Finite Difference Techniques for Arbitrage-Free SABR

29 Pages Posted: 19 Aug 2016

See all articles by Fabien Le Floch

Fabien Le Floch

affiliation not provided to SSRN

Gary J. Kennedy

Clarus Financial Technology

Date Written: August 15, 2016


In the current low rates environment, the classic stochastic alpha beta rho (SABR) formula used to compute option-implied volatilities leads to arbitrages. In "Arbitrage free SABR", Hagan et al proposed a new arbitrage-free SABR solution based on a finite difference discretization of an expansion of the probability density. They rely on a Crank-Nicolson discretization, which can lead to undesirable oscillations in the option price. This paper applies a variety of second-order finite difference schemes to the SABR arbitrage-free density problem and explores alternative formulations. It is found that the trapezoidal rule with the second-order backward difference formula (TR-BDF2) and Lawson-Swayne schemes stand out for this problem in terms of stability and speed. The probability density formulation is the most stable and benefits greatly from a variable transformation. A partial differential equation is also derived for the so-called free-boundary SABR model, which allows for negative interest rates without any additional shift parameter, leading to a new arbitrage-free solution for this model. Finally, the free-boundary model behavior is analyzed.

Keywords: stochastic volatility, stochastic alpha beta rho (SABR), arbitrage, TR-BDF2, finite difference method, finance

Suggested Citation

Le Floch, Fabien and Kennedy, Gary J., Finite Difference Techniques for Arbitrage-Free SABR (August 15, 2016). Journal of Computational Finance, Forthcoming, Available at SSRN:

Fabien Le Floch (Contact Author)

affiliation not provided to SSRN

Gary J. Kennedy

Clarus Financial Technology ( email )

8 Monkwell Square
London, EC2Y 5BN
United Kingdom


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