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In this paper, a general Non-Gaussian Stochastic Volatility model is proposed instead of the usual Gaussian model largely studied. We consider a new specification of SV model where the innovations of the return process have centered non-Gaussian error distribution rather than the standard Gaussian distribution usually employed. The model describes the behaviour of random time fluctuations in stock prices observed in the financial markets. It offers a response to better model the heavy tails and the abrupt changes observed in financial time series. We consider the Laplace density as a special case of non-Gaussian SV models to be applied to our data base. Markov Chain Monte Carlo technique, based on the bayesian analysis, has been employed to estimate the model’s parameters.

The Stochastic Volatility models have been widely used to model a changing variance of time series Financial data [1,2]. These models usually assume Gaussian distribution for asset returns conditional on the latent volatility. However, it has been pointed out in many empirical studies that daily asset returns have heavier tails than those of normal distribution. To account for heavy tails observed in returns series, [

[

This survey of literature proves that we can’t affirm absolutely that one distribution is better than another one. The selection of a density should be based on other parameters. In our work, we consider a general model of non- Gaussian centered error distribution. We prove that the efficiency of a specification of SV model depends on the dispersion of the data base. In fact, we find that when the data base is very dispersed, the Gaussian specification behaves better than the non-Gaussian one. On the contrary, if the data base presents a little dispersion measure, the non-Gaussian centered error specification will behave better than the Gaussian one. For this reason, we propose a general SV model where the diffusion of the stock return follows a non-Gaussian distribution.

Since it is not easy to derive the exact likelihood function in the framework of SV models, many methods are proposed in the literature to estimate these models. The four major approaches are: 1) the Bayesian Markov Chain Monte Carlo (MCMC) technique suggested by [

The rest of the paper is organized as follow: in the second section, we present, in a comparative setting, the usual Gaussian and the general non-Gaussian SV models. Bayesian parameter estimator’s and the MCMC algorithm are described in the third section. The fourth section develops an application of a non-Gaussian centered error density. In particular, we consider the Laplace density as an example of non-Gaussian distribution for the data base that we have studied. We conclude in the last section.

As any nature field, Finance has adopted a simple model, developed over the years, that attempts to describe the behaviour of random time fluctuation in the prices of stocks observed in the markets. This model assumes that the fluctuation of the stock prices follow a log-normal probability distribution function. The simple log-normal assumption would predict a Gaussian distribution for the returns with variance growing linearly with the time lag. What is actually found is that the probability distribution for high frequency data usually deviates from normality presenting heavy tails.

In this section, we present the classical Gaussian SV model and we introduce a non-Gaussian centered error distribution as an extension to the SV models.

The log Stochastic Volatility model is composed of a latent volatility equation and an observed return equation

where

According to the Euler discretization schema, we get

where

In order to consider a more general case of SV model, we propose in the next section, a non-Gaussian centered error distribution for

We consider a Stochastic Volatility model with a non-Gaussian noise where the return

The innovation term in the return equation

A long standing difficulty for applications based on SV models was that the models were hard to estimate efficiently due to the latency of the volatility state variable. The task is to carry out inference based on a sequence of returns

Recently, simulation based on inference was developed and applied to SV models. Two approaches were brought forward. The first was the application of Markov Chain Monte Carlo (MCMC) technique; ([8,16-18]). The second was the development of indirect inference or the so-colled Efficient Method of Moments; ([19-21]).

In our paper, we have chosen the bayesian MCMC approach for the estimation of the parameter’s and the volatility vector.

For the non-Gaussian SV model, we define the parameter set

We can obtain the joint distribution

where

[

where

For

With these posterior densities, the Gibbs sampler is applied and we get a Markov Chain for each parameter and thus the parameter’s estimator. The only difficult step arises in updating the volatility states. According to [

where

as a function of

where

The simulation of the posterior density of the parameters requires the application of the Gibbs Sampler. However, we prove explicit expression for the simulated parameters. In fact, after some simple calculations applied to the posterior density, we find the following expression for the iterated parameters by the Bayesian method in the

For the second parameter

For the parameter

We have considered some statistical model distributions characterized by different density for the noise terms. For each density listed in the second section (Student, Laplace, Uniform), we can formulate one particular specification for SV model. We have conducted a Chi-deux test to select (among these three densities) the appropriate error distribution to our data base. The results indicate that the Laplace density is the suitable distribution compared to the student and the uniform ones. In the second step, we will take the Laplace SV model such as a particular case of non-Gaussian SV model to be compared to the standard Gaussian distribution model that is the Normal one. The next section will present the application study.

In this section, we consider one particular case of non-Gaussian SV model that is the Laplace one. In fact, the application of the Chi-deux test has proved that this model is the more appropriate to our data base.

The study of this density is very interesting. In fact, Laplace density has been often used for modeling phenomena with heavier than normal tails for growth rates of diverse processes such as annual gross domestic product [

The Laplace SV model is defined as:

The innovation term in the return equation

The probability density function of a Laplace density is expressed as:

In order to prove the efficiency of the Laplace model, we conduct a simulation analysis.

Simulation Analysis

This section illustrates our estimation procedure using the simulated data. We generate 1000 observations from the Stochastic Volatility Laplace (SVL) model given by equation (3.12), with true parameters

We draw 6000 posterior samples of MCMC run. We discard the initial

The second step in our simulation study consists of testing the hypothesis that the choice of a specification is based on the dispersion measure of the data base. So, we simulate data bases with different variances. Results prove that when the data base is characterized by a little dispersion measure, the Laplace SV model performs a better specification for the parameters. In fact, the calculation of the Mean Squared Errors for the Laplace and the Normal model prove that this measure is greater for the Gaussian model. So, we should consider the estimators deduced from the Laplace (non-Gaussian) specification.

On the contrary, when the data base is very dispersed, it has been proved that the Gaussian SV model offers more precise estimation for the parameters.

From this simulation study, we can conclude that the selection of a model specification depends on data base characteristics. If we reject the normality assumption for a data base, we can accept it for another one.

Parameter | True | Mean(1) | Stdev.(1) | Mean(2) | Stdev.(2) | MSE |
---|---|---|---|---|---|---|

Laplace Model | ||||||

α | −0.077 | −0.1008 | 7.85 × 10^{−7} | −0.085 | 5.82 × 10^{−7} | 7.85 × 10^{−7} |

[−0.1009, −0.1007] | [−0.086, −0.084] | |||||

β | 0.929 | 0.906 | 5.78 × 10^{−7} | 0.9233 | 5.71 × 10^{−7} | 5.78 × 10^{−7} |

[0.9064, 0.9062] | [0.92, 0.93] | |||||

σ | 0.377 | 0.3593 | 1.46 × 10^{−6} | 0.367 | 1.28 × 10^{−6} | 0.0530 |

[0.3592, 0.3594] | [0.36, 0.37] | |||||

Normal Model | ||||||

α | −0.077 | −0.078 | 2:1 10^{−7} | −0.0927 | 2.02 × 10^{−6} | 2.099 × 10^{−6} |

[−0.079 −0.077] | [−0.0928, −0.0926] | |||||

β | 0.929 | 0.9073 | 2.56 × 10^{−7} | 0.916 | 1.57 × 10^{−6} | 2.56 × 10^{−6} |

[0.9072, 0.9074] | [0.915, 0.917] | |||||

σ | 0.377 | 0.3639 | 1.06 × 10^{−6} | 0.353 | 3.87 × 10^{−6} | 0.0536 |

[0.3638, 0.3640] | [0.352, 0.354] |

Notes: 1. This table provides a summary of the simulation results for the Laplace and the Normal model. 1000 observations were simulated off the true parameters. We report in the third and the fourth column the average (mean(1)) and the standard deviation (Stdev.(1)) of each parameter calculated with 6000 simulated path. After discarding a burn in period of 2000 iterations, we compute the mean (mean(2)), the standard deviation (Stdev.(2)) and the mean squared error for each parameter. Results are presented in the three last columns. We present the confidence interval between brackets. 2. MSE for a parameter

After approving our methodology using simulated data, we apply our MCMC estimation method to daily stock returns data. On our work, we focus on the study of the French stock market index: the CAC40 index returns. The sample size is 5240 observations. The log difference returns are computed as

In

In the last section, we have rejected the Gaussian assumption for the return series of the CAC40 index.

Observations | Mean | Stdev. | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

5240 | 0.0158 | 1.657 | −0.1737 | 24.8566 | −14 | 14 |

Note: This table provides summary statistics for daily return data on the CAC40 French Stoch Exchange index from January 2, 1987 to November 30, 2007.

Therefore, we have proven that the Laplace density is more consistent to our data base. In this section, we will apply the Laplace stochastic volatility model, that we have introduced in Equation (12), to the analysis of the CAC40 index returns. The number of MCMC iterations is 10000 and the initial 2000 samples are discarded.

The estimates of the volatility parameters

In order to test the ability of the Laplace model to predict future returns, we perform an out of sample analysis. We consider 4000 observations for the inference of parameters. We simulate (5240 - 4000) artificial observations from the Laplace model and the Normal one. We compare each simulated vector of returns with the remainder observations.

In

Compared Density | KL Divergence |
---|---|

Data density/Laplace density | 0.0656 |

Data density/Normal density | 1.1653 |

Note: This table summarizes the Kullback Leiber Divergence calculated between the true density and the estimated density: non-Gaussian (Laplace) in the second row, Gaussian (Normal) in the last row.

Parameters | SV Laplace | SV Normal |
---|---|---|

α | −0:0324 (7.5 × 10^{−4}) | 0.1344 (0.0029) |

7.51 × 10^{−4} | 0.029 | |

β | 0.8114 (5.5 × 10^{−6}) | 0.8954 (6.37 × 10^{−5}) |

5:5 10^{−6} | 6.37 × 10^{−5} | |

σ | 0.6563(3:7 0^{−4}) | 0.9216 (0.0032) |

0.0513 | 0.0161 | |

MSE (Model) | 55.7261 | 423.9717 |

Note: Parameter estimates for the CAC40 index data from January 2, 1987 to November 30, 2007. For each parameter, we report the mean of the posterior density, the standard deviation of the posterior in parentheses and the MSE. Estimation for the Laplace SV model are presented in the second column. Results for the Normal model are given in the third column. The last row gives the MSE calculated for the whole model. The first number represents the MSE between estimated returns through Laplace model and the observed returns. The second number represents the MSE for estimated returns with the normal model and observed returns.

SV Laplace | SV Normal | |
---|---|---|

MSE | 9.1438 | 22.7814 |

Notes:1. In this table we present the out of sample analysis results. We take the first 4000 empirical observations to infer the parameter estimates for the Laplace SV model and the Normal SV model. We generate (5240 - 4000) artificial observations with the two model and we calculate the MSE between observed remainder returns and the simulated vector of return with Laplace model and with Normal model. 2. MSE for returns vector is calculated with the following formula:

by the Normal SV model. This result indicates that the Laplace model (non-Gaussian) is able to predict returns better than the Normal one or the standard Gaussian one.

In this paper, we have considered the inference of SV model with non-Gaussian noise. By applying a Chi-deux test, we have chosen the suitable non-Gaussian distribution error for the data base considered in our study among different non-Gaussian distribution that has been considered in last studies (such as: Student, Uniform, Mixture of Normal,

We have performed MCMC technique for the stochastic volatility model when returns follow a Laplace distribution allowing for an important characteristics of returns dynamic: Heavy tails or leptokurticity.

An application to daily CAC40 index returns over the years (1987-2007) illustrates the ability of the Laplace model to deal with heavy tails better than the Log-Normal model. An out of sample analysis proves that the Laplace model better predicts future returns. These results have been reached according to the calculation of the Mean Squared Error calculated between estimated parameters and true parameters.