Estimation of Integrated Volatility in Continuous‐Time Financial Models with Applications to Goodness‐of‐Fit Testing

Abstract

Properties of a specification test for the parametric form of the variance function in diffusionThe test is based on the estimation of certain integrals of the volatility function. If the volatility function does not depend on the variable x it is known that the corresponding statistics have an asymptotic normal distribution. However, most models of mathematical finance use a volatility function which depends on the state x. In this paper we prove that in the general case, where σ depends also on x the estimates of integrals of the volatility converge stably in law to random variables with a non-standard limit distribution. The limit distribution depends on the diffusion process X t itself and we use this result to develop a bootstrap test for the parametric form of the volatility function, which is consistent in the general diffusion model.

Understanding Limit Theorems for Semimartingales: A Short Survey

“…Theorem 3.1, Example 3.2 and Theorem 3.6 of this paper or [4], [17]), or for performing various test procedures (see e.g. [1], [5], [8]). The aim of this paper is to present a short survey of these theoretical results and to carefully explain the main concepts and ideas of the proofs.…”

A focused information criterion for quantile regression: Evidence for the rebound effect

“…This also holds true for alternative model selection methods, such as goodness-of-fit tests, which are proposed by, among many others, DETTE (1999), VETTER (2006), andDETTE (2008). However, conditional on the underlying purpose, some specifications might be better suited than others in terms of estimation efficiency.…”

Edgeworth expansion for functionals of continuous diffusion processes

“…f (x) = f (−x) for all x ∈ R. This type of functionals play a very important role in mathematical finance, where they are used for various estimation and testing procedures; see e.g. [4], [5], [7] and [11] among many others.…”

“…[5,18]), testing approaches for semimartingales (see e.g. [2,7]) or numerical analysis (see e.g. [12]).…”