probability measure, present value, real options, martingale measure, cash flows, real options valuation, probability measures, Uppsala University Sweden, financial options theory, MEMM, M. Rubinstein, Prof Maciej Klimek, Prof Patrick Mangheni, Dr John Mango, Journal of Financial Economics, option pricing, discounted cash flow method, the project, financial options, binomial model, Black and Scholes, Options Valuation, project management, prior probability measure, riskneutral probability, probability space, probability, filtered probability space, probability distributions, Relative Entropy, real option, discount factors, Makerere University, Uppsala University, International Journal of Production Research, constrained optimization, discounted value, constraint equation, African Mathematics Millennium Science Initiative
Content:
Applied
mathematical sciences, Vol. 2, 2008, no. 58, 2875  2890
Valuation of Real Options Using the Minimal Entropy Martingale Measure Cyrus Seera Ssebugenyi Department of Mathematics Faculty of Science, Makerere University
[email protected],
[email protected] Abstract In this article, the problem of real options valuation in multinomial trees is investigated. A concrete single real options value based on the minimal entropy martingale measure is provided. Using the MEMM to valuate options in multinomial lattices is an easy procedure which can easily be implemented by practitioners. Mathematics Subject Classification: 91B28, 91B06, 90B50 Keywords: Real Options Analysis, Minimal Entropy Martingale Measure, Multinomial Lattices
1 Introduction
The traditional approach used to valuate projects is based on the discounted
cash flow method where projected future
cash flows are discounted at a rate
which reflects the riskiness of the project. This gives the present value for
the projected cash flows. For instance, let x = (x1, x2, . . . , xN ) be cash flows
expected at the end of a one period project X. If pj 0, j = 1, . . . , N is the
probability that cash flow xj will occur, then, the present value of the project
is
X0
=
p1x1
+...+ 1+k
pN xN
(1)
where k is the appropriate cost of capital.
This method has been widely criticized for its failure to account for man
agerial flexibility in the lifetime of the project. Management of real world
projects requires flexibility on the part of managers whenever they receive new
information regarding progression of their projects.
It was therefore important to develop valuation models which are able to
capture the value of managerial flexibility over the lifetime of the project. Real
2876
C. S. Ssebugenyi
options analysis is one such methodology that has become very popular in the recent past. According to Klimek [16], in real options analysis (ROA), one attempts to apply the successful methods in financial options theory such as the Black and Scholes' option pricing formula to the world of
project management instead. This is possible because there are a lot of similarities between financial and real options. For example, real options can be classified as calls or puts and their exercise style can be classified as European or American type. Moreover, the gross present value of the expected cash flows does correspond to the current value of the stock and the uncertainty in the project value corresponds to the volatility of the stock. A detailed analogue between financial and real options can be found in [3, 9, 15] and many other sources. There are also several significant differences between real and financial options which may prohibit the direct application of the Black and Scholes's [1] option pricing equation to the area of project management. Copeland and Tufano [4] state that real options are more complex than financial options and no one can expect to capture all the contingencies associated with them in a standard Black and Scholes's option pricing formula. Fortunately, over the years researchers have developed option pricing models which are more appropriate for project management. One such alternative is the binomial models in [7]. In addition to being a perfect approximation to the Black and Scholes's option pricing formula, the binomial model allows for early exercises. Copeland and Tufano [4] note that every node of the binomial lattice is a decision node where managers can incorporate decisions that need to be made over the life of the project. To determine the value of such a decision, you may create a portfolio to replicate decision values at each node. The noarbitrage principle ensures that the value of the replicating portfolio matches the project value. This is the basic principle underlying the CRR [7] binomial model and was used widely in [3]. The argument of replication assumes that the underlying asset is tradeable and this issue brings out another significant difference between financial and real options. For the latter, the underlying asset is not tradeable, therefore replication is not possible. In other words, markets are incomplete. One way to overcome this and still be able to use financial options theory to price real options is to use a surrogate asset. A surrogate asset or a twin security was defined in [14] as a tradeable asset whose price process is closely related to the price process of the nontradeable underlying asset of the real option. Unfortunately, using a surrogate asset can lead to wrong conclusions as was noted in [14]. They showed that using surrogate assets may lead to arbitrary prices which are consistent with absence of arbitrage and riskneutral valuation. Klimek [16], noted that this (negative) phenomenon arises from using surrogate assets whose information structure is somewhat independent from
Valuation of real options
2877
the information of the underlying it is supposed to replace. According to Klimek [16], a surrogate asset is appropriate if it will match the one it replaces in every state of nature. The task of looking for the appropriate surrogate asset is difficult because it is practically impossible to find a financial security whose payoffs in every state of nature matches the value of the project in question. Copeland and Antikarov [3] recommended the use of present value of the project without flexibility as the appropriate twin security. They stated that the present value of the project without flexibility is the best unbiased estimate of the market value of the project. What is more correlated with the project than the project itself? This Marketed Asset Disclaimer (MAD) assumption completes the market and the replication argument similar to that in [7] can easily be implemented in a binomial setting. It was recently established in [8] that the MAD assumption is indeed plausible for real options valuation. In this article, we use the MAD assumption and other considerations to valuate real options in mutinomial lattices, using the minimal entropy martingale measure. The rest of this article is structured as follows. In Section, 2, we discuss that our methods are embedded in rational valuation systems introduced in [16] and also highlight some of the advantages of risk neutral valuation over traditional methods. In Section 3, we introduce the minimal entropy martingale measure on a finite probability space and demonstrate with practical examples how we can use it to valuate real options. Section 4 concludes.
2 Rational Valuation Systems Klimek [16] introduced a more general framework for the valuation of real options. According to Klimek, if (, F, {F}0tT , P) is a filtered probability space with sample space , probability measure P and filtration {F}0tT with F0 = {, } and FT F, then, a rational valuation system on this filtered probability space is defined as a family (st, CFt)0stT where, · CFt = L(, Ft, P) for t = 0, . . . , T, is the space of bounded cash flows. · st : CFt  CFs for 0 s t T is a linear bounded operator representing a valuation projection. These operators map nonnegative functions onto nonnegative functions and satisfy the following consistency conditions: su = st tu if 0 s t u T, tt(CFt) = CFt for t = 0, . . . , T.
2878
C. S. Ssebugenyi
Klimek's rational valuation system is a framework consisting of several valuation models ranging from adjustment of discount factors as in [8] to changing of underlying probability distributions. For instance, if X CFt, then the present value rule can be written as
st (X )
=
1
1 +
rst
EИ[X
Fs]
where rst is the appropriate cost of capital for the period [s, t]. The same project X can be valuated by changing the underlying probability P to a riskneutral probability Q. The rational valuation rule can be written
as
st(X) = EЙ
BsX Bt

Fs
where B is an adapted process representing a riskfree bond.
Thus, according to Klimek, the project can be valuated by changing the
underlying probability structure to a risk neutral one or by making step by
step adjustments in the discount factors, the net present value approach can be
used. As illustrated with Example 2, these two approaches may not in general
be equivalent. The claim that the net present value rule with rightly adjusted
discount rates is appropriate for project valuation, was substantiated in [8]
where a real option valuation formula was formulated based on the present
value of the project without flexibility.
Step by step adjustment of discount factors is a time consuming procedure
and may not be practically feasible. On the other hand, the same riskneutral
probabilities, once determined, can be used throughout the valuation process
and we are inclined to use this alternative.
The riskneutral probability approach to valuation of real options has been
restricted to binomial models. This is partly because of their simplicity but
also that for higher order lattices, markets are generally incomplete and there
is not a unique solution. According to Boute et al [2], the best that can
be done is to derive bounds for real option values and they recommend that
exact solutions be investigated in further research. In this article, we provide
a concrete single real options value based on the minimal entropy martingale
measure. Through a number of
WORKED EXAMPLEs, we show how the minimal
entropy martingale measure can be used in very practical ways.
3 Project Valuation Using the Minimal Entropy Martingale Measure Let us consider the development of the present value of the project without flexibility in a single period. If S0 > 0 is the present value of the project
Valuation of real options
2879
without flexibility, then, with probability pj, which is assumed to be strictly positive, the value moves to ajS0, j = 1, . . . , N, 2 < N < ; N N. We assume that ai > aj if i < j and aj > 0 for all j. Let r be the single period riskfree rate, then, the project is viable if and only if
a1 > 1 + r > aN .
(2)
This condition corresponds to the no arbitrage condition in finance. If 1 + r > a1 then, no one would invest in the project because in every state of nature, the return on the project would be (with positive probability) less than the riskfree return. On the other hand, if aN > 1 + r, then, many players would borrow at a riskfree rate to invest in the project as their returns on investment would exceed the riskfree return. Shortly, this opportunity would be evened out. project's cost of capital. According to the net present value rule, the project is worth investing in if A risk neutral probability measure can be defined as a probability measure under which the expected return on the project is equal to the riskfree rate. In other words, a strictly positive probability measure Q = (q1, . . . , qN ) is said to be a riskneutral probability measure if and only if
EЙ
(
S1 S0
)
=
1
+
r
N aiqi = 1 + r, i=1
(3)
where S1 is the value of the project at the end of the period. Since N > 2, we are operating in an incomplete market where there are many riskneutral probability measures. We choose a specific one that is closest to the objective probability measure in the sense that it minimizes the entropic distance with respect to the prior probability measure. We will need the following definitions.
Definition 1 Let P and Q be probability measures on a general probability space (, F), where the terms carry their usual meanings. A probability measure Q is said to be absolutely continuous with respect to P if P(A) = 0 implies that Q(A) = 0 for all A F. Notation is Q P. Two probability measures are said to be equivalent if each of them is absolutely continuous with respect to the other. The notation is Q P. We will focus on a finite sample space where all positive probability measures are equivalent. We define two sets Me and M as follows:
N
N
M = Q = (q1, . . . , qN ) : Q 0, qi = 1, qiai = 1 + r ,
i=1
i=1
Me = {Q M : Q > 0} .
2880
C. S. Ssebugenyi
The notation Q > 0 implies that qj > 0 for all 1 j N and similarly Q 0 implies that qj 0 for all 1 j N. M is the set of martingale probability measures which are absolutely continuous with respect to the prior probability P and Me is the set of equivalent martingale measures. By the Fundamental Theorem of Asset Pricing (see [6, 11, 12, 13]), the set Me = , since we have assumed an arbitragefree market. In the current setting however, we can deduce from the following proposition that the no arbitrage condition in (2) ensures that Me (equivalently M) is nonempty.
Proposition 1
Let c1, c2, . . . , cn be n (ordered) real numbers such that c1 is the smallest and
cn is the largest and let c R. There exists a strictly positive probability
measure P = (p1, . . . , pn) such that
n k=1
pk ck
=
c
if
and
only
if
c1
<
c
<
cn.
Proof
First,
let
us
suppose
that
n
=
2
and
c1
<
c
<
c2.
Then,
P
=
(
c2c c2 c1
,
) cc1 c2c1
solves the problem. In this case the probability measure is uniquely determined
by c1, c and c2. Let us suppose that n > 2 and let c1, . . . ck c < ck+1, . . . cN . Let
b1
=
c1
+... k
+ ck
and Then,
b2
=
ck+1 + N
.. 
.+ k
cN
.
c
=
b2 b2
c  b1
b1
+
c b2
 b1  b1
b2.
If
we
let
1
=
b2 c b2 b1
and
2
=
, b2c b2 b1
then,
c
=
b2 b2
c  b1
b1
+
c b2
 b1  b1
b2
=
1b1
+ 2b2
=
k i=1
1 k
ci
N + i=k+1
N
2 
k
ci.
Therefore, we can let the required probability measure to be P = (p1, . . . , pn)
where
1 k
,
i = 1, . . . , k,
pi =
2 N k
,
i = k + 1, . . . , N.
On the other hand, if c1, c2, . . . , cn are n (ordered) real numbers such that
c1 = min{c1, . . . , cn}, cN = max{c1, . . . , cn} and
n k=1
pk ck
=
c,
then,
n
n
n
c1 = pic1 < pici = c < picn = cN .
i=1
i=1
i=1
Valuation of real options
2881
Definition 2 Relative Entropy (Cover and Thomas [5]) Let Q and P be probability measures on a finite probability space . The relative entropy of Q with respect to P is defined as
I (Q,
P)
=
Q()
ln
Q() P()
.
We understand throughout the paper that 0 ln(0) = 0. Basic properties of relative entropy are well known. For example,
0 I(Q, P) .
(4)
Relative entropy gives a measure of how different two probability distributions are. It is not a metric though. Definition 3 (MEMM, See for example, [10]) The probability measure Q M is called the minimal entropy martingale measure (MEMM) if it satisfies
I(Q, P) = ЙmiЕn I(Q, P).
(5)
For a single periodfinite probability model, we can determine the MEMM
using the method of Lagrangian multipliers. The basic
optimization problemis given as follows:
min
N i=1
qi
ln(
qi pi
)
s.t.
qiai = 1 + r,
N i=1
qi
=
1
and
qi
0
1
i
N.
(6)
It so happens that problem (6) has a unique solution Q = (q1, . . . , qN ) given
by
qi =
pieai
N j=1
pj
eaj
,
i = 1, 2, . . . N
(7)
provided that there exists a constant which satisfies the following equation.
N
N
piaieai = (1 + r) pieai .
(8)
i=1
i=1
The following lemma due to Frittelli [10], links the existence and uniqueness of to the no arbitrage assumption.
Lemma 1 There are no arbitrage opportunities if and only if equation (8) has a unique solution.
2882
C. S. Ssebugenyi
Proof:
The following proof is a modification of what was given in [10].
Let M(x) =
N i=1
aiqi(x)
where
qj (x)
=
pi.exai
N i=1
piexai
,
j
=
1, . . . , N.
Clearly, M C1(R) and is the mean of a random variable a whose proba
bility distribution is given as
P(a = aj) = qj(x) =
piexai
N j=1
pj
exaj
,
i = 1, 2, . . . N.
To show that there exists a unique such that M() = 1 + r, we write the function M as follows:
M (x)
=
p1
a1 ea1 x p1ea1
+ p2a2ea2x + x + p2ea2x + .
. .
. .
. + pN aN eaN + pN eaN x
x
.
(9)
Then,
lim M(x) x+
=
aN
<
1
+
r
<
a1
=
lim M(x). x
(10)
Therefore, by the Intermediate Value Theorem, we conclude that there exists a constant R such that M() = 1 + r. To show uniqueness, it is sufficient to show that M (x) < 0 for all x R.
M (x)
=
N i=1
ai
dqj (x) dx
=

N i=1
a2i qi(x)
+
N aiqi(x) i=1
2 = V ar(a) < 0. (11)
For the converse, let us suppose that there exists a constant satisfying equation (8). Let
qj =
pj eaj
N i=1
pie
ai
,
j
= 1, 2, . . . , N.
(12)
Then, Q = (q1, q2, . . . , qN ) is a probability measure. It is also a martingale probability measure since
N aj S0qj = j=1
N j=1
pj
aj
eaj
N i=1
pieai
= (1 + r)S0.
(13)
The conclusion follows from Proposition 1. As a corollary, if N = 2, then Q = (q1, q2) where
q1
=
1 + r  a2 a1  a2
and
q2
=
a1  1 a1 
 a2
r
,
Valuation of real options
2883
and Q is the unique equivalent martingale measure for binomial models (See [7]). Let x~j, j = 1, . . . , N be the state j payoff of the project X with a real option and let be the value from immediate exercise. Then, the minimal entropy value of the project with a real option is given as
1N
(X) = max 1 + r j=1 x~jqj,
(14)
and the minimal entropy value of the real option is therefore given as
(X)  X0.
(15)
Using the following example from [8], we illustrate the procedure for the evaluation of real options using the MEMM.
Example 1 Abandonment Option An abandonment option can be defined as an option to close out an investment prior to the fulfillment of the original conditions for termination. Let us consider a project [][pp.8]DDV06 X with three possible outcomes, $1.2, $1 or $0.8 and respective probabilities 25%, 50% and $25%. The riskfree interest rate is 5% and the cost of capital is 10%. Using these values, the present value X0 for the project (without flexibility) is found to be
0.25 Ч 1.2 + 0.5 Ч 1 + 0.8 Ч 0.25
X0 =
1.1
= $0.9091.
We want to determine the value of an abandonment option with a payoff of $1 exercisable only at the end of the period. Solution. Using a riskneutral argument, we can derive bounds on the value (X~ ) of the project with options as follows:
ЙinЕf EЙ
X 1+r
(X) ЙsuЕp EЙ
X 1+r
where
M=
Q=
q,
17 22

2q,
q
+
5 22
17 : 0 < q < 44
.
Thus, the option value is found to be between $0.043 and $0.117.
Given this information, the MEMM is found to be Q = (0.149, 0.474, 0.377)
and the minimal entropy value (X) for the abandonment option is found to
be
(X )
=
0.149
Ч
1.2
+
0.474 Ч 1.05
1
+
0.377
Ч
1

X0
=
$0.072.
2884
C. S. Ssebugenyi
The certaintyequivalent version1 of the net present value formula in [8]
gives an option value between $0.063 and $0.076.
The minimal entropy martingale measure is not the only martingale prob
ability measure that can be used to derive an exact solution. Other martingale
probability measures such as the minimal variance martingale measure can be
used.
Let =
N i=1
pi
ai
and
2
=
N i=1
pi(ai

)2.
The
minimal
variance
mar
tingale measure (MVMM) (as in [10]) is defined as Q = (q1, . . . , qN ) where
qj = pj
1
+
 2
r (

aj )
, j = 1, . . . , N.
(16)
If valued using the MVMM, the value of the abandonment option in Example 1 is found to be $0.069.
Remark 1 We remark that the MEMM is more appropriate as a pricing measure because as seen in equation (7), it is always equivalent to the objective probability measure. On the other hand, the minimal variance martingale measure is not in general equivalent to the objective probability measure. For further discussions on this issue, see [10] and references given there.
Example 2 As another example, let us consider an option to contract or shrink a project. This can be achieved by selling or subletting part of the production facilities to another company. When exercised at a strike price K, the project's present value is shrunk by a factor . The single period minimal entropy value of the project with the contraction option is given as
1N
(X) = 1+r
qj max(xj, xj + K),
(17)
j=1
if the option is exercisable only at the end of the period. In Example 1, let us suppose that the project can be contracted by 25% thereby saving $0.28 in operating expenses. What is the value of the contraction option? A riskneutral approach would give an option value between $0.039 and $0.047 and the certaintyequivalent version of the net present value formula proposed in [8] gives an option value between $0.074 and $0.104. We used the minimal entropy martingale measure and obtained a value of $0.042 for the option to contract. Or, the total minimal entropy value of the project with a contraction option was found to be $0.951. 1Using average market returns of 12% with standard deviation of 20%, as in [8].
Valuation of real options
2885
Example 3 Compound Options2 Copano, a chemical firm is considering a phased investment plant. It will cost $60 million immediately for permits and preparations, which will take a year. At the end of year one, the firm can invest $400 million to complete the design phase. Managers believe that once the design phase is over, the firm has a two year window during which it can make a final investment worth $800 million needed to build the plant. This is an example of what is commonly known as compound options or sequential options: A $60 million investment now, creates the right to invest $400 million one year later, which if exercised creates the right to invest $800 million to purchase the plant. Based on NPV calculations, the firm is assumed to be worth $1, 000 million today but future values are uncertain and are random in nature. The volatility of the project is assumed to be 18.23% per annum and the riskfree rate is assumed to be 8% per annum. At the discount rate of 10.83%, if the firm decides to invest in the second year, the present value of costs in year two, will be $1072.2 but if it decides to invest in the third year, the present value of costs in year three, will be $1008.56. Therefore, according to the net present value rule, it is worthless investing in this project. However, investments at the end of year one, two or three are options and will be exercised if deemed worth. We describe how the value of this project can be determined in both the context of binomial and trinomial models. In both cases, we use a riskneutral approach instead of using a replicating portfolio approach as in [4]. For multinomial models we use the MEMM to derive an exact value for the project.
3.0.1 Determining the value of the project using binomial Models
Following Copeland and Antikarov[3], the present value of the project without flexibility will act as the underlying asset and the binomial event tree for the underlying asset is shown in Figure 1. If S > 0 is the value at the beginning of a period, then, with riskneutral probability q > 0 the value at the end of the period will be uS, u > 0 and with riskneutral probability 1  q it will be dS, d > 0 where q, u and d are parameters in the CRR [7] binomial model3. Having constructed a binomial lattice for the value of te project with no option,
2The following example is adopted (with permission) from [4].
3The
CRR
model
is
generally
understood
to
imply
up
probability
q
=
ert d ud
and
jump
amplitudes
u
=
1 d
jump amplitudes
= exp(t) where , they also derived
an
is the volatility alternative up
in annualized terms.
probability
q
=
1 2
1
With the same
+
(r

2 2
) t
.
Using this alternative parametrization, the value of the project is found to be $M 11.74. We
note that these two models are equivalent
2886
C. S. Ssebugenyi
1727.89 1439.94
1000
1199.97 1000
1199.97
833.35
833.35
694.47 578.74 Event Tree for the Underlying Asset
11.08
114.1
699.2
259.26 0 20.78
Option Values
927.89 399.97 33.35 0
Figure 1: Copano's Binomial lattices showing the project and option values.
we then used single step risk neutral pricing relations to determine the value of compound options. The procedure is summarized in the following steps. If Stk, 0 t 3 is the time t value of the project with no flexibility at node k of the binomial model, then
· At the end of the third year, the value of the project with a real option
at node k is
C3k = max S3k  800, 0 , k = 1, . . . , 4.
· At the end of the second year, the value of the project with a real option at node k is
C2k = max
S2k

800,
1
1 +
r (qC3k
+
(1

q)C3k+1)
, k = 1, 2, 3.
· At the end of the first year, the value of the project with a real option at node k is
C1k = max
0,
1
1 +
r
(qC2k
+
(1

q)C2k+1)

400
, k = 1, 2.
· At time zero, the value of the project with a real option is
C0 = max
0,
1
1 +
r
(qC21
+
(1

q)C22)

60
= $11.08 million
3.0.2 Trinomial models: Determining the value of the project using the minimal entropy martingale measure To compute the minimal entropy value for Copano using trinomial lattices, we will need some additional information. In particular, we assume that present
Valuation of real options
2887
value S > 0 of the project without flexibility can move to uS with probability
p1, to S with probability p2 and to dS with probability p3. The corresponding
jump
amplitudes
are
assumed
to
be
u
=
exp(t)
and
d
=
1 u
.
To estimate the probabilities, we use equation (1). That is,
p1uS + p2S + p3dS = S. 1+k Together with the condition that these probabilities add to one, p1 and p3 are computed as follows:
p1
=
(1
+k
 e)  p2(1 e  e

e )
and p3 = 1  p2  p1, where 0 p2 1 is arbitrarily chosen in such a way that 0 p1, p3 1 and k is the appropriate discount factor. With these parameters the MEMM can be derived from equations (7) and (8). We developed a trinomial event tree for the project without flexibility which is similar to the binomial event tree. We then used familiar steps as in the binomial model to determine the value of the project with a real option for varying values of p2. Results are displayed in Table 1.
p2 0.00 0.05 0.10 0.15 0.20 0.25
Project Value $ million 11.08 8.41 5.71 2.95 0.12 0.00
Table 1: Real Option values using the MEMM. Let us note that as p2 0, the minimal entropy value approaches the value computed using binomial models. In fact for p2 = 0 the minimal entropy result coincides with the value computed using binomial models. We remark that trinomial models offer a more realistic development of the underlying asset. It is more natural to assume that the underlying asset can move up, move down or stay unchanged than to assume only up and down movements. However, for the more realistic trinomial models, markets are incomplete and as such, we use the minimal entropy martingale measure for purposes of option pricing.
2888
C. S. Ssebugenyi
The derivation of the minimal entropy martingale measure involves a constrained optimization of relative entropy when the necessary constraint is that the discounted value of the development of the present value is a martingale. The basic objective in this optimization problem is to derive a probability measure which correctly values the project without flexibility and is closest in the entropic distance to the prior probability measure. We point out that we do not have to be limited to the MAD assumption. A surrogate asset may still be used and it can be shown that using the current methods, the arbitrariness of prices as pointed out in [14], will not occur. Moreover, as a surrogate asset, we may use another project which is highly correlated with the one in question. All these securities are benchmark securities which can be included in the constraint equation in (6). Indeed, we can include as many independent constraints as we desire if we want to find a probability measure which correctly prices all these benchmark securities and is closest to the prior in the entropic distance. This procedure is commonly known as marking to market or model calibration (see for example, [17]).
4 Conclusion The minimal entropy martingale measure was used to solve the problem of real options valuation in multinomial lattices. The MEMM yields a concrete single option value which is in some sense optimal. As illustrated by practical examples, the procedure is easy to implement and can be adopted by practitioners. Empirical research is necessary to determine how close minimal entropy prices are to actual values. It was also shown that two approaches; certaintyequivalence and riskneutral valuation can yield a range of option values which are nonoverlapping. The relationship between these two approaches needs to be investigated in further research. Lastly, I have discussed that with the current approach, the price process of any other marketed security (which is relevant to the pricing of the current project in question) can be used in the constraint equation. The chosen security could be the present value of the project without flexibility, the price process of an exchange traded asset or it could be the present value process of another project which shares similar features with the project in question.
Acknowledgments This work is part of the ongoing Sandwich PhD project between the departments of Mathematics at Makerere University in Uganda and Uppsala University Sweden. This collaboration is financially supported by the International
Valuation of real options
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Science programme Uppsala University Sweden. I thank Prof Maciej Klimek , Prof Patrick Mangheni and Dr John Mango for useful comments and support. I also thank the anonymous referee for very useful comments. Financial support from The Norwegian Programme for Development, Research and Education (NUFU) through collaboration with the Department of Mathematics, Makerere University and The African Mathematics Millennium Science Initiative (AMMSI) is acknowledged.
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