Data Analysis and Related Applications, Volume 1

25Pricing Financial Derivatives in the Hull–White Model Using Cubature Methods on Wiener Space

In our previous studies, we developed novel cubature methods of degree 5 on the Wiener space in the sense that the cubature formula is exact for all multiple Stratonovich integrals up to dimension equal to the degree. In this chapter, we apply the above methods to the modeling of fixed-income markets via affine models. Then, we apply the obtained results to price interest rate derivatives in the Hull–White one-factor model.

25.1. Introduction and outline

Calculating stochastic integrals is one of the main challenges in probability. Stochastic integrals cannot always be calculated in the closed form. Therefore, proper numerical methods should be used to estimate the value of stochastic integrals. Monte Carlo estimates are among the popular approaches to estimate the value of stochastic integrals in mathematics and physics. In mathematical finance (and physics), we would like to calculate (estimate) the expected values of functionals defined on the solutions of stochastic differential equations (SDEs).

Without going through the technical details, we mention that the classical idea of cubature methods and consequently cubature formulae can be described as a construction of a probability measure with finite support on a finite-dimensional real linear space which approximates the standard Gaussian measure. For more technical details, see Lyons and Victoir (2002) and Malyarenko et al. ...