Fifty shades of SABR simulation

We compare a large variety (not fifty, but a large amount) of simulation schemes for the SABR model due to Hagan et al. [2002]. Many schemes are inspired by the recent work of Islah [2009], who has shown that, conditional on the integrated variance, an asset in the SABR model can be approximated by, after a suitable transformation, a squared Bessel process. The result is exact for the zero correlation case. In Chen et al. this result has been utilised to arrive at a low-bias simulation scheme for the SABR model.

We show how a scheme for the underlying asset can be combined in a modular way with a scheme for the integrated variance. For the latter we resort to conditioning techniques used in Asian option pricing, but we also look at more simple schemes. For the zero correlation scheme the resulting scheme is almost exact, as we use recent work of Makarov and Glew [2010] on the simulation of squared Bessel processes with absorption at zero. If the correlation deviates significantly from zero, we have to resort to simpler schemes. All schemes are compared in numerical examples, also to the Ninomiya-Victoir with drift scheme recently devised by Bayer, Friz and Loeffen [2012].

Since the initial work, we have applied the insights gained from the SABR model to the free-boundary SABR model, introduced by Antonov, Spector and Konikov [2015]. This can be of particular relevance in the current low-rate environment.

This work has been presented at the 8th World Congress of the Bachelier Finance Society in Brussels, at the 10th Fixed Income Conference in Barcelona (slides here) as well as at the Global Derivatives Trading & Risk Management conference in Amsterdam. The latter presentation can be found here.