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Rough Bergomi turns grey

Antoine Jacquier, Adriano Oliveri Orioles, Zan Zuric

TL;DR

This work introduces the Grey Bergomi model, a generalised grey Brownian motion based extension of rough Bergomi that relaxes lognormality in the variance process to enable joint SPX and VIX calibration. It develops tractable VIX dynamics, VIX futures representations, and semi-closed pricing tools, together with Malliavin-based small-time asymptotics for SPX and VIX smiles. Numerical algorithms and Markovian approximations are proposed to render calibration feasible, and a calibration exercise to market data yields parameter estimates highlighting both improvements and remaining challenges. The model offers a tractable framework that blends features of rough volatility with stochastic volatility of volatility, providing insights into short-time smile behavior and practical SPX-VIX calibration prospects.

Abstract

We propose a tractable extension of the rough Bergomi model, replacing the fractional Brownian motion with a generalised grey Brownian motion, which we show to be reminiscent of models with stochastic volatility of volatility. This extension breaks away from the log-Normal assumption of rough Bergomi, thereby making it a viable suggestion for the Equity Holy Grail -- the joint SPX/VIX options calibration. For this new (class of) model(s), we provide semi-closed and asymptotic formulae for SPX and VIX options and show numerically its potential advantages as well as calibration results.

Rough Bergomi turns grey

TL;DR

This work introduces the Grey Bergomi model, a generalised grey Brownian motion based extension of rough Bergomi that relaxes lognormality in the variance process to enable joint SPX and VIX calibration. It develops tractable VIX dynamics, VIX futures representations, and semi-closed pricing tools, together with Malliavin-based small-time asymptotics for SPX and VIX smiles. Numerical algorithms and Markovian approximations are proposed to render calibration feasible, and a calibration exercise to market data yields parameter estimates highlighting both improvements and remaining challenges. The model offers a tractable framework that blends features of rough volatility with stochastic volatility of volatility, providing insights into short-time smile behavior and practical SPX-VIX calibration prospects.

Abstract

We propose a tractable extension of the rough Bergomi model, replacing the fractional Brownian motion with a generalised grey Brownian motion, which we show to be reminiscent of models with stochastic volatility of volatility. This extension breaks away from the log-Normal assumption of rough Bergomi, thereby making it a viable suggestion for the Equity Holy Grail -- the joint SPX/VIX options calibration. For this new (class of) model(s), we provide semi-closed and asymptotic formulae for SPX and VIX options and show numerically its potential advantages as well as calibration results.
Paper Structure (18 sections, 10 theorems, 82 equations, 11 figures, 1 table)

This paper contains 18 sections, 10 theorems, 82 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The grey Bergomi model
  3. Generalised grey Brownian motion
  4. The grey Bergomi model
  5. VIX
  6. VIX Dynamics
  7. VIX Futures
  8. Upper and lower bounds for VIX Futures
  9. Skew-Stickiness Ratio (SSR)
  10. Numerical algorithms
  11. Numerical implementation of VIX process
  12. Algorithm for SPX options
  13. Asymptotics of the SPX & VIX smiles
  14. Small-time VIX asymptotics
  15. SPX asymptotics
  16. ...and 3 more sections

Key Result

Lemma 2.3

The representation $B_t^{\beta, \alpha} \stackrel{(d)}{=} \sqrt{Y_\beta} B_t^{\frac{\alpha}{2}}$ holds for all $t \geq 0$, where $B^{\frac{\alpha}{2}}$ is a Mandelbrot-Van Ness fractional Brownian motion with Hurst parameter $\frac{\alpha}{2}$ and $Y_\beta$ an independent (one-sided) $\mathrm{M}$-Wr

Figures (11)

  • Figure 1: Upper and lower bounds in all three scenarios.
  • Figure 2: Truncated Cholesky Monte Carlo prices and Monte Carlo standard deviations in rBergomi and gBergomi with the same parameters (except for $\beta$).
  • Figure 3: SPX Call option prices and implied volatilities.
  • Figure 4: SPX implied volatilities.
  • Figure 5: At-the-money VIX level, skew, curvature asymptotics with $(H,\eta, \xi_0) = (0.07, 1.23, 0.235^2)$, taken from jacquier2018vix.
  • ...and 6 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Proposition 3 in Mura2008
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 17 more