Introducing the Power Series Method to Numerically Approximate Contingent Claim Partial Differential Equations

Abstract

We introduce a previously unused numerical framework for estimating the Black-Scholes partial differential equation. The approach, known as the Power Series Method (PSM), offers several advantages over traditional finite difference methods. Our objective is to highlight the advantages of the PSM over traditionally used numerical approximation approaches. To meet this we deploy a numerical approximation scheme to illustrate the PSM. The PSM is more stable than explicit methods and thus computationally more efficient. It is as accurate as hybrid approaches like Crank Nicolson and faster to compute. It is more accurate over a far wider spectrum of time steps. Finally, and importantly, it can be expressed analytically thus offering the capability of performing comparative statics in a far more stable and accurate environment. For a more complex application this last advantage may have wide implications in producing hedge ratios for synthetic replication purposes.

Using the Power Series Method to Evaluate Non-Linear Contingent Claim Partial Differential Equations

“…Buetow and Sochacki [1] demonstrated that PSM can be used to efficiently solve and analyze the forward linear BS options model and is comparable to the explicit finite difference method and the Crank-Nicolson method. Matić, Radoičić and Stefanica [47] show that PSM can be used to efficiently solve the inverse problem for determining the implied volatility in the linear BS options model.…”

“…As in [47], we use auxiliary variables and power series to solve this nonlinear BS options model. In the appendix, we give the symbolic solution for this model similarly to how it was done in [1]. As in [1] and Glover, Duck, Newton [48], we approximate the pay off with a smooth function.…”

“…Buetow and Sochacki [1] use the PSM approach to evaluate linear PDE generated within the contingent claim valuation framework. Here, we extend their approach to nonlinear PDE (NLPDE).…”

“…Traditional numerical schemes that solve contingent claim partial differential equations (PDEs) abound in the literature. Our focus here is to extend Buetow and Sochacki [1] into the NLPDE framework; their paper is the first to introduce the Parker and Sochacki [5] [6] Power Series Method (PSM) method within the contingent claim literature.…”

“…Buetow and Sochacki [1] presented the advantages of the PSM within the contingent claim valuation framework by approximating the plain vanilla PDEs.…”