Pathwise asymptotics for Volterra type stochastic volatility models

Cellupica, M.; Pacchiarotti, Barbara

doi:10.48550/arxiv.1902.05896

Cited by 3 publications

(8 citation statements)

References 21 publications

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“…In [22], a sample path LDP was established for the log-price process under the same conditions. Similar results were obtained later in [7] under more restrictive conditions. We also refer the reader to [2,8,35,38] for applications of sample path LDPs in financial mathematics.…”

Section: Large Deviation Principles In General Gaussian Stochastic Vo...supporting

confidence: 88%

“…In Section 5, Theorem 5.1, we provide a large-deviation style asymptotic formula for the exit time probability function, using Theorem 3.2. A similar result was obtained in [7,22] for less general models. At the very end of Section 5, we mention possible applications of Theorem 3.4 to the study of small-noise asymptotic behavior of option pricing functions and the implied volatility (see Remark 5.4).…”

Section: Remark 15 We Will Assume In the Present Paper That The Modul...supporting

confidence: 85%

“…In Theorems 3.3 and 3.4, small-noise LDPs are obtained. Large deviation principles similar to those in Theorems 3.2 and 3.4 were established earlier in [15,21,22,7] for time-homogeneous models under much stronger restrictions on the drift function, the volatility function, and the volatility process. See also [25] for sample path LDPs in randomised fractional volatility models.…”

Section: Remark 15 We Will Assume In the Present Paper That The Modul...supporting

confidence: 70%

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Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness

Gulisashvili¹

2020

Preprint

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We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a positive function of a Volterra type continuous Gaussian process that may have extremely rough sample paths. The drift function and the volatility function are assumed to be time-dependent and locally ω-continuous for some modulus of continuity ω. The main result obtained in the paper is a sample path large deviation principle for the log-price process in a Gaussian model under very mild restrictions. We apply this result to study the first exit time of the log-price process from an open interval.

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Section: Large Deviation Principles In General Gaussian Stochastic Vo...supporting

confidence: 88%

Section: Remark 15 We Will Assume In the Present Paper That The Modul...supporting

confidence: 85%

Section: Remark 15 We Will Assume In the Present Paper That The Modul...supporting

confidence: 70%

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Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness

Gulisashvili¹

2020

Preprint

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“…It is clear that the components of the process F ε are building blocks of the process X (ε) , and our aim is to use the extended contraction principle (see [10]) to establish large deviation principles for the process → X (ε) − x 0 . Some of the techniques used in such proofs were developed in [15,22,23,24,5] in the case, where the volatility is modeled by a function of a Gaussian process, and in [17] for certain non-Gaussian models. In this section, we borrow some ideas employed in the proof of the sample path LDP in Theorem 4.2 of [24] (see Subsection 5.6 of [24]).…”

Section: Definition 31 a Locally Bounded Functionmentioning

confidence: 99%

“…It is not hard to see that the process Y (1) defined by (1.2) is a continuous Gaussian process with the mean function m(t) = e −qt y 0 + 1 − e −qt m, t ∈ [0, T], and the covariance function of stochastic volatility models, where the volatility is a nonnegative function of a Volterra type Gaussian process (see, e.g., [15,22,23,24,5]). We have chosen only these references here because all of them are related to the main subjects of the present paper, which are sample path and small noise large deviation principles for log-prices in stochastic volatility models.…”

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confidence: 99%

Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model

Gulisashvili¹

2020

Preprint

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We introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being because of a certain volatility misspecification in the classical Stein and Stein model. A version of this model that uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a special example of a stochastic volatility model with reflection. The main results obtained in the present paper are sample path and small-noise large deviation principles for the log-price process in a stochastic volatility model with reflection under rather mild restrictions. We use these results to study the asymptotic behavior of binary barrier options and call prices in the small-noise regime.

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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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Pathwise asymptotics for Volterra type stochastic volatility models

Cited by 3 publications

References 21 publications

Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness

Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness

Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model

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

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