where Z_{t} and b_{t} are sequences of independent, identically distributed random variables with zeromean, variance given by and respectively,

ω, α_{1}, β_{1} and Φ are real parameters, satisfying the following conditions, ω > 0, α_{1} ≥ 0, β_{1}_{ }≥ 0. |Φx| ≤ ω. Note:, and in order to calculate the kurtosis, we observe that. Then, we have the following moment properties

(2.14)

Proof:

where

Using the facts that

Thus, we have the following expression for the Kurtosis of the process.

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

Theorem 2.5. Suppose y_{t} is a modified RCA model with GARCH (p, q) innovations of the form

where b_{t} is an uncorrelated noise process with zero mean and with variance and Z_{t} is an uncorrelatednoise process with zero mean and with variance. Then, we have the following relationship

(2.17)

(2.18)

(2.19)

Proof: Let,

Now we have

Thus,

and

(2.20)

Using the fact that, we have

For convenience let where,

The Kurtosis of the process is given by

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 δ_{3} = 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 y_{t}, where

(2.24)

(2.25)

where and. Then, we have the following moment properties

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 y_{t}, 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

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)

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 S_{t} is the price of the stock, r is the risk-free interest rate, {W_{t}} is a standard Brownianmotion, σ_{t} is the time-varying RCA GARCH volatility process, {Z_{t}} 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 S_{t}, K is the strike price, T is the expiry date, σ_{t} is a stationary process with finite fourth momentand.

Also,

and

are given by:

where N denotes the standard normal CDF, and under the option pricing model with RCA GARCH volatility,

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.

REFERENCES

- T. Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, Vol. 31, No. 3, 1986, pp. 307-327. doi:10.1016/0304-4076(86)90063-1
- A. Thavaneswaran, S. S. Appadoo and M. Samanta, “Random Coefficient GARCH Models,” Mathematical and Computer Modelling, Vol. 41, No. 6-7, 2005, pp. 723- 733. doi:10.1016/j.mcm.2004.02.032
- A. Thavaneswaran, S. S. Appadoo and M. Ghahramani, “RCA Models with GARCH Errors,” Applied Mathematics Letters, Vol. 22, No. 1, 2009, pp. 110-114. doi:10.1016/j.aml.2008.02.015
- S. S. Appadoo, A. Thavaneswaran and S. Mandal, “RCA Models with Quadratic GARCH Innovations Distributions,” Applied Mathematics Letters, 2012, in Press.
- A. Paseka, S. Appadoo and A. Thavaneswaran, “Random Coefficient Autoregressive (RCA) Models with Nonlinear Stochastic Volatility Innovations,” Journal of Applied Statistical Science, Vol. 17, No. 3, 2010, pp. 331-349.
- A. Thavaneswaran and C. C. Heyde, “Prediction via Estimating Functions,” Journal of Statistical Planning and Inference, Vol. 77, No. 1, 1999, pp. 89-101. doi:10.1016/S0378-3758(98)00179-7
- D. Nicholls and B. Quinn, “Random Coefficient Autoregressive Models: An Introduction,” Lecture Notes in Statistics, Vol. 11, Springer, New York, 1982.
- R. Leipus and D. Surgallis, “Random Coefficient Autoregression,regime Switching and Lond Memory,” Advances in Applied Probability, Vol. 35, No. 3, 2003, pp. 737-754. doi:10.1239/aap/1059486826
- S. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 3, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585
- R. Elliot, T. Siu and L. Chan, “Option Pricing for GARCH Models with Markov Switching,” International Journal of Theoretical and Applied Finance, Vol. 9, No. 6, 2006, pp. 825-841. doi:10.1142/S0219024906003846
- H. Gong, A. Thavaneswaran and J. Singh, “A BlackScholes Model with GARCH Volatility,” The Mathematical Scientist, Vol. 35, No. 1, 2010, pp. 37-42.

NOTES

^{*}Corresponding author.