Widening the Range of Underlyings for Derivatives Pricing with QUAD by Using Finite Difference to Calculate Transition Densities—Demonstrated for the No-Arbitrage SABR Model

Abstract

The QUAD method is a fast, flexible numerical pricing technique, widely applicable to many option types in its QUAD I and QUAD II versions where the underlying process has a closed-form density function or characteristic function. In its most advanced version, QUAD III, sacrificing only a little speed, it retains all the flexibility and applicability of earlier versions while covering an even greater range of underlying processes through use of approximations of the density functions. In this article, the authors show how cases without suitable approximations can be handled by using finite difference methods for (only) that part of the calculation. They illustrate with the no-arbitrage SABR model for the underlying.

TOPICS: Derivatives, options

Key Findings

• • Option pricing techniques under the umbrella term QUAD are the fastest generally applicable and flexible numerical methods we have for derivatives pricing.

• • Cases where there is no transition density function or characteristic function can be solved by using an approximation of the particular density function. However, in this article, an alternative approach is demonstrated, substituting finite difference calculations for the approximation.

• • The no-arbitrage SABR model is used as an example, since it is of special interest to practitioners.