On VIX futures in the rough Bergomi model

Abstract The rough Bergomi model introduced by Bayer et al. [Quant. Finance, 2015, 1–18] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen et al. [Finance Stoch., forthcoming].

[1]  Jorge A. León,et al.  On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility , 2006, Finance Stochastics.

[2]  Damien Challet,et al.  The limits of statistical significance of Hawkes processes fitted to financial data , 2014, 1406.3967.

[3]  P. Carr,et al.  Option Pricing, Interest Rates and Risk Management: Towards a Theory of Volatility Trading , 2001 .

[4]  Jim Gatheral,et al.  Pricing under rough volatility , 2015 .

[5]  L. Bergomi.  Smile Dynamics IV , 2009 .

[6]  M. Rosenbaum,et al.  Volatility is rough , 2014, 1410.3394.

[7]  Claude Martini,et al.  The Extended SSVI Volatility Surface , 2017 .

[8]  Jim Gatheral.  Consistent Modeling of SPX and VIX options , 2008 .

[9]  Jim Gatheral,et al.  Arbitrage-free SVI volatility surfaces , 2012, 1204.0646.

[10]  THE NORMALIZING TRANSFORMATION OF THE IMPLIED VOLATILITY SMILE , 2012 .

[11]  Jim Gatheral.  The Volatility Surface: A Practitioner's Guide , 2006 .

[12]  Masaaki Fukasawa,et al.  Asymptotic analysis for stochastic volatility: martingale expansion , 2011, Finance Stochastics.

[13]  D. Dufresne.  The log-normal approximation in financial and other computations , 2004, Advances in Applied Probability.

[14]  Joint modeling of VIX and SPX options at a single and common maturity with risk management applications , 2014 .

[15]  F. Comte,et al.  Long memory continuous time models , 1996 .

[16]  Mikko S. Pakkanen,et al.  Hybrid scheme for Brownian semistationary processes , 2015, Finance Stochastics.

[17]  M. Meerschaert.  Regular Variation in R k , 1988 .

[18]  P. Hagan,et al.  MANAGING SMILE RISK , 2002 .

[19]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[20]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[21]  L. Bergomi.  Smile Dynamics II , 2005 .

[22]  C. Doléans-Dade,et al.  Quelques applications de la formule de changement de variables pour les semimartingales , 1970 .

[23]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[24]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[25]  Thomas Kokholm,et al.  Joint pricing of VIX and SPX options with stochastic volatility and jump models , 2015 .