More flexible volatility models retain exact moment calculations

The modelling question

Volatility describes the variability of asset returns; variance is its square. Financial models also need to describe the leverage effect—the relationship between return shocks and subsequent variance—and variance of variance, meaning uncertainty about future variance itself. Models with similar expected variance can nevertheless imply different probabilities of extreme returns and different option prices.

The paper asks how much flexibility these relationships can gain while retaining exact calculations of moments. Moments include means and higher-order quantities used to describe dispersion, asymmetry and tail behaviour. Cross-moments describe relationships between quantities, including observations at different dates.

What polynomial preservation does

The authors build on a property called polynomial preservation. A polynomial combines variables and their whole-number powers. In a polynomial-preserving process, the conditional expectation of a polynomial in a future state remains a polynomial of no higher degree in the current state.

This restriction closes the calculation: moments of a chosen order depend only on moments of that order or lower. The authors represent the one-step calculations with a finite matrix. Repeated matrix operations then give moments over longer horizons. For a fixed moment order, the matrix does not grow as the horizon increases, although higher orders require larger matrices.

“Exact” here concerns calculations within the assumed model. It does not mean that the model describes markets exactly or that its estimated parameters are free of uncertainty.

Two related model classes

The study introduces variance-polynomial hybrid GARCH–stochastic volatility models, abbreviated VPGARCH-SV, and jointly polynomial models, JPGARCH-SV. GARCH models update conditional variance using past variance and return shocks. The hybrid versions allow variance to receive a separate shock correlated with the return shock.

Both classes preserve polynomial calculations for variance and support exact moments of cumulative variance and squared returns after subtracting their conditional means. JPGARCH-SV additionally preserves the joint return–variance structure needed for exact cumulative-return moments. That extra property does not generally hold for VPGARCH-SV with correlated shocks.

The models extend established NGARCH and HNGARCH specifications. They permit more flexible relationships between the current variance level, the response to shocks and uncertainty about future variance. The paper separately derives parameter restrictions ensuring nonnegative variance and conditions for finite long-run moments; polynomial preservation alone guarantees neither.

The authors also connect the discrete-time models to continuous-time processes. Under two specified scaling regimes, increasingly short time steps lead to diffusion or jump-diffusion limits, with corresponding convergence of polynomial moments. These results require additional positivity, regularity and boundary conditions.

What the empirical exercise found

The empirical study uses 6,965 daily S&P 500 excess returns and 664,592 option observations across 1,443 weekly dates, from January 1996 to August 2023. It tests the GARCH special cases rather than estimating the full hybrid models with separate variance shocks.

Return parameters are estimated by maximum likelihood. Options are calibrated separately each week, minimizing discrepancies between model and market implied volatilities. Implied volatility is the volatility value corresponding to an option price under the Black–Scholes formula. The root mean squared error summarizes those discrepancies, with greater weight on larger errors.

Against HNGARCH, JPGARCH reduces mean weekly implied-volatility error by 41.4% with Gaussian innovations and 7.5% with symmetric normal inverse Gaussian innovations, a heavier-tailed shock distribution. Against NGARCH, VPGARCH reduces the same measure by 11.2% and 13.7%, respectively. VPGARCH has the lowest mean weekly error under both distributions.

Both extensions also improve return likelihood relative to their benchmarks. However, under heavier-tailed innovations, the Bayesian information criterion—which penalizes additional parameters—gives virtually identical results for VPGARCH and NGARCH.

Where the conclusions stop

These are in-sample fitting results, not demonstrations of better future forecasts or hedging performance. The paper identifies hedging evaluation as further work.

Nor are the empirical option prices calculated exactly from the polynomial moments. The calibration uses Monte Carlo simulation with 50,000 paths. Moving from the probability model for observed returns to the risk-adjusted model used for pricing need not preserve polynomiality, and the calibration does not impose the restrictions that would ensure it.

Long-run stability also matters: some fitted models do not satisfy the condition for a finite stationary second moment of variance, although their finite-horizon conditional moments remain finite. The preprint therefore offers a tractable modelling framework and encouraging calibration evidence, rather than a general guarantee of superior market-risk predictions.