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This article seeks to model daily asset returns using log-ARCH-Lévy type model which is expected to reproduce most of the stylized features of financial time series data (such as volatility clustering, leptokurtic nature of log returns, joint covariance structure and aggregational Gaussianity) that are empirically found in different types of market. In addition, unconditional variance of daily log returns in risk neutral world of different conditional heteroscedastic models is derived. A key observation is that liquid markets and illiquid market may not have the same underlying dynamics. For instance empirical analysis based on S&P500 index log returns as a liquid market do not have autoregressive part in their first moments while in Nairobi Securities Exchange NSE20 index there is strong presence of autoregressive dynamics of order three,
* i.e.* AR(3). Higher moments of both markets are serially correlated.

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 [

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 [

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 [

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

where

where

where

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

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

subject to

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

the asymmetric innovations,

efficient to take the leverage effect into account. The properties of APARCH model have been studied, see [

・ ARCH model of [

・ GARCH model of [

・ GJR-GARCH Model of [

・ TARCH Model of [

Note that

For simplicity, we focus on daily closing indices

All return series exhibit strong conditional heteroscedasticity. The Ljung and Box test rejects the hypothesis of homoscedasticity at all common levels both for returns in S&P500 index and AR(3) residuals of linear re- gression in NSE20 share index. We estimate GARCH type models assuming conditional normality. With re- spect to the absolute value of parameter estimates, we find that

Suppose

NSE20 | S&P500 | |||
---|---|---|---|---|

Parameter | GARCH | GJR | GARCH | GJR |

0.18915 (0.024496) | 0.18136 (0.02424) | |||

0.16451 (0.023785) | 0.16245 (0.02352) | |||

0.11388 (0.023413) | 0.11516 (0.02308) | |||

0.03549 (0.006902) | 0.03458 (0.00647) | 0.006577 (0.001645) | 0.01088 (0.00204) | |

0.15023 (0.017978) | 0.18578 (0.02528) | 0.056461 (0.0067528) | 0.00322 (0.00512) | |

0.78763 (0.024753) | 0.79045 (0.02373) | 0.937566 (0.0074845) | 0.93202 (0.0079) | |

−0.07332 (0.02592) | 0.10558 (0.0123) | |||

9.3468 (0.2287) | 8.8337 (0.2648) | 16.5309 (0.08541) | 15.2862 (0.1220) | |

7.1689 (0.5739) | 8.46159 (0.38973) | 6.8918 (0.54835) | 5.9298 (0.6551) | |

lgl | −8363.5 | −8367.7 | −15090.9 | −15090.9 |

n | 2316 | 2316 | 4549 | 4549 |

Notes: standard errors are in parenthesis. lgl is the log likelihood.

Definition 2.1 The probability density function of the one-dimensional Generalized Hyperbolic distribution is given by the following:

where

The mean and variance of GH distribution are given respectively by the followings

and

where

For more information about GH distribution, see [

Let

physical probability measure and

and Lévy proces

Define daily log return as

where

In this section, we construct risk neutral probability measure in the context of [

Definition 3.1 A pricing measure

NSE20 | GH | HY | NIG | S&P500 | GH | HY | NIG |
---|---|---|---|---|---|---|---|

−1.79233 | 1.0000 | −0.5000 | 2.38336 | 1.0000 | −0.500 | ||

0.98225 | 1.15813 | 0.66862 | 0.14671 | 1.68640 | 1.33977 | ||

−0.05226 | −0.06604 | −0.05864 | −0.14279 | −0.14976 | −0.15755 | ||

1.79373 | 0.45207 | 1.18530 | 0.04052 | 1.04004 | 1.59588 | ||

0.12296 | 0.13923 | 0.13014 | 0.14292 | 0.15130 | 0.16032 |

almost surely with respect to measure

For some commonly used assumptions concerning utility functions and distributions of change of con- sumption, [

Consider the general model of daily log returns under the data generating probability measure

where the parameters

The pricing measure

The LRNVR implies that under the risk neutral measure

It follows quite easily that

The following propositions provide the unconditional variance for the process

Proposition 3.1 Consider AR(3) APARCH(1,1) Lévy filter, with

Proof: See Appendix. W

Proposition 3.2 A special case of AR(1)GARCH(1,1)Lévy filter the unconditional variance under the LRNVR equivalent measure

Proof: See Appendix. W

Example 3.1 In case of Hyperbolic distribution we substitute mean and variance respectively into (25). Where the parameters used maximize the likelihood function of Hyperbolic distribution. i.e. Let

Consider a discrete time economy, where interest rates and returns are paid after each time interval of equal spaced length. Suppose there is a price for risk, measured in terms of a risk premium that is added to the risk free interest rate r to build the expected next period return. As in Duan [

The parameters

hence the resulting TGARCH Lévy filter model

Proposition 3.3 The unconditional variance of the GARCH-M Lévy filter model under the LRNVR equivalent martingale measure

Proof: See Appendix. W

Proposition 3.4 The unconditional variance of the TGARCH-M Lévy filter model under equivalent martingale measure

where

and

Proof: See Appendix. W

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

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

Proof of proposition 3.3

It is a special case of proposition 3.4 when we take

Proof of proposition 3.4

Under measure

where

and

Therefore, for positive support