The characteristic function of rough Heston models

Euch, Omar El; Rosenbaum, Mathieu

doi:10.48550/arxiv.1609.02108

Cited by 12 publications

(31 citation statements)

References 29 publications

(72 reference statements)

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“…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]. As a sanity check, let us also point out that Black-Scholes corresponds to H = 1/2, Λ(x) = x 2 /(2σ 2 ) and σ x ≡ σ.…”

Section: Introductionmentioning

confidence: 71%

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: 71%

Precise asymptotics: robust stochastic volatility models

Friz,

Gassiat,

Pigato

2018

Preprint

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“…In particular, the exploding term structure when maturity goes to zero of the at-the-money skew (the derivative of the implied volatility with respect to strike) is readily obtained, see [1,11]. Other developments about rough volatility models can be found in [2,3,8,9,10,12,14,18,21].…”

Section: Introductionmentioning

confidence: 99%

Rough Volatility: Evidence from Option Prices

et al. 2017

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It has been recently shown that spot volatilities can be very well modeled by rough stochastic volatility type dynamics. In such models, the log-volatility follows a fractional Brownian motion with Hurst parameter smaller than 1/2. This result has been established using high frequency volatility estimations from historical price data. We revisit this finding by studying implied volatility based approximations of the spot volatility. Using at-the-money options on the S&P500 index with short maturity, we are able to confirm that volatility is rough. The Hurst parameter found here, of order 0.3, is slightly larger than that usually obtained from historical data. This is easily explained from a smoothing effect due to the remaining time to maturity of the considered options.

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“…2 Note that, in contrast even to classical SV models, the stochastic volatility is explicitly given, and no rough / stochastic differential equation needs to be solved (hence "simple"). Rough volatility not only provides remarkable fits to both time series and option pricing problems, it also has a market microstructure justification: starting with a Hawkes process model, Rosenbaum and coworkers [16,17,18] find in the scaling limit f, g, h such that…”

Section: Introductionmentioning

confidence: 99%

A regularity structure for rough volatility

Bayer¹,

Friz²,

Gassiat

et al. 2019

Mathematical Finance

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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, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.Dedicated to Professor Jim Gatheral on the occasion of his 60th birthday.Contents arXiv:1710.07481v1 [q-fin.PR]

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The characteristic function of rough Heston models

Cited by 12 publications

References 29 publications

Precise asymptotics: robust stochastic volatility models

Precise asymptotics: robust stochastic volatility models

Rough Volatility: Evidence from Option Prices

A regularity structure for rough volatility

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