Raue stochastischer Volatilität und verwandten Themen

• Antragsteller: Privatdozent Dr. Christian Bayer; Professor Dr. Peter Karl Friz
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2015 bis 2019
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 275035933

• Erstellungsjahr: 2019

Zusammenfassung der Projektergebnisse

The project was very successful in establishing a new paradigm for pricing of options in equity market, namely rough stochastic volatility models. Earlier works by Jim Gatheral et al. (2018) showed that time series of the S & P 500 index (and many other important financial indices) exhibit rough volatility, i.e., that the instantaneous variance of changes in the index is H-Hölder continuous in time with 0 < H << 1/2, and that it can be very well described by the exponential of a fractional Brownian motion with Hurst index H. Hence, volatility is rough, i.e., rougher than Brownian motion. This analysis was later extended to many other indices and individual stocks, so that we are now justified in claiming that volatility of equities is rough. In this project, we extended this framework to the option pricing world. We establish a rough stochastic volatility model in the above sense (the rough Bergomi model ), which produces remarkable fits to implied volatility surfaces of the market – in particular, we consider the S & P 500 index – with only three free parameters: the Hurst index H as mentioned above, a volatility-ofvolatility parameter, and a correlation parameter (responsible for the leverage effect). In particular, we can recover the power-law type explosion of the implied volatility skew when the option’s expiry goes to zero. Hence, the rough Bergomi and other stochastic volatility models can simultaneously price options for all different maturities, including very short terms. In addition, we provide: • Highly accurate explicit asymptotic formulas for vanilla options and implied volatilities; • a theoretical framework for analyzing rough volatility models based on Martin Hairer’s theory of regularity structures; • new simulation methods for European and American option pricing in rough volatility models; • a fast and accurate calibration method for rough volatility models based on deep learning; • new numerical methods for solving rough differential equations.

Projektbezogene Publikationen (Auswahl)

• (2020) A regularity structure for rough volatility. Mathematical Finance 30 (3) 782–832
  Bayer, Christian; Friz, Peter K.; Gassiat, Paul; Martin, Jorg; Stemper, Benjamin
  
• From rough path estimates to multilevel Monte Carlo. SIAM Journal on Numerical Analysis, 54(3):1449–1483, 2016
  C. Bayer, P. Friz, S. Riedel, and J. Schoenmakers
  
• Pricing under rough volatility. Quantitative Finance, 16(6):887–904, 2016
  Christian Bayer, Peter Friz, and Jim Gatheral
  
• Deep calibration of rough stochastic volatility models
  Christian Bayer and Benjamin Stemper
  
• Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model
  Christian Bayer, Chiheb Ben Hammouda, and Raul Tempone
  
• Local volatility, conditioned diffusions, and Varadhan’s formula. SIAM Journal on Financial Mathematics, 9(2):835–874, 2018
  Stefano De Marco and Peter K Friz
  
• Option pricing in the moderate deviations regime. Mathematical finance, 28(3):962–988, 2018
  Peter Friz, Stefan Gerhold, and Arpad Pinter
  
• Pricing American Options by Exercise Rate Optimization
  Christian Bayer, Raúl Tempone, and Sören Wolfers
  
• Short-time near-the-money skew in rough fractional volatility models. Quantitative Finance, 19(5):779–798, 2019
  C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath, and B. Stemper