On the Approximation of the SABR Model: A Probabilistic Approach

Abstract

In this article, we derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters that cover the long-dated options and highly stress market condition. This is a different feature from other current approaches that rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large volatility of volatility (Volvol).

Key Words:

• SABR model
• displaced diffusion
• stochastic volatility

Acknowledgement

We thank David Rufino (Citigroup) for the valuable discussions.

Notes

¹ When and , the underlying process is not a martingale.

² We take into account the main criticism of the SABR formula pointed out in Obloj (2008) whilst deriving this formula.

³ Other authors usually use as the perturbation parameter. Theoretically, they require this parameter to be much smaller than 1 to give precise results, for example, Hagan et al. (2002, 2005) and Wu (2010). In practice, such requirement can only be satisfied for very short maturity (<10 years).

¹ We use Hagan's technique with the assumption that the strike K is not so far away from .