Generalized uncorrelated SABR models with a high degree of symmetry

Abstract

A family of generalized driftless uncorrelated SABR-like models are classified according to the dimensions of the symmetry groups of their corresponding backward Kolmogorov equations. This family contains the original uncorrelated SABR models, for arbitrary positive beta, as special cases. New cases with a rich symmetry group appear.

Keywords:

• Non-Gaussian option pricing
• Derivative pricing models
• Stochastic volatility
• Partial differential equations

Acknowledgements

We are grateful for helpful discussions with Marco Avellaneda, Rafael Douady, Bruno Dupire, and André Lesnewski and the participants at Bloomberg LP and Courant Mathematical Finance seminars. We are particularly grateful to Peter Carr for valuable comments and suggestions. We thank the three anonymous referees for their helpful comments and suggestions to improve the paper and the interesting points raised. All errors are our responsibility. The first author's special thanks go to National Chung Cheng University for its generosity in allowing his use of the computer equipment for 8 months after his departure.

Notes

†The reduced symmetry group we are referring to is the subgroup of the symmetry group that leaves invariant not only the equation but also the initial manifold t = 0 and the initial condition, a delta function.

†These vector fields are those whose ∂/∂t coefficients depends on the variable t alone, i.e. neither on x nor on u, and whose ξ ⁱ coefficients do not depend on u.

†Note that {X ₁, X ₂, X ₅} corresponds to the isometry group of Poincaré space, which is SL(2) = {invertible real 2 × 2 matrices of determinant 1} of dimension 3.