A Nonstandard Approach to Option Pricing

Abstract

Nonstandard probability theory and stochastic analysis, as developed by Loeb, Anderson, and Keisler, has the attractive feature that it allows one to exploit combinatorial aspects of a well-understood discrete theory in a continuous setting. We illustrate this with an example taken from financial economics: a nonstandard construction of the well-known Black-Scholes option pricing model allows us to view the resulting object at the same time as both (the hyperfinite version of) the binomial Cox-Ross-Rubinstein model (that is, a hyperfinite geometric random walk) and the continuous model introduced by Black and Scholes (a geometric Brownian motion). Nonstandard methods provide a means of moving freely back and forth between the discrete and continuous points of view. This enables us to give an elementary derivation of the Black-Scholes option pricing formula from the corresponding formula for the binomial model. We also devise an intuitive but rigorous method for constructing self-financing hedge portfolios for various contingent claims, again using the explicit constructions available in the hyperfinite binomial model, to give the portfolio appropriate to the Black-Scholes model. Thus, nonstandard analysis provides a rigorous basis for the economists' intuitive notion that the Black-Scholes model contains a built-in version of the Cox-Ross-Rubinstein model. Copyright 1991 Blackwell Publishers.

Convergence in Incomplete Market Models

“…To compare the measures defined by these densities, we generalize the results of Section 3.1 of [CKW91], where techniques from nonstandard analysis were used to provide alternative derivations of Black-Scholes option pricing theory. We refer to Section 2 of [CKW91] for a primer in hyperfinite probability theory and the basic definitions needed for a nonstandard description of Brownian Motion and related concepts.…”

“…The Nonstandard Universe. We shall assume given (as in [CKW91]) a fixed nonstandard extension * R of the real line R. The extension * R includes elements defined as non-zero 'infinitesimals' (x ∈ * R satisfying |x| < ε for all ε > 0 in R) and their 'infinite' multiplicative inverses. The extension itself is not unique: * R can, for example, be defined as an ultrapower R N /U of the reals by any non-principal ultrafilter U on N (i.e.…”

“…We refer to Section 2 of [CKW91] for a primer in hyperfinite probability theory and the basic definitions needed for a nonstandard description of Brownian Motion and related concepts. However, we shall need to extend the setup somewhat in order to allow consideration of arbitrary cádlág processes rather than restrict the discussion to path-continuous processes as in [CKW91].…”

“…We refer to Section 2 of [CKW91] for a primer in hyperfinite probability theory and the basic definitions needed for a nonstandard description of Brownian Motion and related concepts. However, we shall need to extend the setup somewhat in order to allow consideration of arbitrary cádlág processes rather than restrict the discussion to path-continuous processes as in [CKW91]. Two fundamental references for these developments are [AFHL86] and [HP83]; to set the scene for these extensions we recall (largely following [Cut00]) the principal ideas and definitions here.…”

“…As in [CKW93a] we employ techniques from nonstandard analysis in proving our results, but are able to state these results in standard terms by means of the stronger concept of D 2 -convergence developed in [CKW93a]. The necessary nonstandard tools are described briefly in the primer provided in [CKW91]; in this paper we outline new nonstandard results required for the extension of their results to incomplete market models.…”