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ISSN 1064.9769
38, Special
Issue,
1998,
pages
725-754
Estimating Volatility and Dividend Yield When
Valuing Real Options to Invest or Abandon
GRAHAM
Colorado
A. DAVIS
School
of Mines
The opportunity
to invest in or abandon a project can in princi$e
be valued using real options
techniques.
In practice,
option pricing
has as inputs the volatility
and dividend
yield of the
project, which are in most cases not observable via market data. Current
methods of estimating
these parameters
are lurgely ad hoc, introducing
potential error into the valuation
process. Thti
paper uses simple production
models to formulize
concepts for estitn&ing project volatility and dividend yield in single stochastic variable option mod&
and provides an example of hmu these estimates can be used in a real option valuahon
exercise.
Despite over two decades of real options development
and promotion
by academics, there is still little practical application of real option analysis. For example, the majority of mineral firms, the prototypical industry through which real
options applications were developed, do not use option pricing in their capital
budgeting
decisions (Bhappu and Guzman, 1996). One contributing
factor is
that the level of mathematics is more sophisticated than that of standard net
present value analysis. To remedy this, several “how to” manuscripts
are
attempting
to communicate
the mathematical
techniques of real options in a
user-friendly manner.’ A second problem, which confronts even finance experts,
is the estimation of the six parameters required in real option applications.*
Most textbook examples demonstrating
the practice and benefits of real options
analysis use hypothetical parameter values that conveniently highlight specific
principles, without specifying how the practitioner is to come up with the requisite parameter values in an actual investment valuation. In the most basic application of real option pricing, where the value of the underlying project is taken
to be stochastic, the project’s volatility and dividend yield are the two most problematic parameter inputs. Majd and Pindyck lament that “it may be difftcult or
impossible to estimate them accurately” (1987, p. 25).
725
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This paper formalizes some concepts for estimating a project’s volatility and
dividend yield when valuing options to invest in or abandon the project. The
basis of the analysis is a fixed-output
model of production, from which tractable,
closed-form valuation equations allow the calculation of project volatility and
dividend yield. I begin with a review of the real option framework.
I.
MODELING
THE VALUE
OF A REAL OPTION
Consider a derivative asset whose value, F, is a function of the value of a single
underlying asset, V, and time, t; F = F(V, t). In real options analysis, F might be
the value of an American call option to irreversibly invest in a project that would
currently be worth I’ if installed, or the value of an American put option to
abandon a project currently worth I’, where V is the expected present value of
net income from the project. The goal of real option pricing is to determine
option value F given a stochastic process for I’.
A standard approach to valuing these types of options is to assume that the
value of the underlying project fluctuates according to an Ito Process,
dti = c&v,
tjdt + B,(v,t)dt,
j = I, A,
(1)
where &;(I’, t) is the expected instantaneous rate of drift in the value of the
project, d,(V, t) is the instantaneous rate of volatility (standard deviation) of the
value of the project, and dz is a standard Wiener process. The indexj indicates
that these drift and volatility functions depend on whether the project underlies
the investment option (i = I) or abandonment
option 0’ = A). A popular form of
this Ito Process is geometric Brownian motion,
dti = $vVdt
+ c+‘dz,
(2)
where dr, and c$ are the project’s (constant) rate of drift and volatility. For
example, McDonald and Siegel (1986) value the option of waiting to invest in a
project, and assume that changes in the value of the completed project follow
the process in Equation 2.’
The assumption that the drift and volatility of V are constant is mathematically convenient, but unrealistic. Under a geometric Brownian motion the value
of the project cannot become negative, whereas real projects can have substantial liabilities attached to them and be worth less than zero. Another problem,
which will become evident in Section II, is that the rate of volatility of V, d,, is
likely to be a function of the level of V and time, and therefore not constant.
However, just as financial option pricing is a method of approximation,
real
option pricing, too, provides only an approximation
of real asset value. The
assumption that V cannot become negative will cause the calculated option
value to be too high for some types of projects, while the direction of bias
ESTIMATING
VOLATILITY
AND DMDEND
YIELD
727
injected by assuming that d, is constant depends on the nature of the true
process c$(V, t).
Staying for the moment with the assumption that c$ and c$ are constant,
applying Ito’s Lemma and the no-arbitrage condition to the function F(V, t) creates the partial differential equation
where r is the nominal risk-free rate of interest and d is the instantaneous net
cash flow per unit time paid out to the holder of the call or put option. Normally, 0’ = 0 and @ > 0. The constant dV represents the extent to which the
percentage change in project value, o$, falls short of the percentage return
required on an investment of this risk class, i%$ ; 8” = i& - a$,. For this reason, $, is called the “rate of return shortfall” (McDonald and Siegel, 1984).
Intuitively,
it is equivalent to the percentage dividend yield of the underlying
asset in a financial index option, and to reduce confusion I will use the term dividend yield rather than rate of return shortfall.
Standard numerical techniques, such as binomial lattices or finite difference
methods, are used to solve partial differential
Equation 3 for F, taking into
account boundary conditions for the specific option being valued. & all cases
the solution technique requires that the modeler specify values for oV and dV.4
Since these are generally unobservable in the market, most analysts resort to
rules of thumb, injecting potential error into the calculation of the option value.
The paper presentsduggestions
for the estimation of dV in Section II, and for
the estimation of or, in Section III. Section IV then provides a practical example that uses these results.
II.
ESTIMATING
PROJECT
cry, THE VOLATILITY
OF THE UNDERLYING
In Equation 2 the rate of volatility of I’, dh, reflects the anticipated dispersion
of future levels of I’. One method of estimating (PV is to use the instantaneous
standard deviation of historic changes in project value, calculated as
(4)
where
n + 1 = number
of historic observations
of V
728
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vi
ui
u
z
=
=
=
=
REVIEW
OF ECONOMICS
AND
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project value at the end of the ith interval of time (i = 0, l,.. ., n)
ln(v,/Vi.t), i = 1, ?,.. ., n
the mean of the ui’s
the length of time between measurements.
However, a historical time series of project market values is seldom available,
especially when valuing the option to invest, since in this case the potential
project does not yet exist. If the potential project is identical to others owned
and operated by listed companies, it may be possible to estimate aV from a historical series of unlevered company values. If there are options on these listed
companies, an implied volatility could also be estimated (Hull, 1997).
It is more likely that the project’s value is neither directly nor indirectly
traded, forcing practitioners
to make informed guesses about d, based on
observable market information. In one alternative, McDonald and Siegel (1986),
Majd and Pindyck (1987), and Dixit and Pindyck (1994), who model V as the
value of a generic factory, take o: to be equal to the average percentage standard deviation of stock market equity, about 0.20 to 0.30. In another, Pickles
and Smith (1993), who model the value of developed underground
oil reserves
as V, use reported historical reserve transaction values and Equation 4 to estimate that cri = 0.23. They also note that the volatility oficrude oil prices is
around 0.22, which they feel corroborates their estimate of or, for oil reserves.
Unfortunately,
miss-estimation of C$ can have important impacts on the
calculated option value, F. For example, in his calculations of an optionlto invest
in oil production, Trigeorgis (1990) shows that a 50% increase in oV brings
about a 40% increase in option value. Many authors, recognizing the fallibility of
their volatility estimate, calculate option values using a variety of aV values,
unable to specify which is the more likely (e.g., Mann, wobie, and MacMillan,
1992; Teisberg, 1994). This section demonstrates that d, can be estimated from
the volatility of the unit price of the project’s output, OS, which in many cases is
either revealed in the prices of traded commodity options or can be calculated
from published historic price series using Equation 4.
In real options analysis it is usually assumed that the price of the project’s
output good, like V, fluctuates according to an Ito Process,
dS = as(S,t)dt + cQS,tJdz,
(5)
where S is the unit price of the output good, cr&S, t) is the expected instantaneous drift of price, os(S, t) is the instantaneous volatility of price, and dz is a
standard Wiener process. Since V depends on the current and futures prices of
the output good, and since by Equation 5 future values of S are dependent on
the current value of S, current project value can be written as V(S, t). For example, the value of a gold mine, I/gol& is a function of the current price of an
ESTIMATING
VOLATILITY
AND DMDEND
ounce of gold, SG01d.5With I/ = V(S, t), and with dS represented
5, Ito’s Lemma gives,
d# = g +a,(S,tg ++,
[
Equation
YIELD
759
as in Equation
t$$]dt +ks(S,f)!gjd%.
(6)
6 is in the form
dti = c&v,
t)dt + d,(v,
t)dz,
(7)
where
d&v,t) = g +qs, t); +;o;,s,t)$
(8)
and
From Equation 7 the process for V is indeed an Ito Process, as assumed in Equation 1, the noise in V driven by stochastic movements in S (since V = V(S, t)).
However, the process for V is the geometric Brownian motion in Equation 2
only if the right-hand-sides
in Equations 8 and 9 are constant multiples of V.
Focusing on the variance term dv(V, t) in Equation 7, and since Equation 2
includes a term d,Vdz, rewrite Equation 6 as
dti
=
acB
at + a#,
ati
t)as
1 2
+ p,(S,
t)-
a2ti
as2
Now, comparing
Equations
1[
aA
dt + os(S, t)xv
1
Vdz .
(10)
2 and 10,
(11)
Changes in the price of the project’s output, as represented in Equation 5, are
frequently assumed to have a constant rate of volatility; os(S, t) = 0~s.~ With
this, Equation 11 becomes
(12)
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where d is the price elasticity of the project’s value, or the sensitivity of project
value to changes in the spot price of the project’s output good. Since project
value is increasing in S,
&j-arts
-zp+o.
Equation 12 presents the paper’s first result: c$ = &J,, or the rate of volatility of the project is directly linked to the constant rate of volatility of the price
of the project’s output good, os, via a positive elasticity term Ei.’ Where d is not
constant during the life at the option, and I show below that it is not, dV will
not be constant over the life of the option, in violation of the assumption that I/
follows a geometric
Brownian motion. Nevertheless, Equation
12 supports
researchers’ intuition about a link between OS and c$. Trigeorgis (1990, 1996)
models ;he value of an option on developed underground
oil reserves, and estimates cry to be 0.20, being guided by crude oil price volatilities, <TS,of between
0.20 and 0.40. Paddock, Sieg$ and Smith (1988) and Smit (1997), in valuing
the same option, simply set oV = 0s. As not e d ab ove, Pickles and Smith (1993),
who also value an optiyn on developed oil reserves, are comforted by the fact
that their estimate of or, for reserves is close to OS for crude oil. From Equation
12, these cases assume E’ values ranging between 0.5 to 1.0, which may or may
not be appropriate.
To find legitimate values of d, and to show that it is time-varying, I first
examine the case of projects with no operating flexibility, and then move on to
the case where the project can be temporarily or permanently shut-down should
the price of the project’s output fall.
A.
Estimating
Project Volatility
When There is No Operating
Flexibility
One could, in principle, estimate d using Monte Carlo simulation, but this
would only give a point estimate, and would indicate little about the dynamic
properties. of d, and hence c$,. This section instead produces closed-form solutions for E7using the simple case of a generic fixed-life project for which there is
no operating flexibility such as the option to expand capacity or to shut down.
This technique of solving for d is similar to that of Blose and Shieh (1995) and
Tufano (1998), except that in this paper I assume there are no fixed costs, variable costs are certain but may inflate through time, the firm does not hedge
financial risk, project output may decline over time, and prices follow a geometric Brownian motion and have a positive convenience yield.
I consider two production profiles that provide tractable closed-form valuation equations. In the first, project output is optimally fixed at plant capacity K
over the project life, T. This constant-production
model applies particularly well
to projects such as mining, where capacity is expensive to build and where the
ESTIMATING
option to temporarily
shut-down
down and maintenance expenses
stances, it is optimal to build the
the plant at that capacity (Cairns,
Assume that the unit price
Brownian motion
VOLATILITY
AND
DIVIDEND
Y1EL.D
731
is unattractive because it incurs large shut(Armstrong and Galli, 1997). In these circumminimum acceptable capacity and then operate
1998).*
of the project’s output follows a geometric
dS = asSdt + o$dz,
(13)
and that average unit costs c change over time at a constant rate d,’
dc = Idcdt .
(14)
In the absence of operating flexibility, the current value of a planned, fixed-production project of life T, 0 < T < 00, can be written as the present value of revenues less the present value of costs,
F$= efso-dco,
(15)
where co is the current average cost of production, So is the current unit price of
the project output, and ef and of are positive discount factors. This is the value
of the underlying
asset in the investment option. The time zero value of an
operating project with remaining project life N(t) = T - t is similarly
$ = &)S,- &)c,,
where the parameters d(t) and d(t) vary with time. This is the value of the
underlying asset in the abandonment
option. (See the Mathematical Ap endix
for the derivation of Equations 15 and 16 and the specifications of ef 3 (t), of
and cd(t)).
A second production profile, of particular relevance to oil extraction, is that
of declining production,
where output, qt, declines exponentially
over time.”
As shown in the Appendix,
the values of these projects can similarly be
expressed as
# = edso-codco
(17)
vf =ed(t) od(t)co
(18)
and
for the cases of the investment
and abandonment
option,
respectively.
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Equations 15 through
18 give two valuation formulas for each type of
option. From Equation 12, however, it is clear that the price elasticity of project
value at time zero is either of the form
Ef= es,
- = (1,; >21
%
t
v
v,>o
(1%
or
ws,
A
Et =T
=(1+~)21
v
v,>o.
cw
All else equal, & = E! and oi = cr$. Also, a 1 1, and, from Equation 12, dV L
os. As unit costs approach zero the elasticity approaches 1.0, revealing that elasticities of greater than 1 reflect project operating leverage. Financial leverage
(that is, fixecl costs) would push the elasticity value even higher (Tufano 1998).
Note that d = 1 and dV = os, as is frequently assumed, u&r if there are no
fixed costs and c = 0. This is possibly the case for equity-financed
natural gas
projects, but unlikely for most other production processes.
Figure 1 shows how the elasticity values for a typical planned fixed-production gold mining project depend on the operating margin. The values are calculated using Equations 19 and A3. In this case, for plausible margins, the
elasticity values are at least 2, and the project volatility, a~, is therefore at least
twice the volatility of the gold price, OS, which has historically been about 0.20.
Thus, in modeling real options to develop gold mining projects, crv should be at
least 0.40, which is higher than typically assumed. Figure 2 plots elasticity values
for a typical planned declining-production
oil well project. The values are calculated using Equations 19 and A12. The results again indicate that for inflexible
oil projects elasticity values are at least 2, making a~, the volatility of the value
an oil well, considerably greater than OS, the volatility of the price of crude oil.
Since V is a function of t, Equations 19 and 20 also show that &and
therefore c&--is constant throughout the life of the option, as assumed in Equations 2
and 3, only when operating costs are zero. When operating costs are positive, d,
= d,(V, t), and the true process for V cannot be a geometric Brownian motion.
This is because Equations 2 and 13 are inconsistent when c > 0; if S follows a geometric Brownian motion, V will also follow a geometric Brownian motion only if V
= b.SB.’ ’
There are two approaches to handling the dynamic nature of c$(V, t). First,
numerical solution techniques can explicitly take this into account by using
(21)
ESTIMATING
VOLATILITY
AND DMDEND
YIELD
733
Oold Price (#oz.)
Figure 1.
Eqrctions
Initial elasticity values for a gol$ project, calculated using
(19) and (A3), given r = 0.07, Id = 0.028, T = 15 years, 6, = 0.02,
o = $25010~.
Oil Price
(Ubbl.)
Figure 2. Initial elasticity values for an oil *project, calculated using
Equations (19) and (Alp), given I = 0.07, d = 0, T = 30 years, 6, = 0.15, y =
0.10, and co = $lO/bbl.
and
?tAt
L t1
2
o$V,t)
=
l+
Wkoe
I/
2
=S
(22)
734
QUARTERLY
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OF ECONOMICS
AND FINANCE
for the variance of V t periods into the investment and abandonment
option,
respectively. The values of o and o(t) depend on the production profile, and are
given in the Appendix. p alternative taken by many analysts is to use an average expected value of oV (I’, t) over the life of the option, accepting the valuation bias that this introduces. This is not as unpalatable as it might seem; the
Black-&holes pricing model is known to value with bias due to nonstationarity
of@
(Jarrow and Rudd, 1982), and yet it continues to be a mainstay of Iinancial option pricing. In the above case of a real option on an inflexible project,
where the true variance is inversely related to project value, out-of-the-money
calls and in-the-money
puts will tend to be overvalued, while out-of-the-money
puts and in-the-money calls will tend to be undervalued (Hull, 1997).
In summary, projects with no operating flexibility and either constant or
declining production
profiles have oV 1 OS, with the extent of the difference
between (3~ and OS primarily dependent on the degree of operating leverage. In
general, ctv is not constant, but is dependent on the level of I/ and time and thus
varies through the life of the option.
B.
Estimating
Project Volatility
When There is Operating
Flexibility
The above models of project value assume an inflexible production
schedule. Many projects instead have production as a call option on output, with the
level of production
flexible between a lower bound of 0 and an upper bound of
plant capacity, K (McDonald and Siegel, 1985). These operating options lower
the value of aI’/& and raise the value of V for any given value of S, combining to
lower the value of d. l2 Hence, when there is operating flexibility embodied in
the project, the value of d will be lower than that given in Equations 19 and 20.
The reduction will be especially significant for marginal projects, since this is
where operating flexibility has the most impact on &‘/& and I/. The elasticities
will still be time-varying, however, since only for perpetual options is the elasticity value constant (Zein, 1998).
An example of the difference between the inflexible production and flexible
production elasticity estimates for a planned fiqed-life project is given in Figure
3. The flexible-production
elasticity values, Q, ranging from 1.7 to 2.7, are
taken from Brennan
and Schwartz’s calculations of project option value
(1985).13 The corresponding
fixed-production
elasticity values are calculated
using Equations 19 and A3 given the copper mine parameters used by Brennan
and Schwartz. As expected, inflexible and flexible production models give comparable elasticity estimates for profitable projects, where operating flexibility has
little impact on &‘/&S and I’, but that the two elasticity estimates diverge as the
project becomes marginal.
In em irical work Blose and Shieh (1995) and Tufano (1998) empirically
estimate f8 for gold firms and find similar results.14 Blose and Shieh find that
the elasticity value is statistically greater than 1, while Tufano actually measures
an average annual elasticity value of 1.96, that average varying through time.
ESTIMATING
VOLATILITY
AND
DMDEND
YIELD
735
Elasticities calculated using a constant-production
model similar to the one presented here match the empirically estimated elasticities for profitable firms, but
overestimate the estimated elasticities of firms with narrow margins. Tufano recommends that, for gold mines, fixed-production
elasticities match empirically
observed elasticities for elasticity values of up to about 3.
While Brennan and Schwartz assume a geometric Brownian motion for copper prices, subsequent research by Schwartz (1997) finds strong evidence that
copper prices are mean reverting. Schwartz estimates copper project values
under flrxibility using mean reverting copper prices, and produces results that
imply &o values ranging from 2.0 to 3.0, with an average of 2.4. Since this is
only marginally different from the flexible production results reported in Figure
3, the nature of the divergence between the flexible production
and inflexible
production
elasticity values demonstrated
in Figure 3 appears to be relatively
robust.
All of this indicates that Equations 19 and 20 provide an exact elasticity
estimate only for inflexible projects. For projects with operating flexibility, they
provide an upper bound on the elasticity value, although they do appear to
provide a reasonable elasticity estimate for flexible projects that are profitable.
The above calculations, along with the empirical work by Tufano and the simulations by Brennan and Schwartz, indicate that the price elasticity of mineral
projects, at least, is significantly greater than 1, and probably close to 2 or 3.
The volatility of the output good’s price, OS, therefore under-represents the volatility of the project’s value, ov. The implication for the previous real options
work that either explicitly or implicitly assumed d = 1.0 is that the estimated
project volatilities were too low, and that the options were as a result signifi-
\
\
\
-texibb
\
pnxluctiarl
\
\
----lxd
\
\
pmdwtlon
El
\
\
\
\
\
c.
.N
.S
---_
-----me___
04
0.3
I
0.4
0.5
0.6
Copper
Pria
0.7
0.8
0.9
1.0
(c&s/lb.)
Figure 3. Initial elasticity values for the Brennan and Schwaftz (1985) copper
project, inflexible and flexible production,
given r = 0.10, le’ = 0.08, T = 15
=
0.01,
and
co
=
$0.5O/lb.
ye-,
h
736
QUARTERLY
REVIEW OF ECONOMICS
AND FINANCE
cantly undervalued
(option value generally increases with volatility). This helps
to explain Davis’s (1996) finding that real option valuations of undeveloped
and developed mineral properties have fallen short of market values; the calculated option vapSes may have been underestimated
due to a significant downward bias in c+
III.
ESTIMATING
PROJECT
The solution
project’s
&, THE DIVIDEND
to partial
dividend
differential
yield &, = $,
be possible to estimate
YIELD
Equation
- d,.
dV, the required
OF THE UNDERLYING
3 also requires
If the project
an estimate of the
is a traded asset, it may
rate of return on the project, and c&,
the project’s rate of drift, directly from market data. Calculating the dividend
yield is more difficult when the underlying project is not openly traded, as is
typically the case, since even if the expected drift in project value is known, the
required rate of return is not. Teisberg (1995) suggests looking to a “twin”
asset for an estimate
project.
of tS$, or else estimating
In the absence of more information,
= 0 (e.g., Mann,
Goobie,
and MacMillan,
equal to Ss, the convenience
Others use an arbitrary
the cost of capital
for the
some options analysts assume &
1992; Trigeorgis,
1990), or set &
yield associated with the project’s
output
good.
value for S$, and test the sensitivity of the option calcu-
lation to the value used (e.g., Majd and Pindyck, 1987; Quigg, 1993).
As with project volatility, this paper formalizes the calculation of i$,, again
linking it to observable financial characteristics of the project. I first consider the
option to invest, and then the option to abandon. In both cases I continue to
model only projects with no operating flexibility, such as those with large startup and shut-down costs, and assume that the price of the project’s output follows
a geometric Brownian motion.
A.
The Option
to Invest
The Appendix
shows that at t periods into an investment
option the
project’s dividend yield, which in the case of an investment option is the opportunity cost of not investing, is
&(V,t)
= (6,-r)E:(v,t)+r+
dt
cqe
v
K
z
,
t
(23)
ESTIMATING
VOLATILITY
AND DMDEND
YIELD
737
dt
woe
where af (V, t) = 1 + 1 1 and the value of the constant o depends on
whether producti i n is fi!$d I r declining. Formulas for o and V, for fixed and
declining production projects are given in the Appendix.
Equation 23 is this paper’s second result. Intuitively, the higher the rate of
inflation
of operating
in the project,
project’s
the higher
output
costs, d, the higher
&it. Similarly,
the opportunity
cost of not investing
the higher S,, the slower the rate of growth in the
good, the slower the rate of growth in project value, and hence
S& . If 3~ = 0 and R’ = r, then Sit = 0, as there is no opportunity
cost to delaying investment.
If 3, = $ = 0, 6Lt < 0, and there is an incentive
not to invest and produce, since the value of the project is growing at greater
than the rate of discount (a disequilibrium
condition).
As with volatility, the dividend yield from the project is not constant, but a
function of the level of I/ and time. If partial differential Equation 3 is solved for
F using finite differences, Equation 23 can be applied directly for the dividend
yield. The alternative
is to assume a constant average value $& over the life of
the option, and accept the valuation bias that this incurs.
While Equation 23 may look cumbersome, the right-hand-side
parameters
can be easily estimated. Where the project’s output good is traded, either
directly or as a derivative security, the value of 6, can be estimated from market
data (Gibson and Schwartz, 1991; Hull, 1997).16 Where the project’s output
good is not traded, it is at least closer to the market than the project itself, and
if guesswork is going to be used, guessing about the convenience yield of a final
good is easier than guessing about the convenience yield of a project that is not
yet in operation. The values of R’ and 78 can be obtained from historic cost
series published in engineering magazines and from engineering estimates. For
example, the Marshall and Swift cost index, published in Chemical Engineering,
shows mining and milling costs rising by 2.8% annually.
In general,
Equation
however, where Sit
3;
23 reveals that S& f 6,. There are two special cases,
= 3,. The first is where c = 0, which makes E: = 1.0 and
= 3~. The second is where the ratio of the unit costs to the project
price
(c/S) is constant
project’s
6; = $.l’
operating
output
through
time,
good has no market
implying
d
output
= as. If, in addition,
the
risk, then kS = crs + 6, = r, and again
In all other cases, however, given a risky output
good and non-zero
costs, the value of Sit will not equal S,, as is sometimes assumed.18
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AND
FINANCE
Equation 23 can also be seen as a general version of a dividend yield calculation proposed by Majd and Pindyck (1987), who, for the case of an infinitelived and costless project that will operate at a constant rate K, suggest that 8:
can be estimated by dividing anticipated project cash flows by the current market value of the completed project. Equation Appendix A8 shows that the current market value of this completed project is
g = (1 --WSoK
0
In valuing
the completed
6s
(24)
.
project, the market takes into account the value of 8s.
If one were able to observe, or know, the current value V$ then, by rearranging
Equation
24, 8s =
(1 - tax)SoK
vfo
E: = 1.0, Equation
Now,
with
the
absence of costs making
.
23 gives 8’V = 8s. From this, it follows that
6~ = (1-tax)StK
vt
4
= (1-tax)K
0
(25)
.
For this type of project the dividend yield is constant and equal to the ratio of
project cash flow to project value.
This result has led some real options researchers to always look to anticipated
cash flows as a proportion
of market value as a way of estimating
8;. If the project
is similar to that owned by a listed company, that company’s earnings to price
ratio could provide this estimate (McDonald and Siegel, 1986). The problem, of
course, is that completed projects are seldom infinite-lived and/or costless. When
either of these assumptions
is removed, the relationship
payout ratios is lost, and Equation
between 8: and cash flow
23 must be used to calculate 3:.
In summary, for options to invest in projects with no operating
and whose output good price follows a geometric Brownian motion,
result is that the project dividend yield is the function
dt
6XOe
6’,(lQ)
=
(iqr)E;(V,t)+r+
n
I
,
v
t
flexibility
a general
Alternative
methods
of estimation
that set 6;
equal to the cash flow payout ratio, the price earnings ratios of similar listed
firms, or the convenience yield associated with the output good are unlikely to
produce a reasonable estimate of project dividend yield.
B.
The Option
to Abandon
Consider now the option to abandon or “put” an operating
of value V,, receiving
some salvage value in return.
plest case is a constant-output
flexibility.
lg If the output
project
goods
vy
that has an infinite
project
= (1-dax)
operation,
project
SA,, the sim-
life and no operating
Brownian
motion,
the project’s value at
= es,-6xt,
K
where 9 = (1 - tax)E
project
In estimating
price follows a geometric
and if unit costs rise at rate a? during
time t is (via Appendix Equation A9)
inflexible
and o = (1 -tcwe)-.
%
(r-7cA>
dividend yield is then
(26)
The Appendix
shows that the
nAtxA
6A,(cQ)
= (S,-r)e;(v,t)+l.+WCo;
(27)
,
t
where E;’ (V, t) =
This is the paper’s third result. If the rate of rise of costs during production,
a, is equal to the antonomous inflation of costs during the planning stage, d, as
in the case when there is no physical depreciation of the asset during operation,
then tiV (V, t) = 6: (V, t). If, on the other hand, c = 0, then 6$ = 6; = 6~ =
(1 - tax)S,K
vt
and again the dividend yield is equal to the (constant) ratio of period
’
cash flows to project value. In general, however, projects have positive costs and
physical deterioration
of operating
plant, making a > d and $(V,
t) > $(V,
t).
740
QUARTERLY
REVIEW
When the project
yield is
OF ECONOMICS
has a finite
AND
life, the Appendix
(1
&w
FINANCE
shows that the dividend
-tax)[St-ct]K
=
v
(33)
.
t
This, the paper’s fourth and final result, verifies a result presented intuitively by
Myers and Majd (1990); the dividend yield on a finite-life operating project is
simply the ratio of period cash flows to project value. This ratio increases, and
the dividend
infinite
yield rises, as the project ages.*’ Also, contrary
projects, projects with a fixed life cannot have t&v,
Figure 4 plots current
6‘&, calculated
to certain cases for
t) = &v,
t).
(N = 15) and expected (N = 14, 13, . . .. 1) values of
using Equations A7 and 28, for Brennan
and Schwartz’s (1985)
fixed-production,
fixed-life copper mine example. Here, if the copper price is
expected to rise at 9%, the dividend yield is expected to rise from an initial
value of 0.07 as the project
Note the downward
given these project
ages, averaging
bias incurred
parameters
0.22 over the life of the mine.*l
if one assumes that &$
and assuming a
0.001 for this project, which is considerably
= 3~ = 0.01. Also,
= &, Equation
23 gives Sio =
lower than tiVo. This illustrates
the
magnitude
of the difference between the initial dividend yield for the investment option and that for the abandonment
option on the same underlying
project.
IV. AN APPLICATION:
RESERVE
THE OPTION
TO DEVELOP
A METAL
To illustrate the use of these methods of estimating project volatility and dividend yield, consider the problem of valuing the option to develop a precious
metal reserve.** The owner of the option has 5 years in which to exercise the
option and develop a mine. Development,
if undertaken,
is instantaneous,
development costs cannot be recovered once spent, and there is no flexibility to
shut down or abandon the producing mine. Table 1 gives the valuation parameters associated with the reserve.
This is an in-the-money
American call option on a developed mine. I will
first value this using a standard binomial lattice, and thus need to assume that
changes in the value of the developed mine can be approximated
by a geometric
Brownian motion,
ESTIMATING
VOLATILITY
AND
DMDFND
YIELD
741
1
0.9
0.8
g 0.7
!z 0.8
B 0.5
f 0.4
% 0.3
0.2
0.1
0
-7
0.0
5.0
Remaining
15.0
10.0
Mine Life, N(f) (years)
4. Expected dividend yield, &, for Brennan and Schwartz’s copper
mine example, calculated using Equations (28) and (A7), and given t = 0.10,
18 = 0.08, 6, = 0.01, co = $0.5O/lb., SO = $l.OO/lb., true = 0.5, T = 15 years,
andK=
10,000,000 lbs./year.
Figure
with constant drift and dividend
to value this option:
yield. There are 6 parameters
r, crV
21 , 3;) T’, $,
that are required
and X. The values of T’ and X are given
in Table 1. The risk-free rate, r, can be determined from the yields on treasury
bonds, which I assume to be constant at 6%. Equation A2 gives the current value
of the mine,
$,b
ut to calculate this I need the (constant) convenience
the metal, 3,. Given that S follows a geometric
Brownian
motion,
yield on
the conve-
nience yield is found from futures market data using the standard equation,
6,
t
-[h@+t]
=
t
’
where Ft is the futures price for a contract on the metal maturing in t years
(Hull, 1997). The futures contracts that are currently traded indicate an average
3, of 2.75%. Now, using Equation A2 and the data in Table 1, #
lion, and the net present value of the project
million.
if developed
= $370 mil-
immediately
is $187
742
QUARTERLY
RF&VIEW
Table 1.
OF ECONOMICS
AND
Mine Valuation
Current metal price (SO)
Cm-rent cash cost (co)
Cost escalation rate (r$)
Development cost (X)
Production capacity (K)
Mine life (r)
Effective corporate tax rate (tax)
Option Life (T)
The
remaining
Parameters
fssoloz.
$28510~.
2.8%~~.
$183 million
361,610 oz./yr.
21 years
40%
5 years
a: and Sb, can be estimated using EquaEquation 12 in turn requires that an estimate of
two parameters,
tions 12 and 23 respectively.
E: . Using Equation
life of the option.
FINANCE
19 I calculate E; to be 3.56, but this will change through
Suppose that S is expected
the
to grow at 5% per year (crs =
0.05). By the expiration of the option ~15 is expected to be 2.83. Equation 12
also requires an estimate of 0s. Options on this metal’s futures contracts indicate
an implied volatility of 0.18. Now, using Equation 12, the average expected
-21
value of bV is 0.335 over the life of the option.
From Equation
23 and the current cost and price values, 6bo = 0.016. By
the end of the option,
given the expected rise in S and c, S& will equal 0.019.
Taking an average of these two values gives 6; = 0.0175.
I now have the six parameter values needed to calculate the value of this
option;
-21
r = 0.06, bv
= 0.335, 8; = 0.0175, T’ = 5, $
= $370 million,
and X
= $183 million. Standard options software, such as that provided with Hull
.(1997), permits simple numerical solution of this call option using a binomial
lattice technique. This gives a current option value of these reserves of $243 million, considerably more than their current net present value of $187 million.
The difference reflects the $56 million option premium created by the option to
defer development for up to 5 years.
As noted above, the assumption of a geometric Brownian motion for V is
inconsistent with S following a geometric Brownian motion, and the resultant
option value of $243 is a biased estimate. A second, consistent method of calculating the option
value allows 0’:
and 3: to vary throughout
the life of the
ESTIMATING
option. In this case, I use implicit
differential equation
VOLATILITY
finite
AND
difference
DMDEND
methods
YIELD
743
to solve the partial
= rF
given the standard
functional
boundary
conditions
for an American
forms for cry and $, are substituted
call option.
into Equation
When the
29, the equation
becomes
aF
aF
x +A(Qrv+
1
a2F
$3(t)= rF.
av2
(30)
where A(t) =
(r - 6,)~~ + (r - 6, - rrz,ofc,cfr,]
and B(t) = (1 + ~.&,~$‘/v,)~cr~~.
[
From this, the “true” value of the option is $252 million.
Table 2 presents a comparison of option values calculated using various estimates of volatility and dividend yield. The resultant option values vary significantly. In one case, using rule-of-thumb
parameter values, the project is seen to
have no option premium, while this paper’s approach gives an option premiu2y
of $56 million (by the approximation
method) and $65 million (allowing for cry
Table 2. The Value of the Option to Invest in a Fixed-Production
Precious
Metal Reserve $million, Czalculated Using Vq@ous Methods of Estimating the
Project Dividend Yield, 6,, and Variance, bv (methods given in brackets)
SIV
Variable
(Equation
43)
3.4%
W2:
2I
by
5.57%
2.75%
(5;
($7
= 6s)
= payout
ratio)
204
192
187
243
234
215
= 0;)
33.5%
(Average
of
Equation
21 over
tbe life of the
option)
Variable
(Equation
Notes:
1.75%
(Average
of
I%quation
43
over the life of
the option)
252
4 1)
I. Call option
value equal to the NW
value,
indicating
it is optimal
to develop
the reserves
immediately.
744
QUARTERLY
REVIEW
OF ECONOMICS
Table 3. Summary of Equations
Yield, Option to Invest
Life of
Project
Production
Profile
finite
finite
fixed
exponential
decline
infinite
infinite
fixed
exponential
Operating
Flexibility
Production
Profile
finite
infinite
fixed
fixed
FINANCE
Needed to Estimate Volatility
Value,
V
Elasticity,
aud Dividend
E’ Volatility,
0:
Dividen
P
Yield,
6,
no
644)
(19)
(12)
(23)
no
0113)
(19)
(12)
(23)
no
(‘48)
(19)
(12)
(23)
no
6417)
(19)
(12)
(23)
Table 4. Summary of Equations
Yield, Option to Abandon
Life of
Project
AND
Operating
Flexibility
no
no
Needed to Estimate Volatility
Value,
V
C47)
(AQ)
Elasticity,
(20)
(20)
aud Dividend
6’ Volatility,
(12)
(12)
G$
Dividen
Yield,
t?v
(28)
(27)
and 6; to vary through time). The development
of this project was in fact
delayed, indicating that in this case the paper’s volatility and dividend yield
equations produce an option value that is consistent with observed market
behavior.
v.
SUMMARY
This paper presents a set of equations that can be used to estimate the volatility
and dividend yield parameters necessary to value certain real options. Table 3 provides a summary of the equations needed when valuing the option to invest in a
project with no operating flexibility. Table 4 provides the same information for an
option to abandon an operating project. The equations can be refined for projects
with other operating characteristics, such as fixed costs or hedged production.
Volatility and dividend yield parameter estimation is an important
step in
real options analysis. The approach in this paper, while more rigorous than previous methods of estimation, still entails several approximations.
In particular, it
assumes that the price of the output good follows a geometric Brownian motion.
As with most advances in real option pricing, the paper’s results remain to be
empirically verified by testing the resultant option values and decision rules
against observed market behavior.
ESTIMATING
VOLATILITY
AND
DMDEND
YIELD
745
Finally, as a preliminary result, it appears that for this real option application
the pricing bias from assuming a geometric Brownian motion process for project
value, given a geometric Brownian motion for the project’s output price, is small.
Further research along the lines of Jarrow and Rudd (1982) is needed to investigate the pricing biases incurred across a spectrum of real options problems.
MATHEMATICAL
A.
Derivation
APPENDIX
of Equations
15 through
l&f4
In the option to invest, the current (t = 0) value of a fixed-output, finite-life
project is the present value of expected cash flows. If prices move according to a
geometric Brownian motion (Equation 13), the project’s current value can be
written as
where K is plant capacity, co is current average unit cost, So is current unit price,
a! is the constant rate of change of average cost that will occur during the
project’s operaion, r is the risk-free interest rate, hS is the risk-adjusted discount rate commensurate with the risk in S, 6, = (a, - as) is the constant convenience yield on the project output, tax is the effective corporate tax rate, and
T is the project life. Integrating Equation Al and simplifying,
-S,T
$
= (l-tax)
So&;
-coK* -e-(-“I
(r-7cAA> 1
S
[
,
(fw
which can be written as
$ = ofso-ofGo,
where e’ = (1 - tax)K
-6,T
l-e
643)
1 _ e-+&T
andof=
(1 -tux)K
. The value of
(r-7cA)
the project t periods into the option can be expressed as
%
If = dS,-wf~~e”‘~
.
(A4)
746
QUARTERLY
REVIEW
OF ECONOMICS
AND
FINANCE
In the option to invest the project life remains fixed through time, and ef
and of are constants. For the option to abandon, the project is operating, and
remaining project life is N(t) = T - t. Initial project value is
or
vf =d(o)so-d-(o)co
)
where d(O) = (1 - tux)K
l-e
646)
-6$‘(O)
1 _ e-(r - nAW(O)
and d(O)
= (1 - tux)K
%
CT--
Project value t periods into the option
?PrK1
_ e-P
- $)W)
(r--
A
)
1
t-47)
.
ecome the constant (1 - tux)x , (6s> 0 ), &and
6s
AsT+m,Bfandf$(t)b
K
become the constant (1 - tax) (r-?cA)
If
4
= (1 -tax)
= (1 -tax)
d(t)
, (r/r@) and
Stf
[
[
*
1
can be expressed as
- toe
= d(t)S, - J(t)coenAt
A
-coen”~];
S
St;
S
(r-7rAA>
-coeXAtL]
<r-7cAA>
648)
WV
The initial value of a project with an exponentially
declining production
profile, such as a developed oil field, is the present value of expected cash flows,
Vf
= (1 - tux);[S,e4”
0
= (1 - tux);poecr
0
- c,e-rrlq7dz
-‘ss)T - coenA7]qoe-yTe-‘Td~,
(A101
JSI’IMATING
VOLATILITY
AND DMDEND
where y is the constant exponential rate of decline of output,
period. Integrating
Equation A10 and simplifying,
YIELD
747
Q, in percent per
1_,-(6,+YV
(6 +y) -co40
(Al
1)
S
which can be written
as
qy = edSo-codco,
where ed = (1 - tux)qo * -(i,
(A153
46, + Y)T
1 -,-(r+y-RAP
and od = (1 - tux)qo
+ y)
A
b-+y-x
-
The
1
value of the project t periods into the option can be expressed as
vy
= eds,- OdcOe+
6413)
For an option to invest, the project life remains fixed through time, and ed
and od are constants. For the option to abandon, the project is already operating, and remaining project life is iV(t) = T - t. Initial project value becomes
1~e-(~+Y-hw)
- Co40
A
(r+-r-n
which can be written
1
(A141
’
1
as
vt” = Bd(0)S,
- od(0)co,
where
f@(O) =
(1
-
tax)qo
1 _ 4, + Y)N(O)
e
(6, + Y)
and
6415)
c&O)
=
(1
-
tax)
1 -,-(r+Y-hW1
Qo
(r+.y-R
expressed as
A
The value of the project t periods into the option
’
)
VI” = ed(t)st - cod(t)coe~“t,
where
et)
=
(1
1 _ e-(r + y - ff)N(t)
tNq0
(r+y-nA)
-
46, + w(t)
tax)qo 1 -&
+y)
can be
(Al@
and
cd(t)
=
(1
-
748
QUARTERLY
REVIEW
OF ECONOMICS
AND
FINANCE
As T -+ @J,sd and cd(t) become the constant (1 - &)&)
40
become the constant (1 - tax)
A , and
1
(r+y-n:
= (1 -tax)
vy
40
Derivation
of Equation
e
O
Qo
(r+Y-n
q”
(r+Y-n
s&j-co2’l
[
B.
x’t
s--c
Vs+Y>
= (l-tax)
, wd and c&t)
S
A
1
1
’
)
A
.
>
23
Assume that the price of the project’s
Brownian motion
output
good follows
dS = (hs - G,)Sdt + o,Sdz
a geometric
(Al91
and that average unit costs autonomously
inflate with certainty over time at a
constant rate dc = dcdt. From Equation A4 or Al3 the value of the project t
periods into the option is I’! = W, - cect, where 8 and o are constants defmed
above for either a fixed or declining production
profile. Using this and Ito’s
Lemma, the process for I/ is
dti
= (CISt(&s - 6,) - wc,x’)dt
+ osS,edz .
However, given Equation A19 and the fact that S and V are linearly related, this
can also be written as the more general diffusion process
dV
1
I
I
= avt(S, t)dt + mvt(S, t)dz
= a&( V, t)V,dt + c&( V, t)V,dz
= (&:,(V, t)-6:,(V,
(Ml)
t))V,dt + c&(C’, t>V,dz.
Setting the drift terms in Equations A20 and A21 equal,
es,@,
- 6,) - coCtkZ = (l&V,
t)-6:,(V,
t))V, .
W2)
E!STIMATING
VOLATILITY
AND DMDEND
YIELD
749
From multi-factor pricing of real assets (Hull and White, 1988) we know that
if the risk in V is spanned (which is assumed in the derivation of Equation 3),
and since the two assets S and V have the same Wiener process, an absence of
arbitrage requires
I
qrt(K
t) - r
I
qq(K
Rearranging
&s-T
=
6423)
%
t)
Equation A23 and substituting
,.I
a,(V,t)
Substituting
=-.
CL: = o’,/os,
I
(a -‘s -r&(V,t)+r
6424)
.
Equation A24 into A22 and rearranging,
&V,t)
= (&S-I)Ef(V,t)+r-
8St(&ts - 6,) - WtXZ
Vt
acOe
= (6, -&v,t)+r+
dt
x
I
vt
c425)
’
where q’(V, t) =
C.
Derivation
of Equation
The derivation
27
here is identical
to that of the derivation
of Svz above, only
with average unit costs inflating over time at a constant rate #. Given an infinite
life project, Vtf* = OS, - ext, where, from Equation A9, 0 and w are the constants
K
and (1 - tax)-
respectively.
From this the derivation
above
<r-n*)
yields
i/t$
&(v,t)
=
@,-~)E;(V,t)+~+
rnOe
v
9
t
where q*(V, t) =
0433)
750
QUARTERLY
D.
Derivation
RJRIEW
OF ECONOMICS
of Equation
AND
FINANCE
28
From Equation A7 above, the value of an operating constant-output
plant
with N(t) years of production
remaining
is VfA = e’ct)S, - t&t)+ From Ito’s
Lemma, and with &V(t)/& = -1,
de _ $(W,(%
- 6,)4t)cp*
dt+asStd(t)dt
+
s,(ad(t,/at)
-c,(aw’ct,/at)
-L
I
=
$(t)S,(b,
- 6,) - tJ(t)cg
- (1 - tax)S,Ke
+ (1 - tax)c,Ke
Following similar manipulations
dend yield is
-Q’(t)
-(r - d)N(t)
dtosStd(t)dz
+
I
as in the case of the option
.
6427)
to invest, the divi-
d-(t)ct7cA
vt I+
&w=((t+)ef(V,t)+r+
(1 - tax)StKe-4N(t)-
(1 - tax)ctKe-(T-‘“)N(t)
Vt
After considerable
algebraic manipulation
(16A,W)
=
ww
this simplifies to
- c,lK
tax)[St
v
t
.
WV
Acknowledgment:
I would like to thank Damien Balmet, Alexis Dodin, Imad Elhaj,
David Moore, George Pinches, participants
at the 1998 Midwest Finance Association
Conference
and participants
at the 2nd Annual Conference
on Real Options for helpful
comments on earlier drafts of the paper. David Moore also provided valuable assistance
with the numerical
calculations.
NOTES
*Direct all correspondence
ness, Colorado School of Mines,
to: Graham A. Davis, Division of Economics and BusiGolden, CO 80401-1887.
E-mail [email protected]>.
1. Examples are Dixit and Pindyck (1994) and Trigeorgis
(1996). The most recent
attempt at making real option pricing techniques transparent
to petroleum
industry decision makers is given in the Energy Journal, I9( l), 1998.
2. These are the volatility,
dividend
yield and current price of the underlying
asset, the risk-free rate, the time to maturity of the option, and the strike price or exercise price of the option.
3. Among others who make thii assumption
about the process for &‘j are Dixit
and Pindyck (1994), Majd and Pindyck (1987). Myers and Majd (1990), Paddock, Siegel,
and Smith (1988), Pickles and Smith (1993), Quigg (1993), and Trigeorgis
(1996). Teisberg (1994) has the expected rate of drift in V as a deterministic
function of the level of
V, dV’ = av(V)Vu2 + o,V&.
4. This is true even when the option value has a closed form solution, as with the
Black-Scholes equation for European options.
5. The value of the mine is also a function of extraction costs and interest rates,
but these are assumed certain in this analysis, making gold price and time the only state
variables.
6. For example, Schwartz (1997) finds strong evidence that traded mineral commodities have a constant rate of volatility.
7. This result is by no means new; Geske (1979) derived Equation
12 in a model
of compound
European financial options. Brennan and Schwartz (1985) relate ov to OS
using Equation
12, and McDonald
and Siegel (1985) and Dixit and Pindyck (1994),
among others, have incidentally
produced
this result in their option valuation
derivations. What seems to have gone unnoticed by many, however, is the concept of an elasticity term relating the two volatilities in real option applications.
8. Tufano (1998) funds that fixed-production
models provide reasonable intuition
about the economics of gold mines.
9. For the option to invest, 1$ includes pure cost inflation and any increasing costs
due to competitive
erosion, as with increased marketing
costs as entry into a brand-loyal
market is delayed (Trigeorgis,
1996, Chp. 9). In the option to abandon,
the rate of
change of average costs for a producing
project, a, incorporates
cost inflation
and any
physical depreciation
or obsolescence that causes costs to rise over time. In mineral
projects, cumulative production
effects are likely to make the latter significant.
Offsetting
this is any learning by doing effects (Dixit and Pindyck, 1994, pp: 205-207, 339-345).
10. A standard assumption
in oil production
modeling
is that production
declines
exponentially
with time as pressure in the well is depleted. This type of production
profile would also apply to a project whose capital “rusts” over time, causing output to
decline as the project ages.
11. The inconsistency
is similar to the problem
of valuing an option on a stock,
when the stock itself is an option on a levered firm (Geske, 1979).
12. This is a standard result of operating
flexibility
(Trigeorgis,
1996).
13. In column (6) of their Table 2, Brennan and Schwartz (1985) report initial values for ov based on simulation
results, From this, I back out ~0 using Equation 12.
14. The elasticity is an abandonment
elasticity since the projects owned by the firms
are already under production.
The non-gold
related assets of gold mining
Iirms will
cause the measured
elasticities
to somewhat underestimate
those of individual
gold
projects (Blose and Shieh, 1995).
15. For example, Paddock et al. (1988) use an elasticity value of around 0.75 to calculate option values of undeveloped
gulf oil reserves. These option values turn out to be
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only marginally
higher than the reported net present value estimates, and still 70% below
the observed transaction values.
16. Brennan (1991), Gibson and Schwartz (1991), and Schwartz (1997) show that 8,
is not constant for some mineral commodities.
However, using an average value of 8, calculated from market data provides a good first step towards calculating St!.
17. Oil reserves may be an example where 6 vz = 8~. Schwartz (1997) finds that the
market price of oil risk is zero. Paddock et al. (1988) and Pickles and Smith (1993) assert
that the ratio of oil prices to oil lifting costs is constant, implying
a! = as. The result,
from Equation 23, is that S$ = S, when valuing options on oil reserves.
18. Brennan (1990) has commented
that, with a convenience
yield on gold of zero,
a gold mine has no dividend yield, and thus gold mines should, according to option theory, never be developed prior to the expiration
of the option to develop. The above derivation shows that this implicitly
assumes that gold has no market risk, which is
reasonable,
and that operating
costs inflate at the expected drift of gold, which, being a
risk-free asset, is the risk-free rate. Mining costs have in fact inflated at 2.8% annually
over the past decade, well below the risk-free rate. Equation 23 thus allows that 3: > 0
even for gold projects, providing
a reason why it can be optimal to develop a gold mine
prior to expiry of the development
option.
19. Derivations
for a declining-output
project follow closely, and are omitted here
in the interest of brevity.
20. It is possible that with a declining
production
profile the payout rate would stay
constant, as assumed by Myers and Majd (1990).
21. Schwartz (1997) finds no market price of risk associated with copper, meaning
that the total return to holding copper, the sum of the dividend yield and the drift in
price, must equal the risk-free rate.
22. The numbers used in this example are taken from an actual mineral project.
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