The principle of not feeling the boundary for the SABR model

Abstract

The stochastic alpha–beta–rho (SABR) model is widely used in fixed income and foreign exchange markets as a benchmark. The underlying process may hit zero with a positive probability and therefore an absorbing boundary at zero should be specified to avoid arbitrage opportunities. However, a variety of numerical methods choose to ignore the boundary condition to maintain the tractability. This paper develops a new principle of not feeling the boundary to quantify the impact of this boundary condition on the distribution of underlying prices. It shows that the probability of the SABR hitting zero decays to 0 exponentially as the time horizon shrinks. Applying this principle, we further show that conditional on the volatility process, the distribution of the underlying process can be approximated by that of a time-changed Bessel process with an exponentially negligible error. This discovery provides a theoretical justification for many almost exact simulation algorithms for the SABR model in the literature. Numerical experiments are also presented to support our results.

Keywords:

• SABR model
• Probability of hitting zero
• Principle of not feeling the boundary
• Time-changed Bessel process

JEL Classification:

• C63
• G13

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 For example, if the correlation is zero and the parameter ‘beta’ is less than 1/2 (see the SDE (1(1) $\begin{array}{rl}d{F}_t& =A_t{F}_t^{\beta }[\sqrt{1-{\rho }^2}d{B}_t+\rho d{W}_t],\\ d{A}_t& =\nu A_{t}d{W}_t,\end{array}$(1) ) for the SABR model), then the mass at zero is positive.

2 $\{F_t;0\le t\le T\}$ has a finite moment (Andersen and Piterbarg 2007, Lions and Musiela 2007) for $\beta \in [0,1)$, thus $S=\underset{n\to \mathrm{\infty }}{lim}\{t>0:F_t\le 1/n,\mathrm{or}{F}_t\ge n\}$.

Additional information

Funding

Nan Chen is supported by the Research Grant Council of Hong Kong SAR Government [grant number CUHK14201114]. Nian Yang gratefully acknowledges the financial support of the China National Social Science Fund [grant number 15BJL093].