An Alternative Stochastic Volatility Model: Jumps and Diffusion

Renzo G. Avesani and Luigi A. Cefis
Department of Economics, University of Trento
rga@opo-rth.opoipi.it

Abstract

On the basis of the following considerations:

a)

presence of ``volatility clustering": speculative prices seem to be characterized by the fact that large deviations are followed by large deviations and small variations tend to be followed by small variations [Mandelbrot1963];

b)

the second moments of most financial asset returns have very complex dynamics, especially when they are observed at very high frequency [Bollerslev, Chou and Kroner1992],

Fong and Vasicek [Fong and Vasicek1991], Hull and White [Hull and White1987], Longstaff and Schwartz [1992] and Scott [Scott1987], point out that the Cox-Ingersoll-Ross model is not able to describe such complex dynamics.

Hence ``stochastic volatility models" have been proposed, where the diffusion coefficient is itself supposed to follow a diffusion process; a specification of this model can be described by the following system of stochastic differential equations:

A difficult problem arising from the statistical estimation of this model, is given by the fact that the stochastic process can not be observed directly.

Among the different solutions which have been proposed to solve this estimation problem, we remember the following ones:

• methods based on the indirect inference approach [Gourieroux, Monfort and Renault1993],
• methods based on GARCH models [Nelson and Foster1994].

Moreover Avesani and Bertrand [Avesani and Bertrand1995] propose a non parametric estimator in order to test the adequacy of the specification given in equation (1). The estimator is then used it on Italian financial time series.

In this paper the function:

is estimated using a kernel type estimator.

If were true, then the estimates of should have very small deviations from a constant estimated value computed on the entire sample.

The nonparametric estimation of suggests the idea of a volatility process characterized by time periods of quite constant variability broken by unexpeted jumps.

Keeping all the above results in mind, we suggest the following probabilistic model:

where is an -stable Lévy motion.

Parametric estimation of the model parameters is then realized using indirect inference methods [Gourieroux, Monfort and Renault1993].

References

Avesani and Bertrand1995

Avesani R. and Bertrand P., 1995. `Jumps and diffusions in volatility: it takes two to tango', mimeo.

Bollerslev, Chou and Kroner1992

Bollerslev T., Chou R., and Kroner K., 1992. `ARCH modelling in finance: a review of the theory and empirical evidence', Journal of Econometrics 52, 5-59.

Fong and Vasicek1991

Fong H. G. and Vasicek O. A., 1991. `Fixed-income volatility management', The Journal of Portfolio Optimization, Summer issue.

Gourieroux, Monfort and Renault1993

Gourieroux C., Monfort A., and Renault E., 1993. `Indirect inference', Journal of Applied Econometrics 8, 85-118.

Hull and White1987

Hull J. and White A., 1987. `The pricing of options on assets with stochastic volatility', The Journal of Finance 3, 281-300.

Mandelbrot1963

Mandelbrot B., 1963. `The variation of certain speculative Prices', Journal of Business 36, 147-165.

Nelson and Foster1994

Nelson D.B. and Foster D.P., 1994. `Asymptotic filtering theory for univariate ARCH models' 62, 1-41.

Scott1987

Scott L., 1987. `Option pricing when the variance changes randomly: theory, estimation and testing', Journal of Financial and Quantitative Analysis 22, 419-438.