Paper: arXiv 2508.08773

Authors: Paolo Foschi

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical generalization of the Hobson-Rogers model using quadratic variance functions and multi-factor exponential polynomial filters, complete with formal proofs and SDE analyses. However, empirical rigor is limited to theoretical qualitative illustrations and numerical exercises rather than empirical backtesting against real market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Hobson-Rogers model with multi-factor quadratic variance] --> B[Model Formulation: QHR Model with Markovian Property]
    B --> C[Analytical Derivations: ARMA structure & Pearson IV distribution]
    C --> D[Numerical Simulation: Calibrate variance parameters]
    D --> E[Key Outcomes: Implied Volatility Surface & Forward Variance Term Structure]