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.

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