Volatility has to be rough

Fukasawa, Masaaki

doi:10.48550/arxiv.2002.09215

Cited by 3 publications

(5 citation statements)

References 23 publications

(35 reference statements)

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“…In the seminal paper Gatheral et al [20] conducted empirical study on stochastic volatility and discovered that the log-volatility behaves like a FBM with Hurst exponent H < 1/2 (mostly between 0.08 and 0.2), and thus proposed a "rough" volatility model. For the recent developments in the empirical studies of the Hurst parameter of financial data and the microstructure of leverage effects, see also [1,30,18] and papers listed on the webpage [44]. In particular, for a given asset with log-price taking the form dP (t)…”

Section: Applications In Financial Modelsmentioning

confidence: 99%

Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance

Ichiba¹,

Pang²,

Taqqu³

2020

Preprint

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We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019). We discuss the applications of the GFBM and its mixtures in financial models, including stock price models, arbitrage and rough volatility. The GFBM is self-similar and has nonstationary increments, whose Hurst parameter H ∈ (0, 1) is determined by two parameters. We identify the region of these two parameter values in which the GFBM is a semimartingale. We also establish the p-variation results of the GFBM, which are used to provide an alternative proof of the non-semimartingale property when H < 1/2. We then study the semimartingale properties of the mixed process of an independent Brownian motion and a GFBM when the Hurst parameter H ∈ (1/2, 1), and derive the associated equivalent Brownian measure. INTRODUCTIONSemimartingale and non-semimartingale properties of the standard fractional Brownian motion (FBM) B H and its mixtures are well understood. These properties are important in modeling stock price [23,34], constructing arbitrage strategies and hedging policies [33,41,37,13], and modeling rough volatility [20,7, 44]. The standard FBM B H captures short/long-range dependence, and possesses the self-similar and stationary increment properties, as well as regular path properties. It may arise as the limit process of scaled random walks with long-range dependence or an integrated shot noise process [32].A generalized fractional Brownian motion (GFBM) X, introduced by Pang and Taqqu [31], is a selfsimilar Gaussian process, but does not have the stationary increments property. See the definition of the process in (2.1). The Hurst parameter H ∈ (0, 1) is determined by two-parameters (α, γ) in the range shown in Figure 1. The GFBM X is derived as the limit of integrated power-law shot noise processes in [31] (see a brief review in Section 6.1). We have studied in [21] some important path properties of the GFBM X, including the Hölder continuity property, the differentiability and non-differentiability properties, and functional and local law of the iterated logarithm, see Section 2 for a summary of its fundamental properties.

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Section: Applications In Financial Modelsmentioning

confidence: 99%

Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance

Ichiba¹,

Pang²,

Taqqu³

2020

Preprint

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“…It has been shown in recent years that RoughVol models provide great fits to observed volatility surfaces [4] capturing fundamental stylized facts of implied volatility in a parsimonious way. Specifically, this class of models can reproduce the steep short end of the smile, displaying exploding implied skew [3,23,24], and they are the only models consistent with the power law of the skew [4,40] not admitting arbitrage [25]. RoughVol is also supported by statistical and time series analysis [28,26,10] and by market microstructure considerations [15].…”

Section: Introductionmentioning

confidence: 76%

Short dated smile under Rough Volatility: asymptotics and numerics

Friz¹,

Gassiat²,

Pigato³

2020

Preprint

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In [Precise Asymptotics for Robust Stochastic Volatility Models; Ann. Appl. Probab. 2020] we introduce 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, using the framework [Bayer et al; A regularity structure for rough volatility; Math. Fin. 2020]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence.

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“…We attempt here to understand the short time behavior of a log-modulated rough stochastic volatility model without considering the small vol-of-vol regime, but just the short time asymptotics. For this, we adapt Fukasawa's framework of [13,6,14] to log-fractional volatility, using the "regular variation" language. Let us consider, similarly to (5.6), a stochastic volatility model of the form .…”

Section: Asymptotic Skew Under Log-fractional Volatilitymentioning

confidence: 99%

“…Here, we also include an asymptotic expansion for the rough Bergomi model, when driven by a log-fBm. In Section 6 we consider a slightly more general kernel and the asymptotics for the skew at the Edgeworth CLT regime, that holds for any vol-of-vol parameter, generalising a result in [14]. At last, in Section 7 we provide some details on numerical simulations and computations of the skew.…”

Section: Introductionmentioning

confidence: 99%

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Log-modulated rough stochastic volatility models

Bayer¹,

Harang²,

Pigato³

2020

Preprint

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We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index H. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for H = 0. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range 0 ≤ H < 1/2 without the need of further normalization. We obtain skew asymptotics of the form log(1/T ) −p T H−1/2 as T → 0, H ≥ 0, so no flattening of the skew occurs as H → 0.

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Volatility has to be rough

Cited by 3 publications

References 23 publications

Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance

Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance

Short dated smile under Rough Volatility: asymptotics and numerics

Log-modulated rough stochastic volatility models

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