On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Abstract

In this paper, we study the stochastic alpha beta rho with mean reversion model (SABR-MR). We first compare the SABR model with the SABR-MR model in terms of future volatility to point out the fundamental difference in the models’ dynamics. We then derive an efficient probabilistic approximation for the SABR-MR model to price European options. Similar to the method derived in Kennedy, J. E., Mitra, S., & Pham, D. (2012). On the approximation of the SABR model: A probabilistic approach. Applied Mathematical Finance, 19(6), 553–586., we focus on capturing the terminal distribution of the underlying asset (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 wide range of parameters that cover the long-dated option and different market conditions.

Key Words:

• SABR
• SABR-MR model
• displaced diffusion
• stochastic volatility
• forward smile

Acknowledgement

We thank the referee for the insightful and valuable comments on this paper.

Notes

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