Approximate and PDE Solution to the Boundary Free SABR Model - Applications to Pricing and Calibration

Abstract

Considering the current interest rate environment it has become necessary to extend option pricing models for 0 and negative strikes. We consider the recently proposed free boundary SABR model, Antonov A., Konikov, M., and Spector, M. (2015). In their paper the authors provide a pricing formula for European Call options based on numerical integration and Markovian projection. Since it is necessary for practitioners to calibrate the model to market data fast approximation methods together with benchmark methods for their performance are essential. In this note we consider the PDE solution for pricing European Call options as well as two approximation formulas for the Bachelier, aka Normal volatility, produced by this model. The latter numbers can then be plugged into the Bachelier pricing formula to get the corresponding option prices.

We have to stress two facts. First, the PDE method can be seen as a benchmark for the approximate solutions and, second, the approximation formulas can serve for calibration purposes, where fast calculation methods are essential, especially, if one wishes to calibrate to implied Bachelier volatilities. In the approach proposed by Antonov A., Konikov, M., and Spector, M. (2015) the implied volatilities have to be inferred from option prices.

Finally, we stress the fact that the PDE or approximate solutions can be used to effciently apply a mixing approach to control the shape of the surface, especially the wings.

Keywords: SABR, Free Boundary SABR, Approximation, Calibration, Partial Dierential Equation

JEL Classification: C13, C63