Pricing Under Rough Volatility

Bayer, Christian; Friz, Peter K.; Gatheral, Jim

doi:10.2139/ssrn.2554754

Cited by 48 publications

(132 citation statements)

References 21 publications

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“…As another experiment, we study Monte Carlo option pricing in the rough Bergomi (rBergomi) model of Bayer et al [10]. In the rBergomi model, the logarithmic spot variance of the price of the underlying is modeled by a rough Gaussian process, which is a special case of (2.14).…”

Section: Option Pricing Under Rough Volatilitymentioning

confidence: 99%

“…The process (ξ 0 t ) t∈ [0,T ] is the so-called forward variance curve [10,Sect. 3], which we assume here to be flat, ξ 0 t = ξ > 0 for all t ∈ [0, T ].…”

Section: Option Pricing Under Rough Volatilitymentioning

confidence: 99%

“…The case α = − 1 6 in (1.2) is important in statistical modeling of turbulence [16] as it gives rise to processes that are compatible with Kolmogorov's scaling law for ideal turbulence. Moreover, processes of similar type with α ≈ −0.4 have been recently used in the context of option pricing as models of rough volatility [1,10,18,19]; see Sects. 2.5 and 3.3 below.…”

Section: Introductionmentioning

confidence: 99%

“…Firstly, we examine the finite-sample performance of an estimator of the roughness index α, introduced by Barndorff-Nielsen et al [6] and Corcuera et al [16]. This experiment enables us to assess how faithfully the hybrid scheme approximates the fine properties of the BSS process X. Secondly, we study Monte Carlo option pricing in the rough Bergomi stochastic volatility model of Bayer et al [10]. We use the hybrid scheme to simulate the volatility process in this model, and we find that the resulting implied volatility smiles are indistinguishable from those obtained using a method that involves exact simulation of the volatility process.…”

Section: Introductionmentioning

confidence: 99%

“…We use the hybrid scheme to simulate the volatility process in this model, and we find that the resulting implied volatility smiles are indistinguishable from those obtained using a method that involves exact simulation of the volatility process. Thus we are able to propose a solution to the problem of finding an efficient simulation scheme for the rough Bergomi model, left open in the paper [10].…”

Section: Introductionmentioning

confidence: 99%

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Hybrid scheme for Brownian semistationary processes

2017

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We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel function by a power function near zero and by a step function elsewhere. The resulting approximation of the process is a combination of Wiener integrals of the power function and a Riemann sum, which is why we call this method a hybrid scheme. Our main theoretical result describes the asymptotics of the mean square error of the hybrid scheme, and we observe that the scheme leads to a substantial improvement of accuracy compared to the ordinary forward Riemann-sum scheme, while having the same computational complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we examine the finite-sample properties of an estimator of the roughness parameter of a Brownian semistationary process and study Monte Carlo option pricing in the rough Bergomi model of Bayer et al. (Quant. Finance 16:887-904, 2016), respectively. B M.S. Pakkanen

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Section: Option Pricing Under Rough Volatilitymentioning

confidence: 99%

“…The process (ξ 0 t ) t∈ [0,T ] is the so-called forward variance curve [10,Sect. 3], which we assume here to be flat, ξ 0 t = ξ > 0 for all t ∈ [0, T ].…”

Section: Option Pricing Under Rough Volatilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hybrid scheme for Brownian semistationary processes

2017

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Unifying pricing formula for several stochastic volatility models with jumps

Baustian

Mrázek

Pospı́šil

et al. 2017

Appl Stoch Models Bus & Ind

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In this paper, we introduce a unifying approach to option pricing under continuous-time stochastic volatility models with jumps. For European style options, a new semi-closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro-differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log-normal and a log-uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out-of-the money contracts.Appl. Stochastic Models Bus. Ind. 2017, 33 422-442 F. BAUSTIAN ET AL. allows to have correlated increments of the asset price and the volatility process (as opposed to Stein and Stein [3]), which can mimic a volatility leverage effect observed on many financial markets. However, the model lacks the ability to fit reasonably well-complex option price surfaces [5,6], especially the ones that involve both short-dated and long-dated contracts.To deal with the drawbacks of the first SV models, many modifications have been introduced since, including a dynamic Heston model that involves time-dependent parameters. The case of piece-wise constant parameters in time is studied in Mikhailov and Ngel [7], a linear time dependence in Elices [8] and a more general case is analysed in Benhamou et al. [9]. The latter result introduces only an approximation of the option price. However, Bayer et al. [5] suggest that the general overall shape of the volatility surface does not change in time, at least to a first approximation given by stochastic volatility inspired (SVI) models. Hence, it is desirable to model volatility as a time-homogeneous process. Other generalizations of the Heston model with time-constant parameters include jump processes in asset price, in volatility or in both (e.g. Duffie et al. [10]). As Gatheral [11] notes (and supports by empirical analyses of several authors), a model with jumps in both underlying and volatility, although having more parameter and degrees of freedom, might not provide significantly better market fit than its counterpart with jumps in underlying only. The first SVJD model introduced in [12] adds a log-normally distributed jumps to the diffusion dynamics of the Heston model. Several different jump-diffusion settings were proposed, for example, models postulated by Scott [13] and Yan and Hanson [14] among others.Another possibility to modify standard diffusion SV models is to use a Lévy subordinator as a driving noise of the volatility process. This idea was firstly developed by where both volatility and as...

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Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

2018

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We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter H, its first Karhunen-Loève eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for H result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1 2 and 3 2 , and the estimator for H sees an affine adjustment, while remaining model-free. AMS 2010 Classification: 60G15, 91G20, 40E05.

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Pricing Under Rough Volatility

Cited by 48 publications

References 21 publications

Hybrid scheme for Brownian semistationary processes

Hybrid scheme for Brownian semistationary processes

Unifying pricing formula for several stochastic volatility models with jumps

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

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