# The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S&P 500 Skew

## Abstract

*Risk*, May (2012), pp. 60--66] to calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX futures. Very interestingly, the joint fit selects (1) much larger values of volatility-of-volatility and mean reversion than those previously reported in [L. Bergomi,

*Risk*, October (2005), pp. 67--73] and [L. Bergomi,

*Stochastic Volatility Modeling*, CRC Press, 2016], and (2) fully correlated Brownian motions, thus producing a (Markovian) pure path-dependent volatility model with rough-like paths.

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility*, Finance Stoch., 11 (2007), pp. 571--589.

*On smile properties of volatility derivatives: Understanding the VIX skew*, SIAM J. Financial Math., 13 (2022), pp. 32--69, https://doi.org/10.1137/19M1269981.

*Exponentiation of conditional expectations under stochastic volatility*, Quant. Finance, 20 (2020), pp. 13--27.

*The short-time behavior of VIX implied volatilities in a multifactor stochastic volatility framework*, Math. Finance, 29 (2019), pp. 928--966.

*Consistent modelling of VIX and equity derivatives using a 3/2 plus jumps model*, Appl. Math. Finance, 21 (2014), pp. 299--312.

*Pricing under rough volatility*, Quant. Finance, 16 (2016), pp. 887--904.

*Quadratic Hawkes processes for financial prices*, Quant. Finance, 17 (2017), pp. 171--188.

*Smart expansion and fast calibration for jump diffusion*, Finance Stoch., 13 (2009), pp. 563--589.

*Closed forms for European options in a local volatility model*, Int. J. Theor. Appl. Finance, 13 (2010), pp. 603--634.

*Time dependent Heston model*, SIAM J. Financial Math., 1 (2010), pp. 289--325, https://doi.org/10.1137/090753814.

*Computing the implied volatility in stochastic volatility models*, Comm. Pure Appl. Math., 57 (2004), pp. 1352--1373.

*Smile dynamics II*, Risk, October (2005), pp. 67--73.

*Smile dynamics III*, Risk, October (2008), pp. 90--96.

*The Smile in Stochastic Volatility Models*, preprint, SSRN, 2011.

*Stochastic volatility's orderly smiles*, Risk, May (2012), pp. 60--66.

*Stochastic Volatility Modeling*, Chapman & Hall/CRC Financial Math. Ser., CRC Press, 2016.

*Analytical approximations of local-Heston volatility model and error analysis*, Math. Finance, 28 (2018), pp. 920--961.

*Weak Approximations and VIX Option Price Expansions in Variance Curve Models*, preprint, https://arxiv.org/abs/2202.10413, 2022.

*Consistent variance curve models*, Finance Stoch., 10 (2006), pp. 178--203.

*The CBOE Volatility Index-VIX*, https://www.cboe.com/tradeable_products/vix.

*The fine-structure of volatility feedback I: Multi-scale self-reflexivity*, Phys. A, 410 (2014), pp. 174--195.

*A consistent pricing model for index options and volatility derivatives*, Math. Finance, 23 (2013), pp. 248--274.

*Linking vanillas and VIX options: A constrained martingale optimal transport problem*, SIAM J. Financial Math., 6 (2015), pp. 1171--1194, https://doi.org/10.1137/140960724.

*Volatility Derivatives in (Rough) Forward Variance Models*, presentation at the Bachelier Congress, Dublin, July 2018.

*Arbitrage Pricing with Stochastic Volatility*, preprint, 1993.

*Does the Term-Structure of Equity at-the-Money Skew Really Follow a Power Law?*, preprint, 2022.

*Short-term at-the-money asymptotics under stochastic volatility models*, SIAM J. Financial Math., 10 (2019), pp. 491--511, https://doi.org/10.1137/18M1167565.

*The large-maturity smile for the Heston model*, Finance Stoch., 15 (2011), pp. 755--780.

*The small-time smile and term structure of implied volatility under the Heston model*, SIAM J. Financial Math., 3 (2012), pp. 690--708, https://doi.org/10.1137/110830241.

*Derivatives in Financial Markets with Stochastic Volatility*, Cambridge University Press, 2000.

*Maturity cycles in implied volatility*, Finance Stoch., 8 (2004), pp. 451--477.

*Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options*, Quant. Finance, 18 (2017), pp. 1003--1016.

*Option pricing in the moderate deviations regime*, Math. Finance, 28 (2018), pp. 962--988.

*Asymptotic analysis for stochastic volatility: Martingale expansion*, Finance Stoch., 15 (2011), pp. 635--654.

*Consistent Modeling of SPX and VIX Options*, presentation at the Bachelier Congress, July 18, 2008.

*Implied volatility surface*, in Encyclopedia of Quantitative Finance, R. Cont., ed., John Wiley and Sons, 2010, pp. 926--930.

*Volatility is rough*, Quant. Finance, 18 (2018), pp. 933--949.

*The quadratic rough Heston model and the joint calibration problem*, Risk, May (2020).

*Weak approximation of averaged diffusion processes*, Stochastic Process. Appl., 124 (2014), pp. 475--504.

*Regime-switching stochastic volatility model: Estimation and calibration to VIX options*, Appl. Math. Finance, 24 (2017), pp. 38--75.

*Calibration of local-stochastic volatility models by optimal transport*, Math. Finance, 32 (2021), pp. 46--77.

*Joint modeling and calibration of SPX and VIX by optimal transport*, SIAM J. Financial Math., 13 (2022), pp. 1--31, https://doi.org/10.1137/20M1375905.

*Path-dependent volatility*, Risk, October (2014).

*Bounds for VIX futures given S&P 500 smiles*, Finance Stoch., 21 (2017), pp. 593--630.

*On the Joint Calibration of SPX and VIX Options*, presentation at Imperial College London, March 28, 2018, and QuantMinds 2018, Lisbon, May 16, 2018.

*The joint S&P 500/VIX smile calibration puzzle solved*, Risk, April (2020).

*Dispersion-Constrained Martingale Schrödinger Bridges: Joint Entropic Calibration of Stochastic Volatility Models to S&P 500 and VIX Smiles*, preprint, 2022.

*Inversion of convex ordering in the VIX market*, Quant. Finance, 20 (2020), pp. 1597--1623.

*Dispersion-Constrained Martingale Schrödinger Problems and the Exact Joint S&P 500/VIX Smile Calibration Puzzle*, preprint, http://ssrn.com/abstract=3853237, 2021.

*The smile of stochastic volatility: Revisiting the Bergomi-Guyon expansion*, preprint, http://ssrn.com/abstract=3956786, 2021.

*Volatility Is (Mostly) Path-Dependent*, presentation at QuantMinds, December 7, 2021.

*Complete models with stochastic volatility*, Math. Finance, 8 (1998), pp. 27--48.

*An Introduction to Complex Analysis in Several Variables*, Van Nostrand, 1966.

*Managing smile risk*, Wilmott Mag., September (2002), pp. 84--108.

*A General Asymptotic Implied Volatility for Stochastic Volatility Models*, preprint, SSRN, 2005.

*Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing*, Chapman & Hall/CRC Financial Math. Ser., CRC Press, 2009.

*On VIX futures in the rough Bergomi model*, Quant. Finance, 18 (2018), pp. 45--61.

*Joint pricing of VIX and SPX options with stochastic volatility and jump models*, J. Risk Finance, 16 (2015), pp. 27--48.

*Implied and local volatilities under stochastic volatility*, Int. J. Theor. Appl. Finance, 4 (2001), pp. 45--89.

*Option Valuation under Stochastic Volatility*, Finance Press, 2000.

*Approximation and calibration of short-term implied volatilities under jump-diffusion stochastic volatility*, Rev. Financial Stud., 20 (2007), pp. 427--459.

*Asymptotic Expansion Formula of Implied Volatility for Dynamic SABR Model and FX Hybrid Model*, UTMS 2006-29, The University of Tokyo, 2006.

*Smiling twice: The Heston++ model*, J. Banking Finance, 96 (C) (2018), pp. 185--206.

*The exact Taylor formula of the implied volatility*, Finance Stoch., 21 (2017), pp. 661--718.

*A regime-switching Heston model for VIX and S&P 500 implied volatilities*, Quant. Finance, 14 (2014), pp. 1811--1827.

*Extreme-strike comparisons and structural bounds for SPX and VIX options*, SIAM J. Financial Math., 9 (2018), pp. 401--434, https://doi.org/10.1137/141001615.

*Achieving Consistent Modeling of VIX and Equity Derivatives*, presented at the Imperial College Mathematical Finance Seminar, November 2, 2011.

*Stochastic volatility, smile and asymptotics*, Appl. Math. Finance, 6 (1999), pp. 107--145.

*Some results on univariate and multivariate Cornish-Fisher expansion: Algebraic properties and validity*, Sankhyā Ser. A, 50 (1988), pp. 111--136.

*Time reversal invariance in finance*, Quant. Finance, 9 (2009), pp. 505--515.

*Volatility conditional on price trends*, Quant. Finance, 10 (2010), pp. 431--442.

*VIX derivatives valuation and estimation based on closed-form series expansions*, Int. J. Financial Engrg., 5 (2018), pp. 1--18.

*An analytical formula for VIX futures and its applications*, J. Futures Markets, 32 (2012), pp. 166--190.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 30 July 2021

**Accepted**: 15 July 2022

**Published online**: 8 December 2022

#### Keywords

#### MSC codes

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Deep Curve-Dependent PDEs for Affine Rough VolatilitySIAM Journal on Financial Mathematics, Vol. 14, No. 2 | 25 April 2023
- Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX marketsQuantitative Finance, Vol. 22, No. 10 | 16 June 2022