Joint calibration of the volatility surface and variance term structure

Abstract: This article proposes a calibration framework for complex option pricing models that jointly fits market option prices and the term structure of variance. Calibrated models under the conventional objective function, the sum of squared errors in Black-Scholes implied volatilities, can produce model-implied variance term structures with large errors relative to those observed in the market and implied by option prices. I show that this can occur even when the model-implied volatility surface closely matches the volatility surface observed in the market. The proposed joint calibration addresses this issue by augmenting the conventional objective function with a penalty term for large deviations from the observed variance term structure. This augmented objective function features a hyperparameter that governs the relative weight placed on the volatility surface and the variance term structure. I test this framework on a jump-diffusion model with stochastic volatility in two calibration exercises: the first using volatility surfaces generated under a Bates model, and the second using a panel of S&P 500 equity index options covering the 1996-2023 period. I demonstrate that the proposed method is able to fit observed option prices well while delivering realistic term structures of variance. Finally, I provide guidance on the choice of hyperparameters based on the results of these numerical exercises.