AN OPTION VALUATION FRAMEWORK BASED ON ARITHMETIC BROWNIAN MOTION: JUSTIFICATION AND IMPLEMENTATION ISSUES

• Date: 01 September 2017
• Author: Joshua A. Brooks,Robert Brooks
• Published date: 01 September 2017
• DOI: http://doi.org/10.1111/jfir.12129

AN OPTION VALUATION FRAMEWORK BASED ON ARITHMETIC

BROWNIAN MOTION: JUSTIFICATION AND IMPLEMENTATION ISSUES

Robert Brooks

University of Alabama

Joshua A. Brooks

Columbus State University

Abstract

We examine arithmetic Brownian motion as an alternative framework for option

valuation and related tasks. After reexamining empirical evidence, we compare and

contrast option valuation based on one of the simplest forms of geometric Brownian

motion with arithmetic Brownian motion. We identify an enhanced way to handle

negative stock prices within arithmetic Brownian motion that is consistent with

empirical observation. We review numerous strengths and weaknesses of both

approaches. The arithmetic Brownian motion framework allows for the aggregation of

any number of correlated factors for risk analysis.

JEL Classification: G13

I. Introduction

The Black–Scholes–Merton option valuation model (BSMOVM) is deeply embedded in

the heart of modern financial analysis. It is also well known, however, that the BSMOVM

is deeply flawed. The Black–Scholes–Merton (BSM) framework has been extended to

incorporate stochastic volatility, jumps, volatility surfaces, and so forth. Although many

of these extensions have been useful for specific applications, core problems remain,

many of which are rarely identified, much less addressed. For the purpose of this

introduction, we identify two problems with the BSMOVM related to the underlying

lognormal distribution assumption. First, there is zero probability of the stock price being

zero in the future. Second, a simple portfolio of stocks follows no known statistical

distribution. Hence, within the BSMOVM, internally consistent portfolio statistics

cannot be computed, which impairs risk management practice.

1

Thus, the lognormal

distribution suffers because it is not additive and the underlying values are strictly

positive. To address these issues, we propose an alternative framework based on

The authors gratefully acknowledge the helpful comments of Don Chance, John Deeble, Matthew Lambert,

Abbi Ondocsin, Pavel Teterin, Kate Upton, and Kaylee Brooks.

1

An important solution for this aggregation issue and lack of closed-form distributions is the use of the

nonparametric models presented in Stutzer (1996). The maximum entropy approach used is significantly sample-

period dependent and does not have internally consistent portfolio characteristics unless each possible portfolio is

separately computed.

The Journal of Financial Research Vol. XL, No. 3 Pages 401–427 Fall 2017

401

© 2017 The Southern Finance Association and the Southwestern Finance Association

RAWLS COLLEGE OF BUSINESS, TEXAS TECH UNIVERSITY

PUBLISHED FOR THE SOUTHERN AND SOUTHWESTERN

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arithmetic Brownian motion (ABM). To illustrate this framework, we use the most

simple and most canonically recognizable form, which we term the arithmetic Brownian

motion option valuation model (ABMOVM).

The ABMOVM presented here, based on the normal distribution, can easily

address these two problems. All the extensions (such as stochastic volatility, jumps,

volatility surfaces, etc.) can also be easily applied within the ABMOVM. Ultimately, the

appropriate model to deploy in practice depends on the user’s objectives, the underlying

instruments involved, and the empirical support. Given the utility of a nonzero

probability of default and the many applications of a known portfolio distribution, our

purpose is to justify the ABM framework as a reasonable approach to consider.

Additionally, we explore selected implementation issues.

Historically, the shift in financial research from ABM to geometric Brownian

motion (GBM) occurred with Samuelson (1965). Samuelson, based on earlier work by

Osborne (1959), Sprenkle (1961), Alexander (1961), Boness (1964), and many others,

introduced GBM for solving financial derivatives valuation problems. GBM was

introduced, in part, to “correct”prior errors by authors relying on ABM. Samuelson

credits Bachelier (1900) with discovering “the mathematical theory of Brownian motion

five years before Einstein’s classic 1905 paper”(p. 13). Two prior errors identified by

Samuelson include stocks possessing limited liability and warrant values exceeding the

underlying instrument’s value. It is important to note that although stocks do possess

limited liability characteristics, they still have some probability of becoming worthless.

As we address later, both are not weaknesses of ABM, but rather an incomplete

application of ABM to this finance problem.

Additionally, we reexamine ABM by focusing on the empirical evidence related

to the distribution of underlying instruments. Because finance is a social science, every

mathematical framework has both strengths and weaknesses. As Derman (2011) notes in

the context of financial theories, “Only imperfect models remain. The movements of

stock prices are more like the movements of humans than molecules. It is irresponsible to

pretend otherwise”(p. 187). Because modeling human movements remains elusive, it is

important to have multiple, pertinent frameworks available for the array of tasks facing

the finance profession.

Finally, we compare and contrast the s imple BSM-based forms of the

orthodox GBM with the ancient AB M. Our purpose is not to advocate ABM- based

tools over GBM-based tools, but rather to as sert that the ABM should be considered as

a viable framework for analys is in many and varied contexts . As option valuation

models become more complex and nu merous, the risk of combinator ial explosion

becomes more plausible. Corres pondingly, many of the most recent mode ls of option

valuation require calibrat ion within each market where they ar e deployed. Each of

these issues is further exacer bated when a risk manager attem pts to gain insight into

the aggregate position of an entire firm. Our approach gives a simple, closed- form,

option valuation framework that allows a level of aggregation never before seen in this

research space. There have been extraordinary advances in representing the finest

details of an underlying ins trument’s behavior in many of the ne west option valuation

models (e.g., Garleanu, Peders en, and Poteshman 2009; Corsi, Fus ari, and La Vecchia

2013; Fulop, Li, and Yu 2014). As ne w markets come online, expandi ng the toolbox to

402 The Journal of Financial Research