Abstracts

In our work we compare a large variety (not fifty, but a large number) of simulation schemes for the SABR model due to Hagan et al. [2002]. All schemes are inspired by the recent works of Islah [2009] and Chen, Oosterlee and Van der Weide [2011]. Islah has shown that, conditional on the integrated variance, an asset in the SABR model follows, after a suitable transformation, a squared Bessel process. In Chen et al. this result has been utilised to arrive at a low-bias simulation scheme for the SABR model.
To be precise, Chen et al. approximate the dynamics of the squared Bessel process by drawing from the exact distribution when the process is close to its absorbing boundary at zero, and drawing from a quadratic Gaussian process (matched to the approximations of the first two moments) elsewhere. For the integrated variance process, approximate first two moments are derived and a lognormal distribution is fitted to approximate the distribution.
Our first set of schemes improves on the work of Chen et al. in a number of ways. First of all, we demonstrate that matching the distribution of the integrated variance is of lesser importance when we use a short-stepped approach such as Chen et al. do, so that simpler approximations, as in Andersen [2008], suffice. In longer-stepped simulations the exact distribution does matter, and we can use conditioning techniques from Asian option pricing to improve on the lognormal approximation.
Second, we derive the exact first two moments of the asset, conditional on the integrated variance, and use a moment-matching scheme with a mixture distribution (a mixture of zero to reflect the absorbing boundary condition, and a distribution resulting in positive values) to fit its distribution.
We also study the recent work of Makarov and Glew [2010] on the simulation of squared Bessel processes with or without absorption at zero. They show that their distribution is that of a randomised Gamma distribution, and devise several schemes for their simulation.
Finally, we also consider a variety of much simpler schemes, such as a simple Euler scheme based on the work of Lord et al. [2010] and an approximation based on Sankaran’s [1963] approximation of the non-central chi-squared distribution.
All schemes are compared in numerical examples for both the CEV model, of which we know an analytical solution, and for the more general SABR case.