The survival probability of the SABR model: asymptotics and application

Abstract

The stochastic-alpha-beta-rho (SABR) model is widely used by practitioners in interest rate and foreign exchange markets. The probability of hitting zero sheds light on the arbitrage-free small strike implied volatility of the SABR model (see, e.g. De Marco et al. [SIAM J. Financ. Math., 2017, 8(1), 709–737], Gulisashvili [Int. J. Theor. Appl. Financ., 2015, 18, 1550013], Gulisashvili et al. [Mass at zero in the uncorrelated SABR modeland implied volatility asymptotics, 2016b]), and the survival probability is also closely related to binary knock-out options. Besides, the study of the survival probability is mathematically challenging. This paper provides novel asymptotic formulas for the survival probability of the SABR model as well as error estimates. The formulas give the probability that the forward price does not hit a nonnegative lower boundary before a fixed time horizon.

Keywords:

• SABR model
• Survival probability
• Probability of hitting zero
• Implied volatility

JEL Classification:

• C63
• G13

Acknowledgements

We are grateful to the conference participants at 20th Conference of the International Federation of Operational Research Societies and seminar participants at The Chinese University of Hong Kong.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

§ In a special case, when the correlation is zero, the SABR model is exactly a time-changed CEV process. Therefore, one can classify whether the forward price can hit zero or not (see, e.g. Karlin and Taylor 1981, Borodin and Salminen 2002). Moreover, it indeed has some positive probability to hit zero for certain parameters. However, it is extremely difficult to fully characterize conditions under which zero can be reached with positive probability for this nontrivial two-dimensional diffusion process.

¶ Thanks to the Managing Editor, who points out that the original asymptotic implied volatility formula derived by Hagan et al. (2002) may not satisfy the model-free bounds given in, say Lee (2004), Rogers and Tehranchi (2010), Fukasawa (2012), and thus is not arbitrage-free even in the log-normal case, where the forward price is always positive.

† For the infinite time horizon, Gulisashvili et al. (2016a, b) derive exact formulas for the probability of hitting zero for the normal and uncorrelated SABR models, respectively. These two probabilities of hitting zero for infinite time horizon cannot be covered by our asymptotic formula.

† If , the forward price is a lognormal process, which is always positive. So the case that and is not needed to be considered.

† For more details, please refer to conditions in theorem 1.1 of De Marco et al. (2017). Assuming that there exists such that as K tends to zero.

‡ The stability of the finite different schemes and the convergence of the Monte Carlo simulation around zero are not (yet) available because of the singularity at zero. A thorough analysis of the convergence of these numerical algorithms is beyond the scope of this paper. Nevertheless, to compute the probability of hitting zero, we use both of the two methods to produce reference values. The numerical experiments show that the performance of our formula is comparable to that of the two numerical methods.

Additional information

Funding

Nian Yang gratefully acknowledges the financial support of the China National Social Science Fund [grant number 15BJL093]. Xiangwei Wan was supported by the National Natural Science Foundation of China (NSFC) for the financial support [grant number 71401103], [grant number 71421002].