Multifactor Quadratic Hobson and Rogers models

TL;DR

This paper introduces a multi-factor quadratic extension of the Hobson and Rogers model, enhancing flexibility in variance modeling while preserving key properties, and provides explicit formulas and numerical illustrations for financial applications.

Contribution

It develops a quadratic variance function extension of the HR model, enabling explicit expressions for forward variance and analyzing stationarity and autocorrelation structures.

Findings

The QHR model maintains Markovian properties with greater flexibility.

Explicit formulas for forward variance are derived.

Numerical examples illustrate implied volatility and skew structures.

Abstract

A multi-factor extension of the Hobson and Rogers (HR) model, incorporating a quadratic variance function (QHR model), is proposed and analysed. The QHR model allows for greater flexibility in defining the moving average filter while maintaining the Markovian property of the original HR model. The use of a quadratic variance function permits the characterisation of weak-stationarity conditions for the variance process and allows for explicit expressions for forward variance. Under the assumption of stationarity, both the variance and the squared increment processes exhibit an ARMA autocorrelation structure. The stationary distribution of the prototypical scalar QHR model is that of a translated and rescaled Pearson type IV random variable. A numerical exercise illustrates the qualitative properties of the QHR model, including the implied volatility surface and the term structures of forward variance, at-the-money (ATM) volatility, and ATM skew.