Rough volatility models

So-called rough stochastic volatility models constitute the latest advancement in option price modeling. In contrast to popular bivariate diffusion models such as Heston, here the driving noise of volatility is modeled by a fractional Brownian motion (fBM) with scaling in the rough regime of Hurst parameter H < 1/2. A major appeal of such models lies in their ability to parsimoniously recover key stylized facts of market IV surfaces such as the exploding power-law behaviour of the ATM volatility skew near zero, a crucial feature Markovian models fail to reproduce. On the flipside, as a consequence of fBM being neither a semimartingale nor a Markov process for H not equal to 1/2, most currently prevalent numerical pricing and calibration routines do not (easily) carry over to the rough setting. This thesis addresses this problem and contributes to the existing literature as follows. In chapter 2, we sharpen the large deviations results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviations (MD) estimates, only recently introduced in the option pricing context. In particular, we derive small-time asymptotic formulae for log call prices and Black-Scholes implied volatility. This in turn allows us to push the applicability range of known ATM skew approximation formulae from CLT type log-moneyness deviations of order t^1/2 to the wider MD regime. In chapter 3, we present a novel Monte Carlo (MC) pricing scheme for rough volatility models based on a Karhunen-Loève-style approximation of White Noise. This complements theoretical results by Bayer et al. (2017). Our numerical experiments confirm a theoretical strong rate of H for a central object of interest and indicate a weak rate of 2H for the option price. In chapter 4, we introduce a novel model calibration routine for (rough) stochastic volatility models dubbed deep calibration. Standard model calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, rendering calibration of many (rough) stochastic volatility models prohibitively expensive since often the map can only be approximated by costly MC simulations. As a remedy, we propose to combine the popular Levenberg-Marquardt optimization algorithm with neural network (NN) regression, replacing expensive MC simulations with cheap forward runs of a NN trained to approximate the implied volatility map. Numerical experiments confirm the high accuracy and speed of our approach.

Author dc:creator

• Stemper, Benjamin Marco

Advisor dc:contributor.advisor

• Friz, Peter Karl

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Identifier URI

http://dx.doi.org/10.14279/depositonce-8422

OAI identifier oai:identifier

oai:depositonce.tu-berlin.de:11303/9364