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Pathwise large deviations for the rough Bergomi model

Antoine Jacquier, Mikko S. Pakkanen, Henry Stone
2018 Journal of Applied Probability  
We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model.  ...  Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets.  ...  In Section 4, we give the proofs of the main results, and Section 5 elucidates the analogous large deviations result for the uncorrelated rough Bergomi model.  ... 
doi:10.1017/jpr.2018.72 fatcat:bx5scfu4szbphhofeqjcehd7ou

Pathwise large deviations for the rough Bergomi model: Corrigendum

Stefan Gerhold, Antoine Jacquier, Mikko Pakkanen, Henry Stone, Thomas Wagenhofer
2021 Journal of Applied Probability  
AbstractThis note corrects an error in the definition of the rate function in Jacquier, Pakkanen, and Stone (2018) and slightly simplifies some proofs.  ...  To make the estimate work for t = 0, confine ε to the finite interval [0,1] instead of R + in line −4 of p. 1088. ρ in line 4 of p. 1083 and in the proof of [1, Theorem 2.1] on p. 1088.  ...  Injectiveness of I follows from the Titchmarsh convolution theorem. We have (I ) * μ = 1 · (·, t)μ(dt) for any measure μ ∈ (C 2 ) * .  ... 
doi:10.1017/jpr.2020.109 fatcat:xfdzsbzalzcj3gddmokixpoqem

Asymptotics for volatility derivatives in multi-factor rough volatility models [article]

Chloe Lacombe, Aitor Muguruza, Henry Stone
2020 arXiv   pre-print
We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle.  ...  Interestingly, we identify the class of models that generate non-linear smiles around-the-money.  ...  Corollary 3.4 can be applied to a large number of existing results on large deviations for (rough) variance processes to get a large deviations result for the integrated (rough) variance process; for example  ... 
arXiv:1903.02833v3 fatcat:dqk33d3wgngq7dztj443lzsefq

Asymptotics for volatility derivatives in multi-factor rough volatility models

Chloe Lacombe, Aitor Muguruza, Henry Stone
2021 Mathematics and Financial Economics  
AbstractWe study the small-time implied volatility smile for Realised Variance options, and investigate the effect of correlation in multi-factor models on the linearity of the smile.  ...  Additionally, we establish small-noise asymptotic behaviour of a general class of VIX options in the large strike regime.  ...  The authors are grateful to Antoine Jacquier, Mikko Pakkanen and Ryan McCrickerd for stimulating discussions. AM and HS thank the EPSRC CDT in Financial Computing and Analytics for financial support.  ... 
doi:10.1007/s11579-020-00288-5 fatcat:cxcji2abq5g3rhkzxgotitkz7u

Large and moderate deviations for stochastic Volterra systems [article]

Antoine Jacquier, Alexandre Pannier
2020 arXiv   pre-print
We provide a unified treatment of pathwise Large and Moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution  ...  We show in particular how this framework encompasses most rough volatility models used in mathematical finance and generalises many recent results in the literature.  ...  ] , while more elaborate pathwise LDPs were derived for the rough Stein-Stein model with random starting point [56] , for the rough Bergomi model [58] , and small-time LDPs for the multifactor rough  ... 
arXiv:2004.10571v2 fatcat:6lzi7me2xbd3veytlxnq3auws4

Short dated smile under Rough Volatility: asymptotics and numerics [article]

Peter K. Friz, Paul Gassiat, Paolo Pigato
2021 arXiv   pre-print
We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility.  ...  , using the framework [Bayer et al; A regularity structure for rough volatility; Math.  ...  A case study: the rough Bergomi model 4.1. The rough Bergomi model.  ... 
arXiv:2009.08814v2 fatcat:ly44nazctnapnnxndgtoaxn3ay

Precise asymptotics: robust stochastic volatility models [article]

Peter K. Friz, Paul Gassiat, Paolo Pigato
2020 arXiv   pre-print
known large deviation asymptotics.  ...  When applied to rough volatility models, e.g. in the setting of [Forde-Zhang, Asymptotics for rough stochastic volatility models, 2017], one obtains precise asymptotic for European options which refine  ...  Unified large, moderate and rough deviations We now put forward our basic large deviation assumption.  ... 
arXiv:1811.00267v2 fatcat:4gudex772vbitmx45bgs75hr2u

A regularity structure for rough volatility [article]

Christian Bayer, Peter K. Friz, Paul Gassiat, Joerg Martin, Benjamin Stemper
2017 arXiv   pre-print
and powerful tool to analyze rough volatility models.  ...  A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models  ...  The second set of results concerns large deviations for rough volatility.  ... 
arXiv:1710.07481v1 fatcat:vzpxkmkcsbftpdugk23wtz3pxe

Rough volatility models: small-time asymptotics and calibration

Henry Stone, Antoine Jacquier, Mikko Pakkanen, Engineering And Physical Sciences Research Council
2020
We prove a pathwise large deviations principle for a small-noise version of the model, and use this result to establish the small-time behaviour of the rescaled log stock price process.  ...  In Chapter 3 we present small-time implied volatility asymptotics for realised variance (RV) options for a number of (rough) stochastic volatility models via a large deviations principle.  ...  Corollary 3.2.3 can be applied to a large number of existing results on pathwise large deviations for rough variance processes to get a large deviations result for the integrated rough variance process  ... 
doi:10.25560/81768 fatcat:mke4gdxdxvgufod3iq5kg5a5xa

Multivariate Stochastic Volatility Models and Large Deviation Principles [article]

Archil Gulisashvili
2022 arXiv   pre-print
The LDP for log-processes is used to obtain large deviation style asymptotic formulas for the distribution function of the first exit time of a log-process from an open set and for the price of a multidimensional  ...  We establish a comprehensive sample path large deviation principle (LDP) for log-processes associated with multivariate time-inhomogeneous stochastic volatility models.  ...  The previous model is similar in structure to the rough Bergomi model introduced in [3] , and it may be called the super rough Bergomi model. More details can be found in [33] and [4] .  ... 
arXiv:2203.09015v2 fatcat:bl5svd6kgjg27f4neuun2hpqwm

Large deviations for rough and complete stochastic volatility models

Chloe Alice Lacombe, Antoine Jacquier
2020
the (multi-factor) rough Bergomi model for realised variance options.  ...  This thesis is concerned with deriving the asymptotic behaviour of these two classes of stochastic volatility models using large deviations techniques, in order to understand the behaviour of the implied  ...  Acknowledgments First and foremost, I would like to thank my supervisor, Doctor Antoine Jacquier, whose experience, passion and continuous support guided me through the completion of this thesis.  ... 
doi:10.25560/82258 fatcat:obk6u34mlnhu3hhaohcwezrghi

A regularity structure for rough volatility

Christian Bayer, Peter K. Friz, Paul Gassiat, Jorg Martin, Benjamin Stemper, Technische Universität Berlin
2020
and powerful tool to analyze rough volatility models.  ...  A new paradigm has emerged recently in financial modeling: rough (stochastic) volatility.  ...  , 1∕2). (5) (Later on, we will allow for explicit time dependence of in order to cover the rough Bergomi model .)  ... 
doi:10.14279/depositonce-10842 fatcat:wx4ws2uhg5dl3pqh2arwcp66om

Affine and quadratic models for volatility and interest rates markets

Elise Gourier
2013
We use a large dataset of S&P 500 and VIX index and option prices with wide ranges of maturities and moneynesses, and analyze the empirical performance of affine jump-diffusion models for S&P 500 returns  ...  We analyze the out-of-sample performance of sub-models depending on which products and markets are considered in the in-sample estimation procedure.  ...  Markus Leippold, my thesis supervisor, for his support and advice, and for the great work atmosphere I could benefit from.  ... 
doi:10.5167/uzh-94023 fatcat:mdkgwndfizg5llugfaxbvexcla

Volatility and Correlation Modelling for Equity Indices

Chris Bardgett
2014
As stated above, we already have pathwise uniqueness for the first n components of the process X. It remains to show it for the remaining components.  ...  Even though the process m is more stable than v, the approximation made by the SVJ model of the process m being a constant is too rough.  ...  The correlation matrix process C t for the Brownian motions (W 1 , · · · , W n ) defined by the model (1) -(3) -(5) is continuous and almost surely positive definite on any interval [0, T ], for any T  ... 
doi:10.5167/uzh-108295 fatcat:4gd6gxjclzhuff5e2uyfon6sfa