A Functional Version of the ARCH Model

TL;DR

This paper introduces a functional extension of the ARCH model to handle high-resolution financial data as continuous processes, establishing theoretical properties and demonstrating practical applicability.

Contribution

It proposes a novel functional ARCH model, providing theoretical foundations and empirical validation for modeling high-frequency financial data as functions.

Findings

Established conditions for stationarity of the functional ARCH model

Derived weak dependence and moment conditions

Showed consistency of estimators and matched model with real data

Abstract

Improvements in data acquisition and processing techniques have lead to an almost continuous flow of information for financial data. High resolution tick data are available and can be quite conveniently described by a continuous time process. It is therefore natural to ask for possible extensions of financial time series models to a functional setup. In this paper we propose a functional version of the popular ARCH model. We will establish conditions for the existence of a strictly stationary solution, derive weak dependence and moment conditions, show consistency of the estimators and perform a small empirical study demonstrating how our model matches with real data.