No-Arbitrage Dynamics for a Tractable SABR Term Structure Libor Model

20 Pages Posted: 2 Oct 2007 Last revised: 22 Jun 2016

See all articles by Massimo Morini

Massimo Morini

Banca IMI; Bocconi University

Fabio Mercurio

Bloomberg L.P.

Date Written: September 1, 2007

Abstract

The SABR closed-form formula (Hagan et. al 2002) is the standard for smile-consistent pricing in the swaption market. Here we address the issue of turning SABR assumptions into a consistent and arbitrage-free term structure model in the BGM/Libor Market Model framework. We compute the joint dynamics followed by Libor rates and stochastic volatility of SABR kind under the general pricing measures used for interest rate derivatives, and we observe that the volatility dynamics is non-standard. Based on the analysis of the equation found, we develop and justify theoretically a few approximations aimed at making these no-arbitrage dynamics compatible with the use of the SABR closed-form formula. Then the formulas developed above are confronted both with alternative numerical implementations and with market data. We verify that the formulas for no-arbitrage corrections are acceptably precise, maintain good fitting, and produce regular Libor parameters. Finally we verify that the no-arbitrage corrections to the volatility dynamics make the out-of-calibration-sample prices implied by the model closer to market quotations, compared to prices implied by a trivial multivariate SABR neglecting such corrections.

Keywords: stochastic volatility, SABR, no-arbitrage,libor market model, BGM

JEL Classification: G13, C63

Suggested Citation

Morini, Massimo and Mercurio, Fabio, No-Arbitrage Dynamics for a Tractable SABR Term Structure Libor Model (September 1, 2007). Bloomberg Portfolio Research Paper No. 2010-03-FRONTIERS, Available at SSRN: https://ssrn.com/abstract=1018026 or http://dx.doi.org/10.2139/ssrn.1018026

Massimo Morini (Contact Author)

Banca IMI ( email )

Corso Matteotti 6
20121 Milano, 20100
Italy

Bocconi University ( email )

Via Sarfatti, 25
Milan, MI 20136
Italy

Fabio Mercurio

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

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