Robust Numerical Calibration for Implied Volatility Expansion Models

Abstract

Abstract. Implied volatility expansions allow calibration of sophisticated volatility models. They provide an accurate fit and parametrization of implied volatility surfaces that is consistent with empirical observations. Fine-grained higher order expansions offer a better fit but pose the challenge of finding a robust, stable and computationally tractable calibration procedure due to a large number of market parameters and nonlinearities. We propose calibration schemes for second order expansions that take advantage of the model's structure via exact parameter reductions and recoveries, reuse and scaling between expansion orders where permitted by the model asymptotic regime and numerical iteration over bounded significant parameters. We perform a numerical analysis over 12 years of real S&P 500 index options data for both multiscale stochastic and general local-stochastic volatility models. Our methods are validated empirically by obtaining stable market parameters that meet the qualitative and numerical constraints imposed by their functional forms and model asymptotic assumptions.Key words. implied volatility expansions, numerical calibration, parameter reductions, nonlinear least squares AMS subject classifications. 91G20, 91G60, 90C30, 60-081. Introduction. Research into implied volatility modelling both in academia and industry is focused on addressing the unrealistic assumption of constant asset volatility in the well-known Black-Scholes framework for pricing financial options (see [13] for an overview). A robust model parametrization of implied volatility needs to capture both characteristics and dynamics of the implied volatility surface along different strikes and maturities (or expiries). In addition, volatility models need to balance parameter stability with quality of fit over market data. Practitioners often prioritize tight fits and re-calibrate daily due to overfits that are not stable over multiple days. Meanwhile, existing literature focuses more on consistency. The downside of models that focus on consistency is that they can be computationally expensive to calibrate or are narrowly applicable in pricing a range of derivative contracts. The literature around volatility models is substantial, and we refer the reader to [10] for an extensive overview and comparison of volatility model classes.Models that consistently capture detailed volatility dynamics typically require calibration via numerical-based approaches that are often computationally expensive or lead to numerical instabilities. In order to address these issues, theoretical results have been proposed that translate model formulations into explicit implied volatility expansions for an alternative faster parameter calibration. Fouque et al [8,9] studied Stochastic Volatility (SV) models with multiple factors driving volatility on different time scales. These multifactor models have been validated by empirical studies that show that the deterministic volatility assumption is inconsistent with real data [2]. Instead, studies find that vol...