From Discrete to Continuous Financial Models: New Convergence Results For Option Pricing

Abstract

In this paper we develop a new notion of convergence for discussing the relationship between discrete and continuous financial models, D 2 ‐convergence. This is stronger than weak convergence, the commonly used mode of convergence in the finance literature. We show that D 2 ‐convergence, unlike weak convergence, yields a number of important convergence preservation results, including the convergence of contingent claims, derivative asset prices and hedge portfolios in the discrete Cox‐Ross‐Rubinstein option pricing models to their continuous counterparts in the Black‐Scholes model. Our results show that D 2 ‐convergence is characterized by a natural lifting condition from nonstandard analysis (NSA), and we demonstrate how this condition can be reformulated in standard terms, i.e., in language that only involves notions from standard analysis. From a practical point of view, our approach suggests procedures for constructing good (i.e., convergent) approximate discrete claims, prices, hedge portfolios, etc. This paper builds on earlier work by the authors, who introduced methods from NSA to study problems arising in the theory of option pricing.