It is well known that the stock price changes are neither independent nor identically distributed. There are linear and nonlinear dependencies between successive price changes. Distributional assumptions concerning risky asset log returns play a key role in option pricing. According to research finding of Mandelbrot [1] , evidence indicates that the empirical distributions of daily stock returns differ significantly from the traditional Gaussian model. In search of satisfactory descriptive models for financial data, large number of distributions have been tried (see for example, [2] - [6] ).

The deviations from normality become more severe when more frequent data are used to calculate stock returns. Various studies have shown that the normal distribution does not accurately describe observed stock return data. Over the past several decades, some stylized facts have emerged about the statistical behavior of speculative market returns such as aggregational Gaussianity, volatility clustering, etc see [7] [8] . On the same note, most of the literature for example [9] - [12] and references therein, assume that daily log returns, can be modeled by exponential Lévy processes and geometric Lévy process.

There are two important directions in the literature regarding these type of stochastic volatility models. Continuous-time stochastic volatility process represented in general by a bivariate diffusion process, and the discrete time autoregressive conditionally heteroscedastic (ARCH) model of [13] or its generalization (GARCH) as first defined by [14] . Option pricing in GARCH models has been typically done using the local risk neutral valuation relationship (LRNVR) pioneered by [15] . The crucial assumptions in his construction are the con- ditional, normal distribution of the asset returns under the underlying probability space and the invariance of the conditional volatility to the change of measure. The empirical performance of these normal option pricing models has been studied extensively, for example in [16] , [17] .

The main focus of this paper is to develop a ARCH type Lévy model which attempts to capture some of the stylized features observed in demeaned log returns from any market data. More so we derive unconditional variance of daily log returns in risk neutral world of different ARCH type models, and an in-depth empirical study in liquid and illiquid market. All parameters are estimated from historical data, i.e. for S&P500 index from January 3, 1990 to January 18, 2008 and NSE20 index from March 2, 1998 to July 11, 2007.

The article is organized as follows. Section 2 provides a brief overview of ARCH type models and Lévy increments resulting to parameter estimation of observed salient features. In Section 3 which is our major con- tribution, unconditional variance of different ARCH type models is presented. Filtered Leptokurtic residuals of Lévy increments are calibrated. Conclusions are drawn in Section 4. Appendix is in the last section.

ARCH-type models are in general, discrete models used to estimate volatility of financial time series data such stock returns, interest rates and foreign exchange rates. Let

where denotes the price of stock at time. Define the following equation

where

where is the GARCH(p, q) volatility process. If then is ARCH(p). [18] and [19] provide a general specifications of volatility dynamic that nest most ARCH type models. In this connection volatility dynamics can be written as

where is the innovation function. Different GARCH models are mainly characterized by the following specifications of the innovation function.

The innovation function is used to model asymmetry and news impact to say the least. These GARCH models can be generalized to allow non-linearity of volatility dynamics by using Box-Cox transformation as follows

which implies modeling news and power, will nest most of the proposed GARCH models in Literature. Note that the leverage parameter shifts the innovation function, the news parameter tilts the innovation, and the power parameters and flatten or steepen the innovation function. Such a model (4) is the Asym- metric Power Autoregressive Conditional Heteroscedastic model i.e. APARCH model defined in (5).

The APARCH(m, n) model of can be written as follows

subject to for, , for. and

The model introduces a Box-Cox power transformation on the conditional standard deviation process and on

the asymmetric innovations, , adds flexibility of a varying exponent with an asymmetry co-

efficient to take the leverage effect into account. The properties of APARCH model have been studied, see [20] . The model nests seven other ARCH extensions as special cases.

・ ARCH model of [13] when, and

・ GARCH model of [14] when, and

・ GJR-GARCH Model of [21] when

・ TARCH Model of [22] when

Note that denote the conditional mean given the information set available at time t − 1. The innovation process for the conditional mean is then given by with corresponding unconditional variance and zero unconditional mean. The conditional variance is defined as

Let be a stochastic basis describing the uncertainty of the economy. We refer to as the

physical probability measure and represent the information flow driven by Brownian motion

and Lévy proces. Let be the price of a stock at time adapted to the natural filtration.

Define daily log return as It is well known from our empirical studies that can he represented as where is a mean function and are the two components of the error term. Moreover, define a order autoregressive process with APARCH(m,n) error as

where and are identically and independently distributed random variables. A general time series model for log returns would be

This article develops an log-ARCH-Lévy type risk neutral model. The proposed method delivers predictive dis- tribution of the payoff function for a given econometric model. As a result, the probability distribution could be useful to market participants who wish to compare the model predictions to the potential prices in liquid and illiquid markets.

Any effective option pricing model is expected to be consistent with distributional and time series properties of the underlying asset. The proposed model accommodates most of the observed stylistic fact about financial time series data i.e. skewness and leptokurtic nature of demeaned GARCH filtered log returns and perhaps aggregational Gaussianity. In summary,

・ developed markets and emerging markets may not have the same underlying dynamics. It would be incorrect to assume that a universal model for the underlying process for all markets.

・ The presence of linear autoregressive dynamics AR(3)-GARCH(1,1) effects in NSE20 index affects the un- conditional variance in risk neutral world. S&P500 index was found to follow GARCH(1,1) plus leptokurtic residual which was calibrated in one class of generalized hyperbolic distributions,say for example, Normal inverse Gaussian (NIG).

・ The presence of autoregressive dynamics, i.e. AR(3)-GARCH(1,1) model of NSE20 index as an example of illiquid market would have an impact in pricing options, if the index were to be used as an underlying process.

The log-ARCH-Lévy model is very tractable compared to other jump-diffusion or stochastic volatility models. It attempts to addresses the drawbacks of local volatilities. Further refinements and extensions are left for future research.

Comments from the Editor and the anonymous referee are acknowledged. Financial support from International Science Progam (Sweden)/EAUMP is greatly appreciated.

Proof of proposition 3.1

Given We note that and

after rearranging and simple algebra

Thus under stationarity, the unconditional expectations are independent of

Therefore, the unconditional variance of AR(3)GARCH(1,1)Levy filter model under LRNVR equivalent mar- tingale measure is

Proof of proposition 3.2

This is a special case of (3.1) with and.

Proof of proposition 3.3

It is a special case of proposition 3.4 when we take and

Proof of proposition 3.4

Under measure

where is the risk premium and

and denoting the cumulative standard normal distribution. Note that and

Therefore, for positive support