The Large-Maturity Smile for the Sabr and Cev-Heston Models

Forde, Martin; Pogudin, Andrey

doi:10.1142/s0219024913500477

Cited by 15 publications

(20 citation statements)

References 26 publications

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“…For example an explicit formula (involving a one dimensional integral) for the transition probability density function of the SABR model when 0 β = or 1 β = and ( ) 1,1 ρ ∈ − has been obtained in [4]. Similar results are contained in [15] when 1 β = , 0 ρ ≤ and in [7] for a modified SABR model. In [16] an option pricing problem is studied.…”

Section: Introductionmentioning

confidence: 57%

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The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices

Fatone¹,

Mariani²,

Recchioni³

et al. 2014

JAMP

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How to cite this paper: Fatone, L., et al. (2014) AbstractThe SABR stochastic volatility model with β-volatility β є (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are ex-

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Section: Introductionmentioning

confidence: 57%

“…This last kernel has already been used in [4] to express the transition probability density function of the SABR model when 0 β = or . Previously the same kernel has been used in the study of the transition probability density function of the time integral of a geometric Brownian motion (see [15] [20]). …”

Section: Introductionmentioning

confidence: 99%

The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices

Fatone¹,

Mariani²,

Recchioni³

et al. 2014

JAMP

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“…Second, regarding long maturity asymptotics, Forde and Pogudin (2013) study the SABR and CEV-Heston models in different strike regimes, but mainly assuming null-correlation.…”

Section: Comparison With the Literaturementioning

confidence: 99%

Analytical approximations of local‐Heston volatility model and error analysis

Bompis

Gobet

2017

Mathematical Finance

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This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin calculus. The error analysis, which requires a careful treatment because of the lack of weak differentiability of the model, is interesting on its own. Moreover, in the particular case of call–put options, we also provide expansions of the Black–Scholes implied volatility that allow to obtain very simple formulas that are fast to compute compared to the Monte Carlo approach and maintain a very competitive accuracy.

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“…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%

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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The Large-Maturity Smile for the Sabr and Cev-Heston Models

Cited by 15 publications

References 26 publications

The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices

The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices

Analytical approximations of local‐Heston volatility model and error analysis

Dirichlet Forms and Finite Element Methods for the SABR Model

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