Pathwise Asymptotics for Volterra Type Stochastic Volatility Models

Cellupica, M.; Pacchiarotti, Barbara

doi:10.1007/s10959-020-00992-4

Cited by 13 publications

(28 citation statements)

References 26 publications

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“…Remark 2.10. A similar LDP was recently obtained in [7] under more restrictive assumptions. For instance, it is supposed in [7] that the volatility function σ is α-Hölder continuous and bounded from above on R, while we assume in Theorem 2.9 that σ is locally ω-continuous for some modulus of continuity ω.…”

Section: Large Deviations: β =supporting

confidence: 77%

“…In Section 2, we characterize the leading term in the asymptotic expansion of the exit time probability function, using the large deviation principle obtained in Theorem 2.9 (see Theorem 2.16). A similar result was obtained in [7] under more restrictions on the volatility function σ. Note that a large deviation principle for the process ε → X ε,0,H explode (see part (ii) of Theorem 6.11).…”

Section: Introductionsupporting

confidence: 82%

“…In Section 2, we prove a sample path large deviation principle (LDP) for the log-price process ε → X ε,0,H (see Theorem 2.9). A similar sample path LDP was obtained in a recent preprint [7] of Cellupica and Pacchiarotti under more restrictive assumptions on the volatility function σ. It is common to apply small-noise sample path large deviation principles in the study of the asymptotic behavior of the exit probabilities.…”

Section: Introductionsupporting

confidence: 62%

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Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

Gulisashvili

2019

SSRN Journal

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In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models. AMS 2010 Classification: 60F10, 60G15, 60G18, 60G22, 41A60, 91G20. Keywords:Gaussian stochastic volatility models, Volterra type models, sample path large and moderate deviations, central limit regime, moment explosions, implied volatility asymptotics. T with state space R was earlier established in Forde and Zhang [17] in the case, where the function σ satisfies the global Hölder condition, while the process B is fractional Brownian motion. In [25], we proved the Forde-Zhang LDP under very mild restrictions on σ and B. We formulate the latter result in Section 2.If 0 < β < H, then the model is in the moderate deviation scaling regime (see, e.g., [4,13,20], and the references therein for more information on moderate deviations). In Section 3, we prove a sample path moderate deviation principle (MDP) for the process ε → X ε,β,H (see Theorem 3.1), and derive a corresponding MDP for the process ε → X ε,β,H T (see Corollary 3.5). As it often happens in the theory of moderate deviations, the rate function in Corollary 3.5 is quadratic. At the end of Section 3, we explain how to pass from small-noise large and moderate deviation principles to small-time ones under the condition that the volatility process is self-similar.The case where β = H corresponds to the central limit (CL) scaling regime. In Section 4, we characterize the limiting behavior on the path space of the distribution function of the process ε → X ε,H,H (see Theorem 4.1), and also that of the process ε → X ε,H,H T in the space R (see Theorem 4.3). The results in the CL regime can be considered as degenerate MDPs with the rate function equal to a constant (see Remark 4.4 in Section 4). An example 3 of a CL scaling can be found in [22]. The volatility of an asset in [22] is modeled by the process δ → σ(δ U), where σ is a smooth function, while U is the stationary fractional Ornstein-Uhlenbeck process (our notation is different from that used in Section 3 of [22]). The CL scaling in [22] corresponds to the following values of the parameters: H = 1 and β = 1 (our notation).It follows from what was said above that the class of small-noise parametrizations of the log-price process in a Gaussian stochastic volatility model (see formula (1.4)) can be split into three disjoint subclasses, which correspond to large deviation, moderate deviation, and central limit scaling regimes. An interesting discussion of certain differences between those regimes can be found in [13]. Gaussian stochastic volatility models and their scaled versions were stud...

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Section: Large Deviations: β =supporting

confidence: 77%

Section: Introductionsupporting

confidence: 82%

Section: Introductionsupporting

confidence: 62%

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Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

Gulisashvili

2019

SSRN Journal

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“…

We study multidimensional stochastic volatility models in which the volatility process is a positive continuous function of a continuous multidimensional Volterra process. The main results obtained in this paper are a generalization of the results due, in the one-dimensional case, to Cellupica and Pacchiarotti in [6]. We state some large deviation principles for the scaled log-price.

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supporting

confidence: 58%

“…The last few years have seen renewed interest in stochastic volatility models in which the volatility process is a positive continuous function σ of a continuous stochastic process B, that we assume to be a Volterra type Gaussian process. The goal of this paper is to extend to the multidimensional case and in a more general asset a problem of large deviations for the log-price process of Volterra type stochastic volatility models, studied in the one-dimensional case in [6], [17] and [18] (time-inhomogeneous case). Large deviations theory deals with the exponential decay of probabilities of "rare events", i.e.…”

Section: Introductionmentioning

confidence: 99%