Nikolay Nikolaev, Goldsmiths College, University of London

Stochastic Volatility Modelling

SV

Stochastic Volatility

The changing variance (volatility) in time series of returns is a measure of their uncertainty, which is analysed in order to price various financial instruments and to reduce risk. A contemporary approach to finding this uncertainty is provided by stochastic volatility (SV) models. Assuming that log-prices follow a Brownian motion, their dynamics is described using stochastic differential equations, whose Euler approximation leads to discrete SV models.

A stochastic volatility model consists of a couple of equations that describe how the returns depend on the volatility. Such a SV model can be explained as follows: 1) the returns are a nonlinear function of the latent volatility, and their distribution is often non-Gaussian with fat tails (and small positive kurtosis); 2) the volatility is an autoregressive function with a specific persistance parameter, and its distribution is typically normal with unknown variance. An attractive characteristic of SV models is that they allow perturbations by random shocks independent from past information, which makes them a promising alternative to the popular GARCH models.

Finding the latent (unobserved) stochastic volatility in series of returns involves learning of nonlinear models. Our research implements various such approaches for the univariate mean-reverting autoregressive Hull-White model (Hull and White, 1987), like:

- Bayesian Markov Chain Monte Carlo (MCMC) using Gibbs and integration sampling;

- Monte Carlo Likelihood (MCL) Estimation through filtering and simulation.

- Particle filtering (PF) using Auxiliary PF, Parzen PF, and Mixture PF techniques.

- Quasi-Maximum Likelihood (QML) Estimation using Kalman filtering and BFGS optimisation.

- Nonlinear Maximum Likelihood (NML) Estimation using Gauss-Hermite quadratures.

Currently nonlinear NML treatment methods are under active investigation, as they allow to integrate numerically the distributions of interest, which helps to calculate analytically the exact likelihood function. Nonlinear models are processed by special moment matching filters and smoothers, and their parameters are estimated using the Expectation Maximisation (EM) algorithm. This approach to inference of SV models enables to make easily extentions for handling heavy-tailed noise using the Student-t distribution, accomodation of leverage effects, as well as jumps. Soft and hard regime switching models are also under development.

Nonlinear SV models operate extremely fast and are suitable for development of dynamic factor modeling for portfolio selection. The SV models allow us to evaluate the changing variances in large number of time series, which can be aggregated into a small number of common factors for generating more accurate forecasts of returns. This is important for practical financial applications, such as hedging porfolios, calculation of value-at-risk, and other risk assessment tasks.