A regularity structure for rough volatility

Bayer, Christian; Friz, Peter K.; Gassiat, Paul; Martin, Joerg; Stemper, Benjamin

doi:10.48550/arxiv.1710.07481

Cited by 12 publications

(29 citation statements)

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“…The main interest of our theorem, of course, lies in the rough regime H ∈ (0, 1/2), where in particular it refines RoughVol large deviations studied by Forde-Zhang [22]. Granted some minimal moments assumptions, it applies to rough volatility models as discussed in [7,8] and notably the rough Bergomi model (with log-normal fractional volatility and negative correlation). The previous remark on Heston applies mutatis mutandis to the rough Heston model [20].…”

Section: Introductionmentioning

confidence: 60%

“…can be removed so that the exponential form of the volatility function ("rough Bergomi" [7]) is covered. In fact, this follows immediately from assumption (A3a-c) below, which hold in the RoughVol setting, due to [8]. But see also [41,33] for related results.…”

Section: Basic Large Deviation Assumption (A1)mentioning

confidence: 78%

“…and the (log)price process under rough volatility can be written as its continuous image Φ(W), see [8]. (All this is reviewed in Appendix A.1.2 for the reader's convenience.)…”

Section: Exact Call Price Formulamentioning

confidence: 99%

“…Evaluation of Wiener functionals ("pricing") under rough volatility goes back to [2], followed by [29,7,30,22,19] and now many others. It was recently seen [8] that Hairer's theory of regularity structures [36], a major extension of Lyons' rough path theory [46], provides a robust formulation of rough volatility models. This was used to derive large deviation estimates, extending the results of [22].…”

Section: Introductionmentioning

confidence: 99%

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Precise asymptotics: robust stochastic volatility models

Friz,

Gassiat,

Pigato

2018

Preprint

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We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of [Bayer et al; A regularity structure for rough volatility, 2017]. In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, Inahama-Kawabi on rough path space. 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 known large deviation asymptotics. Contents 1. Introduction 2. Notation 3. Preliminaries on Black-Scholes asymptotics 4. Unified large, moderate and rough deviations 4.1. Basic large deviation assumption (A1) 4.2. Moment assumption (A2) and large deviation option pricing 5. Exact call price formula 5.1. The case of SDEs and (classical) StochVol 5.2. The case of rough volatility 5.3. Assumptions (A3-5): robust model specification and control theory 5.4. The call price formula 6. Precise asymptotic pricing 6.1. Precise large deviations 6.2. Precise moderate deviations 7. The case of RoughVol 8.

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Section: Introductionmentioning

confidence: 60%

Section: Basic Large Deviation Assumption (A1)mentioning

confidence: 78%

“…and the (log)price process under rough volatility can be written as its continuous image Φ(W), see [8]. (All this is reviewed in Appendix A.1.2 for the reader's convenience.)…”

Section: Exact Call Price Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Precise asymptotics: robust stochastic volatility models

Friz,

Gassiat,

Pigato

2018

Preprint

Self Cite

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“…Adapting the classical notion of rough paths, Gubinelli and Tindel [27] dealt with the mild formulation of rough evolution equations associated to analytic semigroups, which corresponds to infinite dimensional Volterra equations with kernels given by the semigroups. More recently, Bayer et al [7] showed the existence of a solution to a specific rough Volterra equation modelling the 'rough' volatility appearing on financial markets, using Hairer's theory of regularity structures [28].…”

Section: Introductionmentioning

confidence: 99%

Paracontrolled distribution approach to stochastic Volterra equations

Prömel¹,

Trabs²

2018

Preprint

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Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional rough paths. The existence of such rough paths above a wide class of stochastic processes including the fractional Brownian motion is shown. As applications we consider various types of rough and stochastic (partial) differential equations such as rough differential equations with delay, stochastic Volterra equations driven by Gaussian processes and moving average equations driven by Lévy processes.

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Moment explosions in the rough Heston model

Gerhold

Gerstenecker

Pinter

2019

Decisions Econ Finan

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We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arXiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). We show that the critical moments are finite for all maturities. For negative correlation, we apply our algorithm for the moment explosion time to compute the lower critical moment. arXiv:1801.09458v3 [q-fin.MF]

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A regularity structure for rough volatility

Cited by 12 publications

References 0 publications

Precise asymptotics: robust stochastic volatility models

Precise asymptotics: robust stochastic volatility models

Paracontrolled distribution approach to stochastic Volterra equations

Moment explosions in the rough Heston model

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