Journal of Computational Finance

Finite difference techniques for arbitrage-free SABR

Fabien Le Floc’h and Gary Kennedy

  • TR-BDF2 and Lawson-Swayne schemes fast and stable on the Arbitrage-Free SABR problem.
  • Various boundary conditions, including Antonov's free boundary SABR, easily realised.
  • Lamperti transform leads to efficient finite difference implementation.


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.

To continue reading...

You need to sign in to use this feature. If you don’t have a account, please register for a trial.

Sign in
You are currently on corporate access.

To use this feature you will need an individual account. If you have one already please sign in.

Sign in.

Alternatively you can request an indvidual account here: