Boundary conditions for computing densities in hybrid models via PDE methods

Abstract

Motivated by a practical problem arising in equity derivatives modelling, we look into boundary conditions for forward equations related to Feller-type diffusion processes. We adopt the approach of revisiting the original work of Ref. [13] and build upon some of the ideas and methods developed therein. As our main result, we present an extension (in the weak form) of the Feller boundary condition to local stochastic volatility model with the Heston variance.

Keywords:

• Feller process
• forward equation
• boundary conditions

Keywords:

• Primary 60J70

Acknowledgements

I wish to thank A. Khaled, C. Burgard, J. Busquets and V. Piterbarg for their comments on the earlier version of this work.

Notes

^1. Without attempting a rigorous discussion, for convenience we summarize the relevant features of the Feller classification (for a comprehensive treatment the reader is referred to Chapter 15 of [23]). A boundary is called regular if the process can both enter and leave the boundary. An attracting boundary is one that can be reached prior to other any interior point with positive probability. An attractive boundary is also attainable if its first passage time is finite with positive probability. An (instantaneous) reflecting boundary acts as a barrier from which a particle (instantaneously) ‘bounces back’ towards the interior.

^2. It was assumed that integrability for every fixed of and , together with uniform integrability of the latter on every interval .

^3. For instance, the flux of a solute diffusing in one dimension is the rate at which the solute passes through the boundary that is of exactly the same form (see Section V.2 of [2] or [33]). The concept of flux was extended to a more general class of one-dimensional Markov processes in Ref. [19].

^4. It is also interesting to note that the Feller's result can be used to recover directly the Laplace transform obtained in Ref. [5]. To this end, following the reasoning of Ref. [4] we write

so that

hence by the Girsanov theorem (a simple proof of applicability of the Girsanov theorem here can be found in [28])

where and

is a Wiener process under the probability measure ′. Since for and

we may use (3) to complete the calculation of the Laplace transform. Similar procedure can be used to obtain the Heston characteristic function, hence in principle there is no need for solving PDEs for Feller-related characteristic functions beyond the work of Feller.

^5. The approach to reflecting diffusions sketched here is due to Stroock and Varadhan. It was subsequently generalized by a number of authors, ultimately leading to the construction of the ‘patchwork martingale problem’, where the underlying operator is defined separately on the boundaries and the interior. The reader is referred to Ref. [24] and references therein for full details on martingale problem approach to reflected processes.

^6. It should be noted that there is a slight difference in notation here and in the popular Ref. [17].