On the probability of hitting the boundary for Brownian motions on the SABR plane

Gulisashvili, Archil; Horvath, Blanka; Jacquier, Antoine

doi:10.1214/16-ecp26

Cited by 9 publications

(6 citation statements)

References 30 publications

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“…If one starts from suitable single-asset volatility models, such as for example the Heston model [24], and correlates their shocks by resorting to instantaneous co-variation among Brownian motions driving different components, one does not have a manageable dynamical model for the 1 INTRODUCTION 2 index/basket leading to a closed form solution for the index smile. This holds for the large majority of single-asset smile models, including practitioners models such as SABR (see for example [1] for an introduction, and [16,19] for an in-depth study of mathematical properties of the SABR model). On the other hand, modelling directly the multivariate distribution without an underlying dynamics could lead to arbitrageable models.…”

Section: Introductionmentioning

confidence: 94%

The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles

Brigo

Rapisarda²,

Sridi

2017

IISE Transactions

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We introduce a new arbitrage-free multivariate dynamic asset pricing model that allows us to reconcile single name and index/basket volatility smiles using a tractable and explicit dependence structure that goes beyond instantaneous correlation. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. After introducing the model, we derive tractable index option smile formulas resulting from the model and related closed form solutions for multivariate densities taking the form of multivariate mixtures. Using Markovian projection techniques, we relate our model to a multivariate uncertain volatility model and show a consistency result with geometric baskets with hints on possible uses in investigating triangular relationships between foreign exchange rates and the related smiles in practice. We also derive closed form solutions for a number of terminal statistics of dependence, and derive a precise relationship with a simpler but less tractable model based on a basic instantaneous correlation structure. Finally, closed form solutions for volatility/assets correlations illuminating the relationship with the uncertain volatility model are introduced. The model tractability makes it particularly suited for calibration and risk management applications, where speed of calculations and tractability are essential. A few numerical examples on basket and spread options pricing conclude the paper.

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Section: Introductionmentioning

confidence: 94%

The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles

Brigo

Rapisarda²,

Sridi

2017

IISE Transactions

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“…In certain parameter regimes the exact density has been derived for the absolutely continuous part (on (0, ∞)) of the distribution of X: in the uncorrelated case ρ = 0, formulae were obtained in [3,28] by applying time-change techniques. The correlated case is much harder, and approximations have been derived using projection methods in [3,4], and using geometric tools in [23]. Barring computational costs, availability of the distribution of the SABR process is equivalent to computing any European prices.…”

Section: Introductionmentioning

confidence: 99%

Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

Gulisashvili

Horvath

Jacquier

2018

Quantitative Finance

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We study the mass at the origin in the uncorrelated SABR stochastic volatility model, and derive several tractable expressions, in particular when time becomes small or large. As an application-in fact the original motivation for this paper-we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.

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“…However, it seems that these refinements have not fully resolved the arbitrage issue near the origin. Recent results [29,38,39] focus on the singular part of the distribution and suggest an explanation for the irregularities appearing at interest rates near zero; and [38,39] provides a means to regularize Hagan's asymptotic formula at low strikes for specific parameter configurations, based on tail asymptotics derived in [26,37].…”

Section: Introductionmentioning

confidence: 99%

“…Although the exact distribution of the CEV process is available [55], simulation of the full SABR model based on it can in many cases become involved and expensive. In fact, exact formulas decomposing the SABR-distribution into a CEV part and a volatility part are only available in restricted parameter regimes, see [6,32,48] for the absolutely continuous part and [38,39] for the singular part of the distribution. A simple space transformation (see (1.5) below) makes some numerical approximation results for the CIR model (the perhaps most well-understood degenerate diffusion) applicable to certain parameter regimes of the SABR process.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dirichlet Forms and Finite Element Methods for the SABR Model

Horvath

Reichmann²

2018

SIAM J. Finan. Math.

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We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the process both in moderate interest rate environments and in near-zero interest rate regimes such as the currently prevalent ones. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding nonstandard partial differential equations, for which conventional pricing methods -designed for non-degenerate parabolic equations-potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form, its dual space, and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present a finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a θ-scheme in time and provide an error analysis for this discretization.The time derivative of a function u in the appropriate Bochner space is understood in the weak sense: For u ∈ L 2 (J; V), its weak derivative inu ∈ L 2 (J, V * ) ∩ H 1 (J; V * ) is defined by the relation (2.16) J (u(t), v) V * ×V ϕ(t)dt = − J (u(t), v) V * ×Vφ (t)dt, see [44, Sections 2.1 and 3.1] for definitions and properties of Bochner spaces.

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On the probability of hitting the boundary for Brownian motions on the SABR plane

Cited by 9 publications

References 30 publications

The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles

The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles

Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

Dirichlet Forms and Finite Element Methods for the SABR Model

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