Another Look At Option Valuation Based on Arithmetic Brownian Motion

40 Pages·2013·2.3 MB·English

by  Robert Brooks

Another Look At Option Valuation Based on
Arithmetic Brownian Motion

Robert Brooks*
Joshua A. Brooks

We take another look at arithmetic Brownian motion as a framework for
developing financial derivatives valuation and risk management tools. We
compare and contrast option valuation based on geometric Brownian motion and
arithmetic Brownian motion. An alternative way to handle negative stock prices
within the arithmetic Brownian motion approach is identified that is more
consistent with empirical observation than the traditional approach of assuming
no possibility of zero prices within the geometric Brownian motion. After
comparing a wide array of sensitivities with both models, we discuss numerous
strengths and weakness of both approaches. Option valuation based on arithmetic
Brownian motion appears to be a credible additional tool for financial quantitative
analysis.

March 4, 2013

COMMENTS WELCOMED

* Corresponding author: Robert Brooks, Department of Finance, The University of Alabama, 200 Alston Hall,
Tuscaloosa, AL 35487, [email protected], [phone]. The authors gratefully acknowledge the helpful
comments of Kate Upton and Matthew Lambert.

JEL Classification Code: G13
Key words: Arithmetic Brownian Motion, Geometric Brownian Motion, Option Valuation, Black-Scholes-Merton
option pricing model

© 2013 Robert Brooks and Joshua A. Brooks. All Rights Reserved. May not be scanned, copied or duplicated, or posted to
a publicly accessible website, in whole or in part.

Electronic copy available at: http://ssrn.com/abstract=2228288
Another Look at Option Valuation Based on Arithmetic Brownian Motion

Samuelson [1965], based on earlier work by Osborne [1959], Spenkle [1961], Alexander
[1961], Boness [1964] and many others, introduced geometric Brownian motion (GBM) for solving
financial derivatives valuation problems. GBM was introduced, in part, to “correct” prior errors by
authors relying on arithmetic Brownian motion (ABM). Samuelson [1965] credits Bachelier [1900]
with discovering “the mathematical theory of Brownian motion five years before Einstein’s classic
1905 paper.” (13) Two prior errors identified by Samuelson include stocks possessing limited liability
and warrant values exceeding the underlying instrument’s value. As we will address later, both are not
weaknesses of arithmetic Brownian motion, rather an incomplete application of arithmetic Brownian
motion to the particular finance problem.
The purpose of this paper is to take another look at arithmetic Brownian motion as a framework
for developing financial derivatives valuation and financial risk management tools. Because finance is
a social science, every mathematical framework has both strengths and weaknesses. We compare and
contrast the orthodox GBM with the ancient ABM. Our purpose is not to advocate ABM-based tools
over GBM-based tools, rather to advocate that having more tools is better. Ultimately, it is an
empirical question which tools are superior in any given market at any particular season of time. As
seasons change and new markets come on line, expanding the toolbox seems a reasonable endeavor.
The remainder of the paper is organized as follows. Section I discusses the relevant literature.
We do not provide an exhaustive history rather we seek to explain the historical context for GBM
becoming the dominant framework. In Section II, we provide a general representation of option
valuation models based on ABM and GBM as well as selected “greeks.” Section III reports the various
strengths and weaknesses of ABM and GBM. Section IV presents our conclusions.
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Electronic copy available at: http://ssrn.com/abstract=2228288
I. Relevant Literature
There is an extensive literature that addresses option valuation and related problems. This
diverse body of research ranges from the seminal paper by Bachelier [1900] to the vast number of
theoretical and empirical papers available today. A recent search of the Social Sciences Research
Network identified over 1,400 papers containing “option pricing” since 1994. We divide our brief
literature review into three epochs, ancient history to 1950, 1950 to 1972 and 1972 to present.
Although there are numerous option valuation and risk management frameworks available today, the
vast majority are encapsulated within the GBM framework.
A. Ancient History to 1950
According to Kruizenga [1952], the “earliest recorded time-transaction which seems to have
been in nature an option was the one from Biblical times in which Jacob took an option on his wife-to-
be, Rachel.”1 Chance [1995] provides a brief history of derivatives where he documents option trading
has been around for a very long time.
Kruizenga [1952] also identifies Thales the Milesian as both a philosopher and option trader.
According to Diogenes Laërtius, Lives of Philosophers, around 580 B.C., Thales foresaw the scarcity
of olives and made a corner in olive oil. Thales the Milesian (from Aristotle's Politics) buys options on
olive presses and profits exceedingly from a bumper olive crop. Copleston, History of Philosophy
(Vol. 1, p. 22, 1985) states this is probably untrue but the telling of such a story gives validity that such
activity occurred.
Kruizenga [1952] also mentions the possible existence of option trading during the Roman
empire from around 46 B.C. to 476 A.D. Selected other ancient evidence of option trading identified
by Chance [1995] include: Dutch tulip bulb trading resulted in defaults on tulip bulb contingent

1See Genesis 29:15-30. To be an option, Kruizenga points out that Jacob’s agreement with Laban assumes Jacob could
renege during the seven-year period.
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Electronic copy available at: http://ssrn.com/abstract=2228288
forward contracts, 1637; option trading began on securities in London, 1690; Barnard's Act declares
options illegal in London, 1733; and option trading on U. S. securities start, 1790.
Chance [1995] notes, “1835 Scottish physicist Robert Brown allegedly observes movement of
particles suspended in water. This came to be known as Brownian motion; later used to model
movements of stock prices and led to the development of option pricing theory.” Chance also
introduces Russell Sage, a railroad speculator around 1850 who bypasses usury laws using put-call
parity. He became known as the grandfather of modern option trading.
The most relevant work for this paper is Louis Bachelier’s dissertation published in 1900 where
he introduces what is now known as arithmetic Brownian motion and derives the first known option
valuation model.
From the 1900s through the 1920s a variety of over-the-counter options trading firms were
established, many defaulting on short option positions. The result is a general distrust for these “bucket
shops.”
Finally, in response to the Great Depression in the United States, the Investment Act of 1934
created the Securities and Exchange Commission and gave it the power to regulate options trading.
During the 1930s and 1940s, option trading is declared illegal in London several times.
B. 1950 to 1972
Chance [1995] notes that in 1951 K. Itô publishes an article related to stochastic differential
equations. One major breakthrough, now known as Itô’s lemma, provides an important tool in option
pricing theory. In chapter 8 of his 1952 doctoral thesis, Kruizenga [1952] provides detailed analysis of
the variance of expected option gains where the very rudiments of delta hedging are apparent.
Kruizenga was aware of Bachelier’s [1900] work. In fact, Kruizenga [1952] provides anecdotal
evidence in support of Bachelier’s model (see pages 185 to 186).
Osborne [1959] provides a detailed statistical analysis of the log of the price relative for
common stock prices. Osborne’s work provides the early justification for the lognormal distribution
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(ultimately GBM). It should be noted, however, that Osborne concedes, “percentage changes of less
than ±15 per cent, expressed as fractions from unity, are very nearly natural logarithms of the same
ratio.” (146) Thus, Osborne’s approach could easily have provided support for ABM.
Sprenkle [1961], in his dissertation on warrant prices, provides both normative and positive
arguments in favor of using the lognormal distribution (ultimately GBM). Sprenkle notes, “We have
now developed rationales for both normally distributed expected stock prices and lognormally
distributed expected stock prices. Is there any a priori reason for preferring one to the other? The
answer is yes; there is a quite important reason for preferring the lognormal distribution. Normal
distributions imply that the investor thinks there is a chance that future stock prices will be negative. ...
The lognormal distribution restricts x [stock prices] to positive values, and an investor with this type
t
of expectation does not think there is any chance of negative stock prices” (underline in original, page
195). As any investor realizes, however, there is always a positive chance of witnessing a zero stock
price, which is not possible in a lognormal distribution.
Based on a simple thought experiment, Sprenkle [1961] asserts that “over periods where stock
prices do change substantially it seems better to assume the investor’s estimate of the chances of given
percentage gains and losses stays constant.” (195) This assertion remains, however, an empirical
question. One can easily argue that percentage changes increase due to a leverage effect. Sprenkle
[1961] does conduct empirical tests and concludes, “that in most cases a lognormal distribution is a
better assumption than a normal distribution.” (196)
Alexander [1961] provides additional analysis of stock price movements. He notes in response
to observing fat tailed distributions that this “sort of situation (leptokurtosis) is frequently encountered
in economic statistics and would certainly overshadow any attempt to test fine points such as the
difference between a logarithmic and a percentage scheme.” (16)
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Boness [1964] offers an option valuation model based on the lognormal distribution and
“compares predictions based on this model with observed prices of put-and-call options traded in the
New York market.” (163) Boness assumes that “the variances of distributions of logarithms of
expected percentage changes in price of a given stock” (167) are linear in investment horizon. Boness
also assumes investors are risk neutral. Boness’ model is the Black-Scholes-Merton option valuation
model if you assume the risk-free interest rate is the appropriate growth rate on stocks.2
Samuelson [1965] publishes an option valuation model where options and stock have different
risk levels. Samuelson appears to have coined the term geometric Brownian motion. He states that this
model accounts for the observed anomaly of long-term warrants increasing in value to infinity under
Bachelier’s model. He also notes that under this model stocks cannot become negative in value
because they are distributed lognormally. One weakness of this model, however, is that under the
lognormal distribution the probability of observing a zero stock price is zero. We observe every year
that some companies go into bankruptcy and have their stock declared worthless. A lognormal
distribution has no way of accounting for this probability. We take a different approach to option
valuation that has the value of stock always being non-negative but with a positive probability of
bankruptcy. Although there are critiques of both GBM and ABM we present both to compare the
viability of each model in different situations.
We follow the tradition of positive economics championed by Friedman [1966] in that we do
not go so far as to say which option valuation model one should use, only how each internally,
coherent model works in different environments. Samuelson and Merton [1969] develop a more
generalized model of Samuelson’s 1965 paper for warrant pricing. They use utility maximization to
characterize equilibrium where risk averse individuals hold a risk-free asset, stock and warrants on the
stock. They derive a general equilibrium solution for the pricing of warrants that depends on the stock
price and the stock’s risk as well as boundary conditions for the price of warrants. They assume that

2See Boness [1964], equation (4) on page 170.
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the price process follows GBM and transform it to a random walk difference equation. Their analysis
shows a number of boundary conditions for warrant pricing. A particularly interesting result from this
paper is that the value of a perpetual warrant converges to the stock price. We re-examine this
particular issue under ABM with limited liability.
C. 1972 to present
Several major changes took place in the early 70s in option valuation. Two papers created the
basis for modern option pricing. Black and Scholes [1973] develop a risk-free portfolio, under constant
rebalancing as pointed out by Merton [1973], and put forward the partial differential equation for the
value of an option. Merton [1973] puts forth a general framework for rational option pricing that
addresses a number of boundary conditions for call options and warrants. His model also relaxes
several constraints in the Black-Scholes-Merton (BSM) framework. Black and Scholes [1972] use their
option valuation model to empirically test their option prices against the prices of traded options. They
use lagged variance as their risk term and find that their model matches fairly well to observed option
prices, but their model tended to overprice options on high variance stocks and under-price options on
low variance stocks. This pattern is remarkably consistent with implied variance under ABM
documented later in this paper. In 1973 the Chicago Board Options Exchange began trading call
options and put options began trading four years later. With this exchange and the options clearing
corporation, options trading moved from bucket shops that issued many different contracts to
standardized, frequently traded contracts. The existence of exchange traded options led to enormous
growth in options (Kairys and Valerio, 1997). The advent of more data led to a number of empirical
papers on the BSM option valuation model (MacBeth and Merville, 1979; Chiras and Manaster, 1978;
Gultekin, et al, 1982; Moore, 2006; and many others). The BSM option valuation framework gave rise
to many other option valuation models.
Geske [1977] develops a compound options framework to value to corporate liabilities that
have multiple payouts. His model uses a lognormal distribution framework. Generalizing his model
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further, Geske [1979] uses a compound option framework that incorporates financial leverage into the
BSM option framework. This allows his model to avoid the leverage effect critique of the BSM option
model. Under GBM the variance increases with the price of the underlying, but it is commonly thought
that highly leveraged firms have greater stock price volatility. This approach is mentioned by Gultekin,
et al, [1982] as a possible reason for the anomalies observed in their data. We examine this issue
further in our comparison of models.
There have been several notable papers that diverge from the lognormal distributions of BSM.
Murphy [1990] develops an option valuation model based on the normal distribution. In order to
control for limited liability he truncates his normal distribution at zero. Empirical tests show that his
model fits observed option pricing data as well as the BSM option valuation model. Poitras [1998]
compares Bachelier and Black-Scholes option pricing approaches in spread options. He notes that
unlike models based on GBM, ABM allows the derivation of closed form solutions for spread options.
We follow this type of compare and contrast framework as we compare our models for ABM and
GBM.
We extend our model of ABM by building on the use of limited liability by Brooks and Gup
[2006]. They develop a theoretical framework for why bank holding companies separately incorporate
their underlying entities. They note that limited liability is itself a latent option held by the
stockholders. In order to give value to this option, they employ ABM having distributional
characteristics consistent with a negative value for the underlying firm. They note that this approach
has numerous advantages for closed form solutions. There are several recent papers that argue against
the use of ABM. Liu [2007] notes that ABM follows a number of the boundary conditions of the
Black-Scholes-Merton model, but under several conditions the no-arbitrage boundary is violated. We
confront these problems in a new way and show that the ABM does fulfill all the aspects of a rational
option pricing model. Choi, et al [2009] introduces a new method for calculating an option’s volatility
under ABM for use in fixed income markets.
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II. Arithmetic and Geometric Brownian Motion-Based Option Valuation Models
In this section, the arithmetic Brownian motion option valuation model (ABMOVM) is
developed. We contrast this model with the traditional geometric Brownian motion option valuation
model (GBMOVM). In an effort to address the most general case, we assume a more general
framework than the Black, Scholes Merton option valuation model. The standard finance
presuppositions hold, specifically the market participants can rely on clear enforcement of the rule of
law; there exist clean property rights and generally a culture of trust. Clearly, without these
presuppositions, any remaining analysis is moot. We now turn to standard finance assumptions.
A. Assumptions
We assume the standard set up for financial models (see, for example, Harrison and Kreps,
1979, and Harrison and Pliska, 1981):
1) [0,Tˆ ] , for fixed 0≤t≤Tˆ , finite time horizon.
2) (Ω,ℑ,Ρ), uncertainty is characterized by a complete probability space, where the state
€
€
space is the set of all possible realizations of the stochastic economy between time 0 and time Tˆ
€
and has a typical element ω representing a sample path, ℑ is the sigma field of distinguishable events
€
at time Tˆ , and Ρ is a probability measure defined on the elements of ℑ.
€ €
3) F={ℑ(t):t∈[0,Tˆ ] } the augmented, right continuous, complete filtration generated by
€ € €
the appropriate stochastic processes in the economy, and assume that ℑ(Tˆ ) =ℑ. The augmented
€
filtration, ℑ(t), is generated by Z. ℑ(0) contains only Ω and the null sets of P.
€
4) F is generated by a K-dimensional Brownian motion, Z(t)=[Z (t),,Z (t)],t∈[0,Tˆ ] is
1 K
€ € €
€ €
defined on {Ω,ℑ,P}, where {ℑ(t)},t∈[0,Tˆ ] is the augmentation of the filtration {ℑZ(t)},t∈[0,Tˆ ]
€
€
generated by Z(t), and satisfies the usual conditions.
€ € €
5) E (⋅) denotes the expectation with respect to the probability measure Ρ.
Ρ
€ 9
€
€
6) All stated equalities or inequalities involving random variables hold Ρ-almost surely.
7) P is common for all agents implying uniqueness of the nature of the stochastic
€
processes.
€
8) Conventional perfect market conditions are also assumed, such as no transaction costs,
no taxes, unrestricted short selling, and no regulatory or institutional constraints.
The key specific assumption relates to the stochastic process for the underlying instrument. The
ABMOVM is based on
dS=µ(S,t)dt+σ (t)dz (1)
a
whereas the GBMOVM is based on
€
dS=µ(S,t)dt+σ (t)Sdz (2)
g
where S denotes some underlying instrument, µ(S,t) denotes the expected growth rate of the underlying
€
instrument that can vary based on S or t, σ (t) denotes the relative volatility for GBM, σ (t) denotes
g a
€
the absolute volatility for ABM that can vary based on maturity time and finally let dz denote the
€ €
standard Wiener process.
€
The key to developing ABMOVM appropriately is carefully defining the underlying
instrument, S. We define the underlying instrument as the equivalent instrument value assuming
unlimited liability. With GBMOVM, this definition does not result in any difference from the
underlying instrument with limited liability. We denote the underlying instrument with limited liability
as S and note that based on geometric Brownian motion S=S . With ABMOVM, the stock value
LL LL
with limited liability is a portfolio of the underlying instrument and a zero strike put option, thus
€ €
S≤S =S+P(S,X=0).
LL
In this paper, we assume the limited liability option lasts only as long as the stock option.
€
Limited liability is not costless or perpetual; it requires legal fees, document filings and results in tax
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