Rough stochastic volatility and related topics

• Applicants: Privatdozent Dr. Christian Bayer; Professor Dr. Peter Karl Friz
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 275035933

The objective of the proposed research project is the development of a new generation of stochastic asset price models, taking a combined numerical and analytical point of view, with strong emphasis on consistency with real market data, both under the physical and the pricing measure. More precisely, we develop a parsimonious joint model for the S&P 500 index (SPX) and the volatility index VIX which combines excellent fit to both historical time series data and observed option prices (for options on both SPX and VIX). To this end, we go away from classical diffusion, or jump diffusion / Levy models, and embrace a view, supported by the data, that volatility is rougher than predicted by any of these models and propose a fractional model for the volatility dynamics. The proposed research is structured in connected tasks as follows.(i) We propose to study the fundamental properties of such a fractional model, both from a theoretical and from a practical perspective. For realistic pricing of VIX-option it will be pivotal to incorporate a non-trivial market price of risk, leading to mathematical and computational challenges due to non-Markovianity of the stochastic volatility component.(ii) We propose highly accurate and fast-to-evaluate (almost) explicit formulas for option prices in such models. Building on expertise gained in previous works, we propose to use of large and moderate deviation theory, combined with Laplace methods in finite and infinite dimensions. Such formulas will also be important in the context of calibration of the model.(iii) We derive accurate numerical methods for option pricing in these rough stochastic volatility models, which are, in particular, necessary for pricing of exotic options. This task is non-trivial since standard PDE methods cannot be used in this context, once again due to absence of the Markov property. Note that exotic option prices also give us an additional handle for comparison with other asset price models, as exotic option prices make comparison of several models calibrated against vanilla options possible, using new techniques from optimal martingale transport.(iv) For final calibration of the model against observed market prices we propose to use an extension of a recent technique based on non-linear PDEs and the corresponding stochastic interacting particle systems. To this end, we shall first provide rigorous mathematical foundations for this non-linear calibration technique in standard (diffusive) stochastic (local) volatility models, before extending both the theoretical and the practical/numerical aspect to the rough stochastic (local) volatility models at hand.

• DFG Programme: Research Grants