TITLE:
Introducing the Power Series Method to Numerically Approximate Contingent Claim Partial Differential Equations
AUTHORS:
Gerald W. Buetow, James Sochacki
KEYWORDS:
Black-Scholes-Merton Options Pricing, Partial Differential Equations, Finite Difference Methods, Crank-Nicolson Method, Power Series Methods
JOURNAL NAME:
Journal of Mathematical Finance,
Vol.9 No.4,
October
25,
2019
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.