In their article ?An Arbitrage-Free Approach to Quasi-Option Value? [J. Environm. Econom. Management 35, 103-125, 1998], Coggins and Ramezani interpreted the concept of quasi-option value introduced by Arrow and Fisher [Quart. J. Econom. 88, 1974, 312-319] as being identical to Dixit and Pindyck?s real option value. This means their approach differs from the approach by Fisher and Hanemann [J. Environm. Econom. Management 14, 183-190, 1987] who formalized the concept of quasi-option value a decade before. By indirectly characterizing Dixit and Pindyck?s real option value Coggins and Ramezani confirmed classic results in the field of real options theory. --

Mensink, Paul

English

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Mensink, Paul
Working Paper
A comment on "An arbitrage-free
approach to quasi-option value" by
Coggins and Ramezani
Economics working paper / Christian-Albrechts-Universität Kiel, Department of Economics,
No. 2004,06
Provided in cooperation with:
Christian-Albrechts-Universität Kiel (CAU)
Suggested citation: Mensink, Paul (2004) : A comment on "An arbitrage-free approach to quasi-
option value" by Coggins and Ramezani, Economics working paper / Christian-Albrechts-
Universität Kiel, Department of Economics, No. 2004,06, http://hdl.handle.net/10419/21982 
 
Economics Working Paper 
No 2004-06 
 
A comment on “An arbitrage-free approach 
to quasi-option value” by Coggins and 
Ramezani  
 
by Paul Mensink A comment on “An arbitrage-free approach to
quasi-option value” by Coggins and Ramezani∗
Paul Mensink †
June 21, 2004
Abstract
In their article “An Arbitrage-Free Approach to Quasi-Option Value” [J.
Environm. Econom. Management 35, 103-125, 1998], Coggins and Ramezani
interpreted the concept of quasi-option value introduced by Arrow and Fisher
[Quart. J. Econom. 88, 1974, 312-319] as being identical to Dixit and Pindyck’s
real option value. This means their approach diﬀers from the approach by Fisher
and Hanemann [J. Environm. Econom. Management 14, 183-190, 1987] who for-
malized the concept of quasi-option value a decade before. By indirectly charac-
terizing Dixit and Pindyck’s real option value Coggins and Ramezani conﬁrmed
classic results in the ﬁeld of real options theory.
∗I would like to thank Jay Coggins, Andreas von D¨ ollen, Joseph Herriges, Dagmar Nelissen, Cyrus
Ramezani, Till Requate and two anonymous reviewers for their critiques and suggestions with respect
to earlier versions of this comment, and Diderik Lund for guidance in the ﬁeld of ﬁnancial economics.
I would like to stress that the persons I thank do not necessarily agree with all claims made in this
comment. Part of the research was done at the University of Kiel, Germany; support by the German
Research Council (DFG) is gratefully acknowledged.
†Independent. Zeye Strasse 30, 24106 Kiel, Germany. E-mail: pgcmensink@yahoo.com
11 Introduction
In the context of irreversible decision making under uncertainty in a dynamic frame-
work, concepts of option values have been developed independently in diﬀerent strands
of literature. Most prominently Dixit and Pindyck (e.g.[6]; for an overview see [14],
section VI) contributed to the investment and ﬁnancial literature by arguing that by
making an irreversible investment that could have been postponed, the option to in-
vest is exercised and therefore its value should be accounted for in the investment
decision.1 Arrow and Fisher [1] - analyzing a similar decision framework as Henry [9] -
triggered a series of publications in the area of environmental and resource economics
by describing the concept of quasi-option value. Their work lead to a formal deﬁnition
of quasi-option value with respect to an irreversible investment under uncertainty by
Fisher and Hanemann [7]. This option value, to be called OV FH here, represents the
value of the information that becomes available when uncertainty is resolved over time
[7,8].
Attempts were made to unify both concepts. Lund [10] reconciled the Fisher-Hanemann
decision framework with ﬁnancial option valuation. Mensink and Requate [12] estab-
lished the relation between the value of the option identiﬁed by Dixit & Pindyck and
OV FH. Finally, Coggins and Ramezani [4] interpreted quasi-option value as a ﬁnancial
option using an arbitrage-free (‘contingent claims’) valuation approach in the standard
binomial model [5].
In this comment I argue that the option value that is the object of study of Coggins and
Ramezani’s article “An arbitrage-free approach to quasi-option value” [4], is equivalent
to Dixit and Pindyck’s real option value, and not equivalent to the quasi-option value
OV FH as deﬁned by Fisher and Hanemann [7].
The paper is organized as follows. First, I present a decision framework to deﬁne Fisher
and Hanemann’s quasi-option value OV FH (section 2). Then I show that the ‘quasi-
option value’ Coggins and Ramezani [4] analyze is equivalent to Dixit and Pindyck’s
1This option is often referred to as ‘real option’ because it is an option with respect to a ‘real’
investment, and not a ﬁnancial derivative.
2real option value and not Fisher and Hanemann’s quasi-option value (section 3). After
that, I discuss the consequences of this result and draw a conclusion (section 4).
2 The Model
The model presented here is identical to the one in Fisher and Hanemann [7] and
Hanemann [8]2: a ﬁrm has the opportunity to make an investment that can be made
in either of two periods, i = 1,2. By di ∈ {0,1} we denote the decision variable
which indicates whether or not the investment is in place in period i. The investment
is irreversible: this implies d2 ≥ d1. Moreover, the investment can only be made
once: d2 + d1 ≤ 1. The investment generates a beneﬁt B1(d1) in period 1, and a
stochastic beneﬁt B2(d1,d2,θ) in period 2, where θ is a random variable that represents
uncertainty. Now consider two scenarios. In the ﬁrst scenario, the ﬁrm knows the true
value of θ in period 2 before it has to decide for the second time whether or not to
invest. The expected beneﬁt as a function of the investment decision in period 1 is
given by
ˆ V (d1) = B1(d1) + E[max
d2
B2(d1,d2,θ)] (1)
In the second scenario the ﬁrm does not know the realization of the value of θ when
it decides whether or not to invest in period 2. In this case the expected beneﬁt as a
function of the investment decision in the ﬁrst period is determined by
V
∗(d1) = B1(d1) + max
d2
E[B2(d1,d2,θ)] (2)
The quasi-option value as deﬁned by Fisher and Hanemann, OV FH, is the diﬀerence
between these two expected beneﬁts if no investment is made in the ﬁrst period [7,8]:
OV
FH = ˆ V (0) − V
∗(0). (3)
2For an extensive presentation and analysis of this model the reader is referred to [7,8,12]
3OV FH measures the extra expected proﬁt the ﬁrm can gain by taking into account
information that will become available in the future. Formally speaking, OV FH is the
expected value of information with respect to θ conditional on non-investment in the
ﬁrst period [8, p.29]. It can also be interpreted as a shadow tax on investment that
corrects myopic behavior. In other words, this shadow tax corrects the behavior of
those investors that do not take into account information that will become available in
the future [8, p.27].
The ‘real option value’ with respect to an irreversible decision, as formulated by Dixit
and Pindyck (see [6, p.96-97;12]) - expressed in terms of our model - has been deﬁned
as:
OV
DP = max{ˆ V (0), ˆ V (1)} − NPV (4)
with NPV being the maximum of the expected beneﬁt investing in the ﬁrst period and
not investing at all:
NPV = max{V
∗(1), B1(0) + E[B2(0,0,θ)]} (5)
The real option value, OV DP, is the value of the opportunity to postpone the invest-
ment decision. OV DP is not equal to OV FH in general, though it includes OV FH,
which is that part of the value of waiting that is related to information. For a detailed
discussion see [12].
OV DP is traditionally analyzed either using dynamic programming techniques or - if
the return on investment can be replicated by a traded portfolio - by means of an
arbitrage free argument in the tradition of Black, Scholes and Merton [6,13].
3 Quasi Option Value in Coggins and Ramezani [4]
To formally deﬁne the concept of quasi-option value as it was introduced by Arrow and
Fisher [1], Coggins and Ramezani ﬁrst describe the ’expected net present value rule’
4(ENPV) in the context of a dynamic decision making problem as
‘Replacing future stochastic variables with their expected values, discount-
ing costs and returns back to the present, and investing whenever ENPV is
positive [...]’ [4,p.103-104].
In terms of the model in section 2, their ENPV decision rule says3:
d1 = 1, d2 = 1 if B1(1) + B2(1,1,E[θ]) ≥ B1(0) + B2(0,0,E[θ]) (6)
d1 = 0, d2 = 0 otherwise. (7)
Following [4], we deﬁne W0 as the expected value of the investment if the decision
maker would act according to this ENPV decision rule. The authors use this deﬁnition
to claim quasi-option value in [1] is equal to:
‘[...] the extra value that can be gained if one eschews ENPV analysis in
favor of the fully dynamic alternative.’ [4, p.104].
This means that Coggins and Ramezani claim that the concept of quasi-option value
decribed by Arrow and Fisher [1] is equal to ˆ V (ˆ d1)−W0, with ˆ d1 = argmax ˆ V (ˆ d1). At
the bottom of page 107 they write that O∗ = max{0,W ∗
0 − W0} ≡ max{0, ˆ V (0) − W0}
is the deﬁnition of quasi-option value.
Neither of these expressions is equivalent to the quasi-option value OV FH as deﬁned
and analyzed by Fisher and Hanemann [7,8] who stylized the framework of irreversible
investment under uncertainty developed by Arrow and Fisher [1] and Henry [9]. Please
recall that Fisher and Hanemann deﬁne OV FH as follows:
3Although the deﬁnition of this rule leaves room for the decision-pair d1 = 0,d2 = 1 if B1(1) +
B2(1,1,E[θ]) < B1(0)+B2(0,0,E[θ]) and B2(0,1,E[θ]) > B2(0,0,E[θ]), equation (1) in Coggins and
Ramezani [4] makes clear that d1 = 0,d2 = 1 is not considered a possibility under this decision rule:
(1) only takes d1 = 0,d2 = 0, and d1 = 1,d2 = 1 into account, though d1 = 0,d2 = 1 might be a
rational choice when A > (P0 − c) and u >> d.
5OV
FH = ˆ V (0) − V
∗(0). (8)
It is easy to see that (8) is neither equal to ˆ V (ˆ d1) − W0, nor to O∗: the diﬀerence is
most obvious when W0 < V ∗(0). This is the case when even without knowing the true
value of θ, investing in the second period is still more attractive than either investing in
the ﬁrst period or not investing at all. The crucial diﬀerence between OV FH and both
ˆ V (ˆ d1) − W0 and O∗, lies in the fact that, in their ENPV rule, Coggins and Ramezani
do not account for the opportunity to invest in the second period in the absence of
extra information.4
In fact, O∗ = ˆ V (0)−W0 is strongly related to OV DP - and for Coggins and Ramezani’s
model they are equivalent as we will show below. Therefore it should not come as a
surprise that Coggins and Ramezani show that O∗ is equal to the value of the right to
delay the investment decision, which after all is the verbal deﬁnition of OV DP. Please
recall that
W0 = max{B1(0) + B2(0,0,E[θ]),V
∗(1)} (eq.1 in Coggins and Ramezani, 1998) (9)
We know that5
4This is conﬁrmed by two mathematical errors in what I interpreted as their attempt to show
equivalence between the Fisher and Hanemann [7] and Hanemann [8] deﬁnition of quasi-option value
and O∗ (footnote 10): neither W1(q)/R = W0(q) nor V ∗(0) = V ∗ = W1(q)/R hold in general. In
stead V ∗(0) = V ∗ = A + (W1(q)/R). W1(q)/R = W0(q) can not hold in general because the term on
the right contains P0 and the term on the left side does not.
5Please note that equation (14) holds because by deﬁnition NPV ≥ V ∗(1) = ˆ V (1) and if NPV >
ˆ V (1) then ˆ V (0) − NPV ≥ 0.
6O
∗ = max{0,W
∗
0 − W0} (10)
= max{0, ˆ V (0) − max{B1(0) + B2(0,0,E[θ]),V
∗(1)}} (11)
(and if B2 is linear in θ, as in Coggins and Ramezani (1998))
= max{0, ˆ V (0) − max{B1(0) + E[B2(0,0,θ)],V
∗(1)}
| {z }
NPV
} (12)
= max{0, ˆ V (0) − NPV } (13)
= max{ˆ V (1) − NPV, ˆ V (0) − NPV } (14)
= max{ˆ V (0), ˆ V (1)} − NPV (15)
= OV
DP (16)
The fact that O∗ is equal to OV DP conﬁrms that O∗ is not equivalent to OV FH, because
OV FH and OV DP are not equal in general [12].6
4 Concluding remarks
This comment showed that when in sections I and II of “An Arbitrage-Free Approach to
Quasi-Option Value” [4] Coggins and Ramezani deﬁne ‘quasi-option value’ they deﬁne
a concept that diﬀers from the ‘quasi-option value’ as it was deﬁned and analyzed by
Fisher and Hanemann [7,8]. This means they re-interpreted the concept of quasi-option
value that was introduced by Arrow and Fisher [1]. What Coggins and Ramezani deﬁne
as ‘quasi-option value’ was shown here to be equivalent to the ‘real option value’ as
formulated and analyzed by Dixit and Pindyck [6,13].
This means that in section III, Coggins and Ramezani express Dixit and Pindyck’s
real option value in terms of a risk free portfolio. This was done before by Brennan &
Schwartz and Pindyck [3; 13, section A]. This also means that in the same section Cog-
gins and Ramezani analyse the eﬀect of increasing uncertainty on Dixit and Pindyck’s
6Bosetti and Messina [2] based their model on Coggins and Ramezani’s and therefore seem to adopt
the same deﬁnition of quasi-option value as Coggins and Ramezani did.
7real option value, and show it is non-decreasing in the level of uncertainty. Thereby
conﬁrming a result by McDonald and Siegel [11].
Conclusion: after re-interpreting the concept of quasi-option value as being equal to
Dixit and Pindyck’s real option value, Coggins and Ramezani [4] conﬁrmed classic
results in the ﬁeld of ﬁnancial economics.
Bibliography
[1] K.J. Arrow, A.C. Fisher, Environmental preservation, uncertainty and irreversibil-
ity, Quart. J. Econom. 88 (1974) 312-319.
[2] V. Bosetti, E. Messina, Uncertainty and Option Value in Land Allocation Problems,
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[3] M.J. Brennan, E.S. Schwartz, Evaluating Natural Resource Investments, J. of Busi-
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[4] C.J. Coggins, C.A. Ramezani, An Arbitrage-Free Approach to Quasi-Option Value,
J. of Environ. Econom. Management 35 (1998) 103-125.
[5] J.C. Cox, S.A Ross, M. Rubinstein, Option pricing: A simpliﬁed approach, J. Finan.
Econom. 7 (1979) 229-263.
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pelled, J. of Environ. Econom. Management 14 (1987) 183-190.
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Econom. Management 16 (1989) 23-37.
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Amer. Econom. Rev. 64 (1974) 1006-1012.
8[10] D. Lund, Financial and Non-ﬁnancial Option Valuation, in D. Lund and B.
Øksendal (Eds.) Stochastic Models and Option Values (Contributions to Economic
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[11] R. McDonald, D. Siegel, The Value of Waiting to Invest, Quart. J. Econom. 101
(1986) 707-727.
[12] P. Mensink, T. Requate, The Dixit-Pindyck an the Arrow-Fisher-Hanemann-Henry
option values are not equivalent. Forthcoming in Resource and Energy Economics. The
working paper version this article can be downloaded at http://ssrn.com/abstract=405062.
[13] R.S. Pindyck, Irreversibility, Uncertainty, and Investment, J. Econom. Lit. 29
(1991) 1110-1148.
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sessment, J. Finan. 55 (2000) 1569-1622.
9

An Arbitrage-Free Approach to Quasi-Option Value,

Continuous-Time Methods in Finance: A Review and an Assessment,

Environmental preservation, uncertainty and irreversibility,

Evaluating Natural Resource Investments,

Information and the Concept of Option Value,

Investment Decisions Under Uncertainty: The “Irreversibility Eﬀect”,

Investment under Uncertainty,

Option pricing: A simpliﬁed approach,

Quasi-Option Value: Some Misconceptions Dispelled,

The Dixit-Pindyck an the Arrow-Fisher-Hanemann-Henry option values are not equivalent. Forthcoming in Resource and Energy Economics. The working paper version this article can be downloaded at http://ssrn.com/abstract=405062.

The Value of Waiting to

Uncertainty and Option Value in Land Allocation Problems,

A comment on "An arbitrage-free approach to quasi-option value" by Coggins and Ramezani

http://econstor.eu/bitstream/10419/21982/1/EWP-2004-06.pdf

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