Mean-Reverting SABR Models: Closed-form Surfaces and Calibration for Equities

TL;DR

This paper derives closed-form solutions for mean-reverting SABR models with different volatility dynamics, enabling efficient calibration to equity options surfaces and revealing insights into model stability and behavior.

Contribution

It provides explicit closed-form formulas for three mean-reverting SABR models and demonstrates their effective calibration to real market data.

Findings

Closed-form expressions are tractable and easily implementable.

Models achieve excellent fit with only five parameters.

CIR-volatility model captures sub-lognormal behavior and shows stable parameters.

Abstract

In this paper, we consider three stochastic-volatility models, each characterized by distinct dynamics of instantaneous volatility: (1) a CIR process for squared volatility (i.e., the classical Heston model); (2) a mean-reverting lognormal process for volatility; and (3) a CIR process for volatility. Previous research has provided semi-analytical approximations for these models in the form of simple (non-mean-reverting) SABR models, each suitably parameterized for a given expiry. First, using a computer algebra system, we derive closed-form expressions for these semi-analytical approximations, under the assumption that all parameters remain constant (but without the constraint of constant expected volatility). Although the resulting formulas are considerably lengthier than those in simpler SABR models, they remain tractable and are easily implementable even in Excel. Second, employing these closed-form expressions, we calibrate the three models to empirical volatility surfaces observed in EuroStoxx index options. The calibration is well-behaved and achieves excellent fits for the observed equity-volatility surfaces, with only five parameters per surface. Consequently, the approximate models offer a simpler, faster, and (numerically) more reliable alternative to the classical Heston model, or to more advanced models, which lack closed-form solutions and can be numerically challenging, particularly in less sophisticated implementation environments. Third, we examine the stability and correlations of our parameter estimates. In this analysis, we identify certain issues with the models - one of which appears to stem from the sub-lognormal behavior of the actual equity-volatility process. Notably, the CIR-volatility model (3), as opposed to the CIR-variance Heston model (1), seems to best capture this behavior, and also results in more stable parameters.