better predictor (having smalller forecast errors when compared with the actual observations) than the linear forecast for the lynx data. It is of interest to note that by defining
, the optimal forecast for
can be obtained as
.
That is, the estimating function method can be used to obtain a nonlinear forecast for a nonlinear models by considering a class of elementary martingale estimating functions generated by nonlinear functions of the observations. Using a similar argument we could also obtain forecasts for various class of GARCH models, see Thavaneswaran and Heyde [6] for details. The main message is RCA models could be used to improve the forecasting performance of stochastic volatility models.
Lemma 2.1. When Zt is a standard normal random variable such that Zt ∼ N(0,1) then,
Now if
then
As a general case for even powers we have ![](https://www.scirp.org/html/2-1490056\af71f475-d9fc-4400-b46a-0894875b8f57.jpg)
2.1. RCA Models
Random coefficient autoregressive time series were introduced by Nicholls and Quinn [10] and some of their properties have been studied recently by Thavaneswaran [3]. RCA models exhibiting long memory properties have been considered in Leipus and Sugailis [8]. A sequence of random variables {yt} is called an RCA (1) time series if it satisfies the equations
![](https://www.scirp.org/html/2-1490056\640d2b51-5cfa-4f35-bc34-cf4d9e54be37.jpg)
where Z denotes the set of integers and 1) ![](https://www.scirp.org/html/2-1490056\c42f67e4-6593-46bc-8002-f52063323482.jpg)
2) ![](https://www.scirp.org/html/2-1490056\b0bf2e9d-9ae9-4939-978a-8fe5eaf44327.jpg)
The sequences {bt} and {εt} respectively, are the errors in the model.
Theorem 2.1. Let {yt} be a modified RCA (1) time series with an absolute value random coefficient satisfying conditions (1) and (2). The modified RCA (1) model is given by
(2.2)
Then we have the following 1) ![](https://www.scirp.org/html/2-1490056\08efa272-a678-4678-aa14-b784c488d19e.jpg)
2) ![](https://www.scirp.org/html/2-1490056\8343138c-4ae0-42a3-930b-8d10b78c16fd.jpg)
3) ![](https://www.scirp.org/html/2-1490056\d1655973-9423-4528-a6f4-add3617db7e8.jpg)
The autocovariance and the autocorrelation functions are given by
and ![](https://www.scirp.org/html/2-1490056\45cffdea-e52b-4fd9-9f87-f4051ea16edc.jpg)
wang#title3_4:spProof:
![](https://www.scirp.org/html/2-1490056\a6c1a676-62fd-4450-a383-6e2745d7ff9e.jpg)
Thus, we have
(2.3)
and we have
![](https://www.scirp.org/html/2-1490056\35e5dd65-1dc8-4b1f-aec5-56e4e13b8275.jpg)
(2.4)
When
, the kurtosis of the process yt converge to K(y) = 3. Thus, the autocorrelation function is given by
![](https://www.scirp.org/html/2-1490056\c7066ae7-cd5f-4d3f-802b-7ba219a96a02.jpg)
where we use the fact that ρ0 = 1.
Theorem 2.2. Suppose yt is an Random Coefficient Moving average process model of the form
(2.5)
where bt is an uncorrelated Gaussian process with zero mean and with variance
. εt is an uncorrelated Gaussian process with zero mean and with variance
. Then, we have the following
Proof:
(2.6)
and
![](https://www.scirp.org/html/2-1490056\3dfef17d-aca1-48a1-8d1c-b05524d415a7.jpg)
We have
(2.7)
and
![](https://www.scirp.org/html/2-1490056\307c8479-01a1-428a-b7f4-9d784ed9c35c.jpg)
Thus we have
![](https://www.scirp.org/html/2-1490056\f7381673-fe17-4f53-acbc-a6f3bfd270a7.jpg)
The kurtosis of the process is given by
(2.8)
When
, the kurtosis of the process yt converge to
and when
, and
the kurtosis of the process yt turns out to be 35.
Theorem 2.3. Suppose yt is an Random Coefficient Moving average process model of the form
(2.9)
where bt is an uncorrelated noise process with zero mean and with variance
. εt is an uncorrelated noise process with zero mean and with variance
. Then, we have the following relationships
![](https://www.scirp.org/html/2-1490056\1abe3bb0-2218-451c-bbf3-ae00600adb69.jpg)
and the autocorrelation functions are given by
(2.10)
wang#title3_4:spProof:
![](https://www.scirp.org/html/2-1490056\ddc20b93-e455-4d17-83e4-1effb7448848.jpg)
![](https://www.scirp.org/html/2-1490056\0828477c-87b5-4fb6-a9bb-1db7722728db.jpg)
Thus,
![](https://www.scirp.org/html/2-1490056\2f382e23-945c-4f32-8e3c-aa0657a10c64.jpg)
The autocorrelation function is given by
![](https://www.scirp.org/html/2-1490056\bd18bdf6-8f37-477c-b208-86e86c6db2cf.jpg)
Theorem 2.4. Let {yt} be a Sign RCA-GARCH (1,1) time series satisfying conditions (i) and (ii) given by
(2.11)
where
(2.12)
(2.13)
where Zt and bt are sequences of independent, identically distributed random variables with zeromean, variance given by
and
respectively,
![](https://www.scirp.org/html/2-1490056\33c83c07-912f-4e8e-81a6-eb2d639cf382.jpg)
ω, α1, β1 and Φ are real parameters, satisfying the following conditions, ω > 0, α1 ≥ 0, β1 ≥ 0. |Φx| ≤ ω. Note:
, and in order to calculate the kurtosis, we observe that
. Then, we have the following moment properties
![](https://www.scirp.org/html/2-1490056\1852d517-6892-41d6-9221-848ff4fc5919.jpg)
![](https://www.scirp.org/html/2-1490056\798a9bda-c4f5-4609-9ea6-52a5acfd9d60.jpg)
(2.14)
Proof:
![](https://www.scirp.org/html/2-1490056\e4a032a2-ef16-463d-ac2c-5c9b5d45a8c2.jpg)
![](https://www.scirp.org/html/2-1490056\7e6b3278-d3a6-4154-a44d-24cba44c1639.jpg)
![](https://www.scirp.org/html/2-1490056\f0b3c5dc-f0b8-4384-ba88-51784db7b599.jpg)
![](https://www.scirp.org/html/2-1490056\41051afc-0e1d-4925-9a42-a323714dd1da.jpg)
where
![](https://www.scirp.org/html/2-1490056\9f1e6a7d-3b41-4f7d-9dfb-93266f1fad68.jpg)
![](https://www.scirp.org/html/2-1490056\25d13cf3-3902-46df-8733-6e0a0541b799.jpg)
![](https://www.scirp.org/html/2-1490056\b58f0ed4-9fac-4aa1-b535-eba9178d069d.jpg)
Using the facts that
![](https://www.scirp.org/html/2-1490056\da85c293-9f63-4bed-b750-652c73242895.jpg)
![](https://www.scirp.org/html/2-1490056\14b9ea14-9adb-4bec-8560-64ee79c328ea.jpg)
Thus, we have the following expression for the Kurtosis of the process.
![](https://www.scirp.org/html/2-1490056\71bc65f2-744b-41dd-a5b4-673b54a6be5c.jpg)
Special cases, for a Normal GARCH (1,1).
(2.15)
Note, that when
, and
in (2.15), the kurtosis of the process converges to
(2.16)
When
in (2.16),the kurtosis of the process converge to ![](https://www.scirp.org/html/2-1490056\c4fcd015-7009-4392-a160-a2b945803b19.jpg)
Theorem 2.5. Suppose yt is a modified RCA model with GARCH (p, q) innovations of the form
![](https://www.scirp.org/html/2-1490056\c3fcba28-bb66-47ef-9185-560bd5ef514b.jpg)
where bt is an uncorrelated noise process with zero mean and with variance
and Zt is an uncorrelatednoise process with zero mean and with variance
. Then, we have the following relationship
(2.17)
(2.18)
(2.19)
Proof: Let, ![](https://www.scirp.org/html/2-1490056\0fc66b23-3fd2-4703-934b-c6de0222ef3f.jpg)
![](https://www.scirp.org/html/2-1490056\4fab2e7f-0028-4bd1-96e0-f23024f285a2.jpg)
![](https://www.scirp.org/html/2-1490056\b27b0faa-485e-4370-80eb-d3666f845abe.jpg)
Now we have
![](https://www.scirp.org/html/2-1490056\44ff80e7-7842-4b30-94d7-4c27dbf01f48.jpg)
Thus,
![](https://www.scirp.org/html/2-1490056\8b857b33-4620-41ab-8857-51ac7065911b.jpg)
and
![](https://www.scirp.org/html/2-1490056\38aeb32a-d5f7-4d5e-b377-b27c72258f10.jpg)
(2.20)
![](https://www.scirp.org/html/2-1490056\1ea90e84-7b42-453b-af80-10ac18b7e4d8.jpg)
Using the fact that
, we have
![](https://www.scirp.org/html/2-1490056\f0c89932-411b-441e-a4aa-ac844d9eba06.jpg)
For convenience let
where,
![](https://www.scirp.org/html/2-1490056\e99464ba-9ef3-4d90-bc49-550bec428c03.jpg)
The Kurtosis of the process is given by
![](https://www.scirp.org/html/2-1490056\1ad46a29-77ac-4c32-809c-8e0834047010.jpg)
Using the fact that
we have
(2.21)
2.2. Quadratic GARCH Model
Besides having excess kurtosis market returns may display seriously skewed distributions. Linear GARCH models cannot cope with such skewness, and therefore we can expect forecast of linear GARCH model to be biased for skewed time series. To deal with this problem non-linear GARCH models are introduced, which take into account skewed distributions. The QGARCH model differs from model the classical GARCH model by
(2.22)
(2.23)
This model reduces to the GARCH (1,1) model when the shift parameters δ3 = 0. The QGARC Hmodel can improve upon the standard GARCH since they can cope with positive (or negative) skewness.
Theorem 2.6. Consider the general class of RCA QGARCH (1,1) Volatility Models for the time series yt, where
(2.24)
![](https://www.scirp.org/html/2-1490056\f6dce6bb-ead1-4831-b7e2-1a5046b5edc0.jpg)
(2.25)
where
and
. Then, we have the following moment properties
![](https://www.scirp.org/html/2-1490056\cebee3bc-09c0-49cc-a5e7-b6f177b7267a.jpg)
![](https://www.scirp.org/html/2-1490056\026715ba-a7fa-4390-8ee5-49cc5c8d2d32.jpg)
Proof: is easy and is omitted.
Theorem 2.7. Consider the special class of RCA QGARCH (1,1) Sign Volatility Models for the time series yt, where
(2.26)
(2.27)
(2.28)
where
, at ∼N(0,σ2a) and
are sequences of independent, identicallydistributed random variables with zero mean, variance given by
and
and
respectively, and
![](https://www.scirp.org/html/2-1490056\9ef47f22-e516-4982-8f2c-11dd64d0d5b3.jpg)
Note:
, and in order to calculate the kurtosis, we observe that
. Then, we have the following moment properties
(2.29)
(2.30)
(2.31)
(2.32)
where
(2.33)
(2.34)
(2.35)
![](https://www.scirp.org/html/2-1490056\19362ca2-1d58-4176-b0c0-13f785ca7789.jpg)
Proof: is easy and is omitted.
3. Option Pricing with Volatility
Option pricing based on the Black-Scholes model is widely used in the financial community. The BlackScholes formula is used for the pricing of European-style options. The model has traditionally assumed that the volatility of returns is constant. However, several studies have shown that assetre turns exhibit variances that change time [9,10,] and others derived closed form option pricing formulas for different models which are assumed to follow a GARCH volatility process. Most recently, Gong et al. [11] derive an expression for the call price as an expectation with respect to random GARCH volatility. The model is then evaluated in terms of the moments of the volatility process. Their results indicate that the suggested model outperforms the classic BlackScholes formula. Here we apply [11] and propose an option pricing model with RCA GARCH volatility as follows:
(3.1)
(3.2)
(3.3)
where St is the price of the stock, r is the risk-free interest rate, {Wt} is a standard Brownianmotion, σt is the time-varying RCA GARCH volatility process, {Zt} is a sequence of i.i.d. randomvariables with zero mean and unit variance and Φ(B), and β(B) have been defined in (1.5). Theprice of a call option can be calculated using the option pricing formula given in [11]. The call priceis derived as a first conditional moment of a truncated lognormal distribution under the martingalemeasure, and it is based on estimates of the moments of the GARCH process. The call price basedon the Black-Scholes model with seasonal GARCH volatility is given by:
(3.4)
where f and g are twice differentiable functions, S is the initial value of St, K is the strike price, T is the expiry date, σt is a stationary process with finite fourth momentand
.
Also,
![](https://www.scirp.org/html/2-1490056\2790d2ba-56a5-421e-bb0f-a1c1791978ce.jpg)
and
![](https://www.scirp.org/html/2-1490056\4c285ea3-215d-4cfb-b234-119d03559b70.jpg)
are given by:
![](https://www.scirp.org/html/2-1490056\bff43ad8-caab-4d0e-b0e3-5e5229bc2efe.jpg)
![](https://www.scirp.org/html/2-1490056\88c092e8-3e9f-4af3-9908-85b7c64bec8b.jpg)
![](https://www.scirp.org/html/2-1490056\d31b1671-bdd1-4422-aaf5-ad184b54c063.jpg)
where N denotes the standard normal CDF, and under the option pricing model with RCA GARCH volatility,
![](https://www.scirp.org/html/2-1490056\a6666610-5b4f-4151-a4b8-6b4023a3313c.jpg)
4. Concluding Remarks
Financial time series exhibit excess kurtosis and in this paper, we propose various classes of RCAGARCH volatility models and derive the kurtosis in terms of model parameters. We consider time series models such as RCA with GARCH errors and quadratic GARCH errors. The models introduced here extend and complement the existing volatility models in the literature to RCA models with quadratic GARCH models by introducing more general structures. The results are primarily oriented to financial time series applications. Financial time series often meet the large data set demands of the volatility models studied here. Also, financial data dynamics and higher order moments are of interest to many market participants. Specifically, we consider the Black-Scholes model with RCA GARCH volatility and show that these moments can be used to evaluate the call price for European options.
NOTES