Tradable Schemes

TL;DR

The paper introduces a tradables-based pricing framework that reformulates derivative valuation PDEs to a drift-free form, enabling a fitted Crank-Nicolson-type finite-difference scheme. By enforcing exact solutions for the basic tradables and a chosen $R(x,\tau)$, the method achieves high accuracy and stability while avoiding drift-related numerical pathologies. Applied to arithmetic Asian options and vanilla options on dividend-paying stocks, it demonstrates fast convergence and substantial accuracy (often to six digits) with practical run-times on modest hardware, and it offers a path to market-conform pricing via price-fitting. The approach also provides a tractable route to analytic $R(x,\tau)$ for several payoff families, and it lays groundwork for extending to implied schemes based on observed prices.

Abstract

In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to $\sim 0.1%$ in about 10ms on a Pentium 233MHz computer and to $\sim 0.001%$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.