Estimation in Financial Models - RiskLab

Introduction
Over the last few years various new derivative instruments have emerged in
nancial markets leading to a demand for versatile estimation methods for
relevant model parameters. Typical examples include volatility, covariances
and correlations. In this paper we give a survey on statistical estimation
methods for both discrete as well as continuous time stochastic models.
The text is organized as follows: in Chapter 1 we rst motivate a model in
which volatility ofa price process is assumed to follow astochastic process.
Out of the variety ofcontinuous time stochastic volatility models introduced
in the literature wechoose two empirically relevant ones, that is an arithmetic
Ornstein-Uhlenbeck process and a square root diusion model. Those two
models serve as reference models in some of the later chapters. As for discrete
time stochastic volatility models, we concentrate on log-AR(p) processes,
ARCH(q) and GARCH(p; q) processes all of which will be discussed in detail
in Section 3.2.
Approximations of diusion models by discrete time models and vice versa
are described in Chapter 2. The convergence result on which these approximations
are based is stated. As applications the diusion limits of GARCH(1,1)-
type processes and of AR(1) E-ARCH processes are derived. In addition, we
present strategies for approximating diusions and briey compare dierent
discretizations of a diusion model.
In Section 3.1 estimation of an unknown parameter in a general diusion
process is discussed. For continuously observed processes we develop the classical
theory of maximum likelihood estimation including properties like consistency
and asymptotic normality. In the case of discrete observations the
maximum likelihood estimator retains all the 'good' properties if the transition
densities of the process are known. However, in most cases relevant for
nance, we do not have explicit expressions for the underlying transition densities
and the use of approximate likelihood functions leads to inconsistent
estimators when the time between observations is bounded away from zero.
We describe alternative estimation methods, as there are martingale estimat-
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