19

Estimation of Continuous-Time

Stochastic Volatility Models

George Dotsis, Raphael N. Markellos and Terence C. Mills

Abstract

This chapter reviews some of the key issues involved in estimating continuous-time stochastic
volatility models. Such models have become popular recently because they provide a rich variety
of alternative specifications which often lead to closed or semi-closed solutions in a variety of
asset-pricing applications. An empirical comparison of various stochastic volatility models is also
undertaken, along with a discussion of some directions for future research.

19.1 Introduction 951
19.2 Volatility specifications 955
19.2.1 Affine diffusions 956
19.2.2 Affine jump diffusions 957
19.2.3 Non-affine diffusions 957
19.3 Inference in stochastic volatility models 958
19.3.1 Simulation-based inference 959
19.3.1.1 Efficient method of moments 959
19.3.1.2 Markov chain Monte Carlo 961
19.3.2 Characteristic function methods 961
19.3.3 Derivatives markets 963
19.3.4 Integrated volatility 964
19.4 Empirical comparison of volatility processes 965
19.5 Conclusions 966

19.1 Introduction

It is now widely accepted that volatility in financial markets evolves stochasti-
cally over time.^1 The stochastic behavior of volatility has important implications
for asset allocation, the pricing and hedging of derivative securities, prudent risk
management and the behavior of financial assets in general. There are two cen-
tral facets to the modeling of time-varying volatility. The first is the estimation
of the model’s parameters, the second is the filtration of latent volatility given
these parameter estimates. The filtration of volatility is particularly important for
applications such as option pricing, value-at-risk and portfolio allocation, all of
which require volatility estimates. One popular approach that tackles both of

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