Podcast: Abi-Jaber and Li on a ‘sticky’ volatility problem

The pair discuss their model to jointly capture Vix, SPX and SSR

Developing a model that jointly calibrates options on Vix and SPX while generating consistent dynamics between spot price and volatility has long exercised quants.

Early studies proposed various ways to solve the joint calibration problem, using rough volatility models, jump processes or proceeding by interpolation, with mixed success on tractability and accuracy. More recent works have improved accuracy and speed.

A more neglected part of the problem involves the skew stickiness ratio (SSR), a quantity introduced by Lorenzo Bergomi that measures how much at-the-money implied volatility moves when the underlying asset’s spot price moves.

Most modern model specifications don’t capture SSR consistently. That means two models taking a snapshot at the same Vix and SPX smiles might project two very different forward volatilities, consequently leading banks to adopt different hedging strategies.

Having a reliable handle on SSR greatly helps efficient hedging.

Eduardo Abi-Jaber (pictured above, right), professor of applied mathematics at École Polytechnique, and Shaun Li, quant strategist on the exotic equity derivatives desk at Morgan Stanley, discuss their solution to this problem in the latest episode of Quantcast.

Quantcast – a Risk.net Cutting Edge podcast · Shaun Li and Eduardo Abi Jaber 22/05/26

 

“S&P 500 options exhibit a steep negative skew, at least for short maturities. At the same time, Vix options have a pronounced upward slope smile,” says Abi-Jaber. “Finding a model that is capable of reproducing both features while reproducing realistic spot-vol dynamics is known to be very difficult ... and this was the starting point of Shaun’s PhD,” he adds, referring to his role as Li’s PhD supervisor prior to his move to Morgan Stanley. 

Li explains why this matters. Even the computation of Greek measures, like the delta of a position, may give different results depending on whether SSR is in the so-called sticky-delta regime or in the sticky-strike regime. “To better control and understand the P&L it is very important to get the SSR right, because it can lead to very different hedging strategies,” Li says.

Related

Capturing smile dynamics with the quintic volatility model: SPX, SSR and VIX 

A new model captures the term structure of SPX & VIX implied volatilities, ATM skew, and the skew-stickiness ratio

The pair describe their model as a two-factor quintic Ornstein-Uhlenbeck model. In essence, it is a degree-five polynomial of the sum of two random processes driven by the same Brownian motion.

As for the quintic term, the authors explain that the choice of a degree-five polynomial is a result of the search for the simplest possible model to describe all the intended features. A degree five captures the necessary complexity, and there’s a structural, mathematical reason for the degree to be an odd number (meaning four and six aren’t suitable), which relates to the process to preserve the properties required for financial problems.

The pair’s two-factor model builds on a previous, one-factor version that Abi-Jaber and Li authored with Camille Illand of BNP Paribas Asset Management. With two factors, the model can capture the full volatility surface, while the one-factor version could only describe short maturities, up to three months, Abi-Jaber says. 

Looking ahead, Abi-Jaber plans to extend the quintic model to include path-dependency, or the inclusion of memory effects in volatility modelling.

Index: 

00:00 Introduction

01:30 Background on the paper and collaboration

07:55 Quintic – why a five-degree polynomial is appropriate

15:40 Implementation and challenges 

24:00 Future of volatility modelling with path signatures 

28:46 Next projects

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