Is Volatility Rough ?

Fukasawa, Masaaki; Takabatake, Tetsuya; Westphal, Rebecca

doi:10.48550/arxiv.1905.04852

Cited by 21 publications

(36 citation statements)

References 28 publications

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“…The first empirical evidences reported in [15] suggest that the logarithm of the asset price stochastic variance can be represented by a fractional Brownian motion (fBM) of Hurst exponent H close to H 0.1 < 1/2. More recent studies based either on quasi-likelihood approach [12] or GMMapproach [9], consistently suggest the H is even closer to H = 0, i.e., H 0.05 for a large panel of equity data. In that respect, it is natural to consider the limit H → 0 in the rough process driving the volatility logarithm.…”

Section: Introductionmentioning

confidence: 80%

“…However, since historical volatility is not directly observable on financial markets, in order to consider a more realistic scenario, we decided to run the experiments directly on a price model. We consider that a "price" Xt is modelled by a Brownian motion whose variance is a log S-fBM measure dM , i.e., where MH,T is the log-fBM defined in (12) while Bt is a Brownian motion independent of MH,T . Let us notice that, when H = 0, Xt is precisely the MRW process introduced in [25,1].…”

Section: Numerical Illustrations and Empirical Performances Of The Gm...mentioning

confidence: 99%

“…Figure 3: Estimation of the correlation function difference D(n) as defined in Proposition 6. D(n) from observations over intervals of increasing size L= L 0 , L = 4L 0 , L = 16L 0 and L = 64L 0 with L 0 = 212 . The remaining parameters are ∆ = 1, H = 0.1, λ 2 = 0.08 and T = 2L.…”

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confidence: 99%

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From Rough to Multifractal volatility: the log S-fBM model

Wu,

Muzy,

Bacry

2022

Preprint

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We introduce a family of random measures MH,T ( dt), namely log S-fBM, such that, for H > 0, MH,T ( dt) = e ω H,T (t) dt where ωH,T (t) is a Gaussian process that can be considered as a stationary version of a H-fractional Brownian motion. Moreover, when H → 0, one has MH,T ( dt) → MT ( dt) (in the weak sense) where MT ( dt) is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal (H = 0) and rough volatility (0 < H < 1/2) models. The main properties of the log S-fBM are discussed and their estimation issues are addressed. We notably show that the direct estimation of H from the scaling properties of ln(MH,T ([t, t+τ ])), at fixed τ , can lead to strongly over-estimating the value of H. We propose a better GMM estimation method which is shown to be valid in the high-frequency asymptotic regime. When applied to a large set of empirical volatility data, we observe that stock indices have values around H = 0.1 while individual stocks are characterised by values of H that can be very close to 0 and thus well described by a MRM. We also bring evidence that, unlike the log-volatility variance ν 2 whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called "intermittency coefficient" λ 2 , which is the product of ν 2 and the Hurst exponent H, appears to be far more reliable leading to values that seems to be universal for respectively all individual stocks and all stock indices.

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Section: Introductionmentioning

confidence: 80%

Section: Numerical Illustrations and Empirical Performances Of The Gm...mentioning

confidence: 99%

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From Rough to Multifractal volatility: the log S-fBM model

Wu,

Muzy,

Bacry

2022

Preprint

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“…From empirical studies, Gatheral et al (2018) find that the log-volatility essentially behaves like a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. Further empirical evidence has been documented in Fukasawa et al (2019), where the authors constructed a quasi-likelihood estimator applied to realized volatility time series, and confirmed that the Hurst parameter is much smaller than half, i.e. volatility is indeed rough.…”

Section: Introductionmentioning

confidence: 82%

Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models

Ma¹,

Yang²,

Cui³

2021

Preprint

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Rough volatility models have recently been empirically shown to provide a good fit to historical volatility time series and implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with Hurst parameter less than half. Due to the challenge that it is neither a semimartingale nor a Markov process, there is no unified method that not only applies to all rough volatility models, but also is computationally efficient. This paper proposes a semimartingale and continuous-time Markov chain (CTMC) approximation approach for the general class of rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semimartingale approximation for the RSLV model and establish its existence, uniqueness and Markovian representation. We propose a fast CTMC algorithm and prove its weak convergence. Numerical experiments demonstrate the accuracy and high efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

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“…H < 1 2 . This empirical observation lead to the construction of stochastic models with strong anti-persistent volatility dynamics, the so called rough volatility models (Gatheral et al, 2018;Takaishi, 2020;Fukasawa et al, 2019;Livieri et al, 2018). One of the most leading models in this category is the rough Bergomi model (hereafter rBergomi), see (Bayer et al, 2016(Bayer et al, , 2019.…”

Section: Rough Volatilitymentioning

confidence: 99%

Multiscaling and rough volatility: an empirical investigation

Brandi¹,

Matteo²

2022

Preprint

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Pricing derivatives goes back to the acclaimed Black and Scholes model. However, such a modeling approach is known not to be able to reproduce some of the financial stylized facts, including the dynamics of volatility. In the mathematical finance community, it has therefore emerged a new paradigm, named rough volatility modeling, that represents the volatility dynamics of financial assets as a fractional Brownian motion with Hurst exponent very small, which indeed produces rough paths. At the same time, prices' time series have been shown to be multiscaling, characterized by different Hurst scaling exponents. This paper assesses the interplay, if present, between price multiscaling and volatility roughness, defined as the (low) Hurst exponent of the volatility process. In particular, we perform extensive simulation experiments by using one of the leading rough volatility models present in the literature, the rough Bergomi model. A real data analysis is also carried out in order to test if the rough volatility model reproduces the same relationship. We find that the model is able to reproduce multiscaling features of the prices' time series when a low value of the Hurst exponent is used but it fails to reproduce what the real data say. Indeed, we find that the dependency between prices' multiscaling and the Hurst exponent of the volatility process is diametrically opposite to what we find in real data, namely a negative interplay between the two.

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Is Volatility Rough ?

Cited by 21 publications

References 28 publications

From Rough to Multifractal volatility: the log S-fBM model

From Rough to Multifractal volatility: the log S-fBM model

Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models

Multiscaling and rough volatility: an empirical investigation

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