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Multidimensional Stochastic Processes as Rough Paths: Theory and Applications
Preface Introduction The story in a nutshell Part I. Basics: 1. Continuous paths of bounded variation 2. Riemann-Stieltjes integration 3. Ordinary differential equations (ODEs) 4. ODEs: smoothness 5.
Pricing Under Rough Volatility
From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum previously showed that log-volatility behaves essentially as a fractional Brownian
A Course on Rough Paths
We give a short overview of the scopes of both the theory of rough paths and the theory of regularity structures. The main ideas are introduced and we point out some analogies with other branches of
A Course on Rough Paths: With an Introduction to Regularity Structures
Introduction.- The space of rough paths.- Brownian motion as a rough path.- Integration against rough paths.- Stochastic integration and Ito's formula.- Doob-Meyer type decomposition for rough
Differential equations driven by Gaussian signals
We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). Gaussian rough paths are constructed with a variety
Regular Variation and Smile Asymptotics
We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The
The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment
Valuation of volatility derivatives as an inverse problem
Ground-breaking recent work by Carr and Lee extends well-known results for variance swaps to arbitrary functions of realized variance, provided a zero-correlation assumption is made. We give a