Advanced Analytics for the SABR Model

Antonov, Alexandre; Spector, Michael

doi:10.2139/ssrn.2026350

Cited by 28 publications

(40 citation statements)

References 11 publications

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“…For example, Fouque, Papanicolaou, and Sircar (2000), Masoliver and Perello (2006) and Perello, Sircar, and Masoliver (2008) studied the equity market and investigated why the mean reverting behaviour of the volatility process is needed to explain the data (S&P 500 and Dow Jones Industrial Average indices). In case of the interest rate market, the works by Kaisajuntti and Kennedy (2014), Piterbarg (2005) and the most recent paper by Antonov and Spector (2012) also agree that the mean reverting volatility set-up is needed especially for large time horizons. A particular form of the SABR-MR model was studied in Kaisajuntti and Kennedy (2014) within the context of swaption data.…”

Section: Introductionmentioning

confidence: 81%

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Kennedy

Pham

2014

Applied Mathematical Finance

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In this paper, we study the stochastic alpha beta rho with mean reversion model (SABR-MR). We first compare the SABR model with the SABR-MR model in terms of future volatility to point out the fundamental difference in the models' dynamics. We then derive an efficient probabilistic approximation for the SABR-MR model to price European options. Similar to the method derived in Kennedy, J. E., Mitra, S., & Pham, D. (2012). On the approximation of the SABR model: A probabilistic approach. Applied Mathematical Finance, 19(6), 553-586., we focus on capturing the terminal distribution of the underlying asset (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a wide range of parameters that cover the long-dated option and different market conditions.

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Section: Introductionmentioning

confidence: 81%

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Kennedy

Pham

2014

Applied Mathematical Finance

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“…Several variations of this formula are used in practice [4][5][6][7][8][9][10][11][12][13], but they all agree to within O ( 2 ) . The SABR model usually fits the observed smile N ( T ex , K ) quite well at any given expiry T ex , but fitting smiles at different expiries T ex usually requires a different set of SABR parameters ( , , ) for each expiry.…”

Section: Echnical Papermentioning

confidence: 99%

Effective Media Analysis for Stochastic Volatility Models

Hagan¹,

Lesniewski²,

Woodward³

2018

Wilmott

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Analysis of the standard SABR model leads to an effective forward equation which has time-independent coefficients, and analysis of this reduced-dimensionality equation leads to explicit asymptotic formulas for the implied normal volatilities of European options. These formulas are accurate to within O( 2 ), and are used extensively in practice for pricing and managing the risks of European options. A recent analysis of the dynamic SABR model leads to an identical effective forward equation, except that the coefficients are time-dependent. Here we use singular perturbation methods to analyze this new equation. For each exercise date T ex , we derive a set of constant coefficients. Replacing the time-dependent coefficients with the constant coefficients, and solving the effective forward equation, yields the correct probability density function -and thus the correct European option values -to within O ( 2 ) . These constant coefficients now let us apply the existing analysis to obtain explicit asymptotic formulas for the implied volatilities of European options, accurate to within O ( 2 ) , for the dynamic SABR model. Current analyses of several other stochastic volatility models, including the -SABR and generalized Heston models, also yield identical effective forward equations with time-dependent coefficients. Therefore, the implied volatilities for these models are also given by the SABR implied volatility formulas.

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“…There exist several refinements to this asymptotic formula: in [62] a correction the leading order term is proposed, and [63] provides a second-order term. In the uncorrelated case ρ = 0 the exact density has been derived for the absolutely continuous part of the distribution of X on (0, ∞) in [6,33,48] and the correlated case was approximated by a mimicking model. However, it seems that these refinements have not fully resolved the arbitrage issue near the origin.…”

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%

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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Advanced Analytics for the SABR Model

Cited by 28 publications

References 11 publications

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Effective Media Analysis for Stochastic Volatility Models

Dirichlet Forms and Finite Element Methods for the SABR Model

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