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Multidimensional Stochastic Processes as Rough Paths: Theory and Applications

- P. Friz, Nicolas Victoir
- Mathematics
- 15 March 2010

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.… Expand

Pricing Under Rough Volatility

- C. Bayer, P. Friz, Jim Gatheral
- Economics
- 28 January 2015

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… Expand

A Course on Rough Paths

- P. Friz, Martin Hairer
- Mathematics
- 2014

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… Expand

A Course on Rough Paths: With an Introduction to Regularity Structures

- P. Friz, Martin Hairer
- Mathematics
- 4 September 2014

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… Expand

Differential equations driven by Gaussian signals

- P. Friz, Nicolas Victoir
- Mathematics
- 2 July 2007

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… Expand

Regular Variation and Smile Asymptotics

- Shalom Benaim, P. Friz
- Mathematics
- 6 March 2006

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… Expand

Multidimensional Stochastic Processes as Rough Paths: Markov processes

- P. Friz, Nicolas Victoir
- Mathematics
- 2010

SMILE ASYMPTOTICS II: MODELS WITH KNOWN MOMENT GENERATING FUNCTIONS

- Shalom Benaim, P. Friz
- Mathematics
- 24 August 2006

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… Expand

Valuation of volatility derivatives as an inverse problem

- P. Friz, Jim Gatheral
- Mathematics
- 1 December 2005

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… Expand

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