On the martingale property in the rough Bergomi model

Gassiat, Paul

doi:10.1214/19-ecp239

Cited by 21 publications

(21 citation statements)

References 47 publications

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“…In order to be able to use the system (1) for pricing purposes, we need to show how to complete the market, and to check for the existence of some probability measure under which the stock price is a true martingale. The latter issue was solved for the rough Heston model by El Euch and Rosenbaum [25], while the rough Bergomi case with non-positive correlation was recently proved by Gassiat [31]. We show that this still holds in our framework.…”

Section: 2supporting

confidence: 56%

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Deep PPDEs for Rough Local Stochastic Volatility

Jacquier

Oumgari²

2019

SSRN Journal

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Section: 2supporting

confidence: 56%

“…Remark 2.5. Following [31], we could in fact partially relax the assumption on the function l(·). Assume for example that l(t, S, v) = L(t, S)ς(v), but with Assumption 2.1 replaced by ς(v) = exp(ηv), for some η > 0.…”

Section: 2mentioning

confidence: 99%

Deep PPDEs for Rough Local Stochastic Volatility

Jacquier

Oumgari²

2019

SSRN Journal

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“…In the Scott model, the volatility process is the Ornstein-Uhlenbeck process, while the volatility function is σ(x) = e x . Gassiat obtained a similar result for a model with ρ < 0, the volatility function satisfying condition (G), and the Riemann-Liouville fractional Brownian motion with H > 1 2 as the volatility process (see Theorem 2 in [24]). We do not assume in part (ii) of Theorem 6.11 that the model is negatively correlated and the volatility function satisfies condition (G).…”

Section: Introductionmentioning

confidence: 66%

“…The previous assertion was first included in the arXiv:1808.00421v4 (September 22, 2018) version of the present paper. The same result in a special case, where the volatility process is the Riemann-Liouville fractional Brownian motion with the Hurst index H > 1 2 , while the volatility function satisfies an additional condition (condition (G) formulated before Theorem 6.11), was obtained independently, but a little later, by Gassiat (see Theorem 2 in the arXiv:1811.10935v1 (November 27, 2018) version of [24]). We also show that in a correlated Volterra type Gaussian model (ρ = 0), all the moments of the order γ > 1 1−ρ 2 7…”

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confidence: 67%

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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“…The most basic form of (lognormal) rough volatility model is the so-called rough Bergomi model introduced in [BFG16]. Gassiat [Gas18] recently proved that such a model (under certain correlation regimes) generates true martingales for the spot process. The lack of Markovianity imposes numerous fundamental theoretical questions and practical challenges in order to make rough volatility usable in an industrial environment.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

2019

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We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around-the-money.Date: March 25, 2019.2010 Mathematics Subject Classification. Primary 60F10, 60G22; Secondary 91G20, 60G15, 91G60. ε > 0, P ε is the joint law and, for each δ > 0, the set ω :and lim sup ε↓0 h ε log P ε (Γ δ ) = −∞, where Γ δ := {(ỹ, y) : d(ỹ, y) > δ} ⊂ Y × Y. Theorem D.2. Let X and Y be topological spaces and f : X → Y a continuous function. Consider a good rate function I : X → [0, ∞]. For each y ∈ Y, define I (y) := inf{I(x) : x ∈ X , y = f (x)}. Then, if I controls the LDP associated with a family of probability measures {µ ε } on X , the I controls the LDP associated with the family of probability measures {µ ε • f −1 } on Y and I is a good rate function on Y.

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On the martingale property in the rough Bergomi model

Cited by 21 publications

References 47 publications

Deep PPDEs for Rough Local Stochastic Volatility

Deep PPDEs for Rough Local Stochastic Volatility

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

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

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