Optimal Stopping Under Certainty and Uncertainty*
Graham A. Davis
Division of Economics and Business
Colorado School of Mines
1500 Illinois St.
Golden, CO 80401
gdavis@mines.edu
Robert D. Cairns
Department of Economics, McGill University
855 Sherbrooke St. W.
Montreal, Canada H3A 2T7
robert.cairns@mcgill.ca
February 17, 2006
Keywords: investment timing, stopping rules, investment under uncertainty, r-percent rule, real options,
postponement value, quasi-option value.
JEL Codes: C61, D92, E22, G12, G13, G31, Q00
Abstract: In investment problems under certainty it is optimal to stop the program such that net
present value is maximized. An equivalent, r-percent stopping rule suggests that the program
should be stopped when the project’s rate of appreciation falls to the force of interest. We extend
the r-percent stopping rule to the case of uncertainty, in which the program is again stopped once
the project’s rate of appreciation falls (in an expectations sense) to an adjusted force of interest.
This rule has all of the intuition of the rule under certainty, and the adjustment to the force of
interest reveals additional insights.
*
Cairns was supported by FCAR and SSHRCC. Davis acknowledges support from CIREQ. The authors would like
to thank Margaret Insley for helpful comments.
Optimal Stopping
At any time in any industry there may be several potential lumpy investment prospects, or more
generally several lumpy economic decisions, that are known but have not been implemented. Some
would have been profitable (would have produced a positive discounted cash flow in complete markets) if
implemented at the time, but are instead optimally held “on the shelf”.
Such lumpy, irreversible investment problems are broadly termed stopping problems. An early
analysis of binary choice stopping problems under certainty was Faustmann’s discussion of the optimal
timing of cutting trees. Another was Wicksell’s wine storage problem. Both appear in modern-day
mathematical economics textbooks (e.g., Chiang 1984, Hands 2004), and there have been extensions of
the tree cutting problem to considerations of uncertainty.1 Lumpy irreversible investments and the
associated stopping problem under certainty and uncertainty are also prevalent more generally, with
ongoing treatments in land development, non-renewable resource extraction, public works projects, and
equipment replacement.2
Traditional analyses suggest that lumpy projects be initiated immediately if their net present value
(NPV) is positive. The real options view of irreversible lumpy investment (e.g., Dixit and Pindyck 1994)
emphasizes that a positive NPV is necessary but not sufficient for optimal immediate investment; the
option of waiting to invest must also be evaluated, bringing intertemporal, equilibrium market forces into
play. Several authors have compared the timing decision under certainty and uncertainty (e.g., Malliaris
and Brock 1982, McDonald and Siegel 1986, Clarke and Reed 1989, 1990a, Dixit and Pindyck 1994, BarIlan and Strange 1995, Capozza and Li 2002,), with the analyses revealing little in common between the
calculus of and the intuition behind the variously derived stopping rules.
1
Malliaris and Brock (1982), Brock, Rothschild, and Stiglitz (1989), Clarke and Reed (1990a, 1990b), Reed and
Clarke (1990), Reed (1993), and Saphores (2003).
2
See, for example, Marglin (1963), Arnott and Lewis (1979), Brennan and Schwartz (1985), Hartwick, Kemp and
Long (1986), Capozza and Li (1994, 2002), Mauer and Ott (1995), Bjerksund and Ekern (1990), Cairns (2001), and
Holland (2003).
2
Optimal Stopping
In this paper we argue that much of the intuition of irreversible investment under uncertainty and the
premium associated with the option to wait can be directly compared with the case of equilibrium
investment under certainty. We begin with a review of investment timing under certainty, derive an
unconventional yet intuitive r-percent stopping rule for that case, and then derive a comparable rule for
investment timing under uncertainty. To allow for closed-form solutions we initially focus on projects
that can be delayed indefinitely, reflecting complete property rights over the investment timing decision.
To avoid “now or never” decisions we also assume that asset values are initially rising at a sufficient rate
to warrant investment delay, but that the rate of rise falls at some point such that investment is triggered in
finite time.
I. Stopping Problems under Certainty
The investment projects we consider involve an irreversible capital expenditure, a plan that specifies
outputs in future time periods, a shut-down time and possible other choices. Typically there is an
underlying “virtual” asset that becomes a “real” asset only upon development at some time t0. In contrast
to financial options, there is no existing supply of underlying real assets that trade hands upon the
option’s exercise; rather, exercising the option creates the underlying asset.3 There are also no cash flows
thrown off by the underlying asset while the agent waits to invest, and its rate of capital appreciation can
exhibit a “rate of return shortfall” (McDonald and Siegel 1984).
Let the present be time t = 0. The level of investment and the production plan depend on the time of
initial investment, t0, and the future equilibrium price path of outputs and inputs, among other things. The
firm need not be a price taker. For simplicity, we assume that investment is instantaneous and fixed in
scale with cost C. A profit-maximizing agent chooses an optimal time of investment tˆ0 such that the
project’s value, as of time t = 0, is maximized.
3
For example, lumber is created when a tree is harvested, adding new lumber supply to the market.
3
Optimal Stopping
Under conditions of certainty, there are at least four approaches to this optimal timing problem.
Analysis can be conducted in either the time or the value domain.
A. Method 1) Direct optimization with t0 as the choice variable (time domain)
Let Y(t0) be the forward NPV received by irreversibly creating an asset worth W(t0) at time t0  0 by
incurring an investment cost C: Y(t0) = W(t0) - C. Without loss of generality we assume for the moment
that C = 0. We also assume that W(t0) > 0 for at least some non-degenerate interval of time, and that W(t0)
is differentiable. The forward value of the project at time t  t0 is
 (t , t0 )  D(t , t0 )W (t0 ) ,
(1)
t0
where D(t, t0) = exp[   r ( s ) ds ] is a riskless discount factor integrated over the spot rates of interest.4
t
Traditional expressions of NPV assume t0 = t whenever W(t) > 0, at which point the forward value is
(t , t )  D(t , t )W (t )  W (t ) (Dixit and Pindyck 1994: 4-5, 145-47). But immediate investment upon W(t)
becoming positive is not necessarily optimal. Maximizing the forward (and current) value by choosing
the optimal time of investment tˆ0 , and assuming an interior solution t < tˆ0 < , we find from (1) that
t0 (t , t0 )  Dt0 (t , t0 )W (t0 )  D(t , t0 )W (t0 )  0 .
(2)
The solution to (2) yields tˆ0 and the relationship
Dt (t , tˆ0 )
W (tˆ0 )
 0
 r (tˆ0 ) .
W (tˆ0 )
D(t , tˆ0 )
(3)
Equation (3) states that at the optimal time of investment the NPV of the asset is rising at the spot rate of
interest. The second-order condition requires that
4
We can also write D(t, t0) = exp[r (t , t0 )(t0  t )] , where r (t , t0 ) defines a yield curve (Ross 2003).
4
Optimal Stopping
r (t0 ) 
W (t0 )
for t0  tˆ0
W (t0 )
(4)
r (t0 ) 
W (t0 )
for t0  tˆ0 .
W (t0 )
(5)
and
Equations (3) to (5) constitute an r% rule for investment timing under certainty: invest only when the
rate of rise of forward NPV of the project falls to the contemporaneous force of interest. If there are
multiple such points, that which provides the greatest value  should be chosen. This r% rule, while
previously being recognized as being satisfied at stopping points in tree-harvesting problems (e.g., Clarke
and Reed 1990a), has not generally been defined as an investment timing rule.5 Where it has been defined
as a timing rule (Clarke and Reed 1989, Reed and Clarke 1990) it has only been applied to tree
harvesting. Yet clearly, it is very general, holding for all types of well-defined projects in all types of
markets.6
The intuition of the rule is also impeccable. At the stopping point the benefit of waiting, the growth
in NPV, is exactly offset by the opportunity cost of waiting, the time value of money. Prior to the optimal
investment time the NPV is rising at greater than the rate of interest, and it is intuitive that waiting is
optimal.
Given the presumption that markets operate according to optimality conditions, the forward (time t)
market or equilibrium (option) value of the (optimally managed) investment opportunity is
tˆ0
 (t , tˆ0 )  D(t , tˆ0 )W (tˆ0 )  exp[   r ( s ) ds]W (tˆ0 ) .
(6)
t
5
Clarke and Reed (1990a), for example, mention only boundaries on value or age as stopping rules. The r% rule in
(3) is a boundary on the rate of asset appreciation.
6
By well-defined, we mean that if there is no investment expense the rate of growth of the asset must not be a
constant greater than the discount rate. If there is an investment expense, the rate of growth of the portfolio of the
asset and the investment liability must be such that the optimal investment time is finite.
5
Optimal Stopping
The difference (t , tˆ0 )  W (t ) is the option premium associated with waiting. To emphasize the
intertemporal nature of this option premium Mensink and Requate (2005) call it pure postponement value
(PPV). Prior to stopping PPV is positive, and at the stopping point PPV is zero.
Differentiation of equation (6) with respect to t yields a result that we will use later,
t (t , tˆ0 )  r (t )(t , tˆ0 ) ;
(7)
at all times prior to investment the market value of the investment opportunity is rising at the spot rate of
interest.
A simple example modeled on Wicksell's insights can make our analysis more concrete. The
example illustrates that the r% rule is applicable to any lumpy economic decision – disinvestments as well
as investment, consumption providing utility as well as investment providing monetary gain.
Example 1. Suppose that a connoisseur has a bottle of wine that can provide one util if served
immediately or can be stored costlessly and served after t0 periods to yield exp( t0 ) utils. Let the
instantaneous utility-discount rate be r, a constant, and let t = 0. If the wine is stored for t0 periods, the
present value of the wine is, in utility terms,
 (0, t0 )  D(0, t0 )W (t0 ) = exp(rt0 ) exp( t0 ) .
(1)
Equation (2) above implies that tˆ0 = 1/(4r2): the connoisseur will wait 1/(4r2) periods before serving the
wine, even though serving it now will provide a positive benefit of one util. We confirm equation (3) by
observing that
0.5
W (tˆ0 ) 0.5(tˆ0 ) exp( tˆ0 )

 0.5(2r )  r .
W (tˆ0 )
exp( tˆ0 )
If r = 0.20, for example, the wine is served at tˆ0 = 1/(4r2) = 6.25. At that time it is worth W( tˆ0 ) =
exp( tˆ0 ) = 12.1825 utils and has a present value of exp(rtˆ0 )exp( tˆ0 ) = 3.49 utils. The option
6
Optimal Stopping
premium or PPV associated with the option to store the wine at time t = 0 is 3.49 – 1 = 2.49 utils, a full
71% of the wine’s value. There is a significant option premium even under certainty.
Figure 1 plots equation (3) for a range of investment times for Example 1 when r = 0.20. In
accordance with the r% stopping rule the program is optimally stopped when the rate of appreciation of
forward NPV first crosses the hitting boundary r from above.
B. Method 2) Direct optimization with W as the choice variable (value domain)
In a related derivation the domain of analysis is value W rather than time. Let W = f(t0) and Wˆ  f (tˆ0 ) ,
where Ŵ  W in an interior solution. We assume that f(t0) is monotone in the relevant interval and so has
an inverse. Then,
t0  f 1 (W )  g (W ),
and
dW  f (t0 )dt0  f   g (W )  dt0  b(W )dt0  0 .
(8)
Equation (8) expresses the rate of capital gain on the underlying asset. Let V (W Wˆ ) be the market
(option) value of the investment opportunity, with the change in notation to V due to the change in
domain to value rather than time. A no-arbitrage or equilibrium condition expressing investors’
willingness to hold the investment opportunity over a small period dt0, given that it has some positive
value, is the first-order linear differential equation
R W V (W Wˆ )dt0  dV (W Wˆ )  V (W Wˆ )dW  V (W Wˆ )b(W )dt0 ,
(9)
where R is now the possibly value-dependent rate of interest. Equation (9) states that the opportunity cost
of holding an investment opportunity that throws off no cash flows, R(W )V (W Wˆ )dt0 , must be offset by
the opportunity’s capital gains dV (W Wˆ ) . Solving (9) for an initial value Ws yields
7
Optimal Stopping
 W R( y ) 
V (W Wˆ )  A exp 
dy  .
 Ws b( y) 
(10)
The constant A is determined by a boundary condition, in this case the value-matching condition


 Wˆ R( y)  ˆ
V Wˆ Wˆ  A exp 
dy   W ,
 Ws b( y ) 
(11)
 Wˆ R( y ) 
A  Wˆ exp  
dy  .
 Ws b( y ) 
(12)
from which
The market value given forward asset value W is obtained by substituting (12) into (10):
 Wˆ R( y ) 
V (W Wˆ )  Wˆ exp  
dy    (W Wˆ )Wˆ  Wˆ
W b( y )


(13)
with equality at W  Wˆ since  (Wˆ Wˆ )  1 .7 Equation (13) is analogous to equation (6), for a different
domain.
From (13) follows Dixit, Pindyck, and Sødal’s (1999) first-order optimality condition,8 derived for
stopping under uncertainty,
VW (W Wˆ )  W (W Wˆ )Wˆ   (W Wˆ )  0 .
(14)
Condition (14) is solved for Ŵ as an asset value stopping boundary.
Example 2: For the wine problem with a constant interest rate R = r,
W = f(t0) = e
t0
;
(15)
g (W )  t0  f 1 (W )  (ln W )2 ;
(16)
dt0  g (W )dW ;
(17)
7
The discount factor is now written  because it is a function of W rather than of time.
8
This is Dixit et al.’s equation (3) for the case of W (our notation)  V – C (their notation).
8
Optimal Stopping
dW 
dt0
dt0
Wdt0


.
g (W ) (2ln W ) / W 2ln W
(8)
Equation (12) becomes,
 Wˆ r

 Wˆ 2r ln y 
V (W Wˆ )  Wˆ exp  
dy   Wˆ exp  
dy 
y
 W b( y ) 
 W

ˆ
 We
 r (ln Wˆ )2 (ln W )2 


(13)
.
The optimality condition (13) then yields
e
 r (ln Wˆ )2 (ln W )2 


ˆ
 2r ln We
 r (ln Wˆ )2 (ln W ) 2 


0,
(14)
from which Ŵ = exp[1/(2r)] = 12.1825 when r = 0.20, as in Method 1.9 In Method 2 the agent waits for
W to rise to 12.1825 before investing, while in Method 1 the agent waits till the instantaneous rate of
appreciation of the underlying asset slows to 20%. Figure 2 plots W against time and shows the optimal
stopping point at W = 12.1825. The option premium (PPV) derived from waiting to serve the bottle of
wine is the vertical distance between the two curves. The wine is consumed when the PPV goes to zero.
C. Method 3) Indirect optimization using value matching and smooth pasting conditions (time domain).
The real options approach to stopping problems determines the stopping point using value matching and
smooth pasting conditions expressed in the value domain, W. Under certainty, comparable value
matching and smooth pasting conditions can be expressed in the time domain.
Consider a candidate interior time t 0* > 0. The market value of the asset at time t  t 0* is defined in
equation (1). The value matching condition is
(t0* , t0* )  D(t0* , t0* )W (t0* )  W (t0* ) .
9
(18)
The second order condition is that VWW (W | Wˆ )  0 ; it holds strictly in this example.
9
Optimal Stopping
As with analyses under uncertainty, the value matching condition is not sufficient for finding the optimal
stopping time, for it yields an infinite number of solutions for t 0* , only one of which is optimal. An
additional, smooth pasting condition is typically specified: the derivatives of both sides of the value
matching condition are equated. In this case, the smooth pasting condition is
 t (t , t0* )
t t0*
 W (t0* ) .
(19)
Given (18) and (19), one can solve for the unique optimal stopping time t0 .
Dividing the smooth pasting condition in equation (19) by the value matching condition in equation
(18) yields that
 t (t , t0* )
t t0*
 (t0* , t0* )

W (t0* )
.
W (t0* )
(20)
By equation (7), the left-hand side of (20) equals r(t), generating the r% stopping rule, equation (3).
Smooth pasting conditions are often specified in the real options literature as subtle and technical (Dixit
and Pindyck, 1994, p. 109), and there are ongoing efforts to explain them in more intuitive terms (e.g.,
Sødal 1998). In the case of certainty, however, they correspond naturally to the r% rule presented in
Method 1 above.
Example 3: The value of the wine when served is (tˆ0 , tˆ0 ) = W( tˆ0 ) = exp( tˆ0 ) . At any time t < tˆ0 the
forward value of the wine (in utility terms) is
 (t , tˆ0 ) = exp[–r( tˆ0 - t)] W (tˆ0 )  exp[r (tˆ0  t )]exp( tˆ0 ) .
(6)
The program is stopped when  pastes smoothly to W. The value matching and smooth pasting
conditions are depicted in Figure 2 for r = 0.20.
Method 3 is a direct link between Method 1, which has the agent waiting until the rate of increase in
forward value W is equal to the interest rate r (the time domain), and Method 2, which has the agent
waiting till W rises to 12.1825 (the value domain). Here, the agent satisfies both conditions incidentally
10
Optimal Stopping
and simultaneously by waiting (6.25 periods) until market value  and NPV W are equal and rising at the
interest rate r.
D. Method 4)Indirect optimization using value matching and smooth pasting conditions (value domain).
Our last stopping calculation involves value matching and smooth pasting conditions when asset value is
the domain. The solution determines the free hitting boundary Ŵ . The process for W is as in equation
(8). The value matching condition is


V W Wˆ  Wˆ
(21)
and the smooth pasting condition is


VW W Wˆ  1 .
(22)
These two conditions are sufficient to solve for Ŵ .
Example 4: In the wine example,



ˆ
ˆ  r (lnW )
V W Wˆ  We
2
(ln W )2 

.
The value matching condition holds trivially. The smooth pasting condition gives

ˆ
ˆ  r (lnW )
2r ln We
2
(ln W ) 2 

 1.
(22)
W Wˆ
As in Method 2, the solution yields Ŵ = 12.1825 given r = 0.20.
Figure 3 shows the wine stopping problem in this characteristic real options formulation, with option
value plotted as a function of underlying asset value (e.g. Dixit and Pindyck 1994, p. 139). The wine
should be consumed once the market value is equal to and rising at the same rate as the forward NPV, at
W = 12.1825 when r = 0.20. As in Figure 2, the option premium (PPV) derived from waiting to serve the
bottle of wine is the vertical distance between the two curves. The wine is consumed when the PPV goes
to zero.
11
Optimal Stopping
E. Discussion
Stopping problems under certainty can be solved using any of the four closely related techniques. While
Method 1 is the most common in optimal timing problems under certainty, Methods 2 and 4, which are
used in real options problems, also provide the optimality conditions. Method 3 is an intuitive addition to
these stopping algorithms, illustrating that value matching and smooth pasting optimality conditions are
embodied within the r% stopping rule. This will prove useful in our derivations of a similar rule under
uncertainty.
Seeing stopping under certainty as an r% rule renders investment analysis dynamic (invest when rate
of change of NPV falls to the rate of discount) rather than static (invest if NPV > 0). One should not stop
the program, even if its NPV is positive, when its value is rising by more than the rate of discount. Given
the option to wait, even projects whose value is negative if implemented now can have a positive market
value due to an option premium called pure postponement value. This option premium is often solely
attributed to uncertainty for reasons that we discuss below. Here we see that the premium also arises for
irreversible investments under certainty.
II. Stopping Problems under Uncertainty
In this section we derive an r% stopping rule under uncertainty for the simplest case, a general,
autonomous Itô process. The rule is remarkably similar to the rule derived under certainty. Then we use
the rule to solve for stopping points for two common stochastic processes for the underlying asset’s
forward value, Brownian motion and geometric Brownian motion. Once again we focus our analysis on
situations that yield an interior stopping point. In the following section we generalize our results to
include non-autonomous and finite horizon problems, and processes with jumps.
Under uncertainty, forward asset prices are described by a density function of which the moments are
assumed to be known. To facilitate closed-form solutions in our examples we represent the asset birth
value as the one-dimensional autonomous diffusion process in stochastic differentiable equation form
12
Optimal Stopping
dW  b(W (t0 ))dt0   (W (t0 ))dz
(23)
over any short period of time dt0, where dz is a Wiener process. As above, W(t0) is the forward NPV of
the asset if brought to life or developed at time t0 for a known and constant stopping cost C  0. The
typical optimal stopping rule given uncertainty is to invest at the time when the forward NPV reaches
some trigger or boundary value Ŵ for the first time:

tˆ0  inf t0 W (t0 )  Wˆ

(24)
(Brock et al. 1989, Dixit and Pindyck 1994).
A main difference between decisions under certainty and uncertainty is that under certainty the
investment decision can be made at time 0, even where investment timing is optimally postponed. In an
interior solution under uncertainty the investment decision is continually deferred until certain stopping
conditions are satisfied (e.g., equation (24)), at which point the investment decision and the timing of the
investment are concurrent. In other words, a stopping rule under uncertainty is both an investment
decision and a timing rule. Our r% rule in Section I was such a rule, and we seek to mimic that here.
Let Y (W (t ); C ) be the forward NPV at time t, with the simplest case being Y = W(t) – C. Let  > 0 be
the risk-adjusted discount rate that incorporates the market risk of the project.10 At time t  t0 , the
project’s forward market (option) value is
tˆ0


V (W (t ) W (tˆ0 ))  E exp[   ds] Y W (tˆ0 )  .


t
tˆ0


With E exp[   ds]   W (t ) W (tˆ0 )  (Dixit et al. 1999), we can write


t
V (W (t ) W (tˆ0 ))   W (t ) W (tˆ0 ) Y W (tˆ0 )  .
10
(25)
As is typical in dynamic programming problems we hold the discount rate constant (Insley and Wirjanto, 2005)
13
Optimal Stopping
To reduce notational clutter we condense V (W (t ) W (tˆ0 )) to V (W ) , V (W (tˆ0 ) W (tˆ0 )) to V (Wˆ ) ,
VW (W (t0 ) W (tˆ0 )) to V (W ) , VW (W (tˆ0 ) W (tˆ0 )) to V (Wˆ ) , and Y (W (t ); C ) to Y (W ) . We assume that Y
and V are twice differentiable.
From Ito’s Lemma the expected rate of capital gain on the option value is
2
E[dV ] b(W )V (W )  12  (W )V (W )
.

V (W )dt
V (W )
(26)
and the expected rate of gain in forward NPV is
2
E[dY ] b(W )Y (W )  12  Y (W )
.

Y (W )dt
Y (W )
(27)
At an interior free boundary Ŵ > W the value matching and smooth pasting and conditions are that
V( Ŵ ) = Y( Ŵ )
(28)
V’( Ŵ ) = Y’( Ŵ ).
(29)
and
In equilibrium,
E[dV (W )]
.
V (W )dt0
Combining this with conditions (26) to (29) yields that at the stopping point there is an adjusted r% rule,


1 2 ˆ
 (W ) V (Wˆ )  Y (Wˆ )
E[dY ]
2
 *   
.
Y (Wˆ )dt0
Y (Wˆ )
(30)
Rule (30) is analogous to the r% rule under certainty, equation (3). The left hand side of (30) is the
expected rate of rise of the project’s NPV. The right hand side of (30) is the opportunity cost of realizing
this gain. This opportunity cost is an “adjusted” force of interest * comprised of the constant riskadjusted discount rate , representing the opportunity cost of money, less a value/time-varying positive
14
Optimal Stopping
term.11 The program is stopped—the decision is made to immediately invest—when the expected rate of
rise of NPV over the next small time dt0 falls to this adjusted force of interest.12 Equation (3), the
stopping rule under certainty, is the limit of equation (30) as 2 goes to zero. In both the certainty and
uncertainty cases the smooth pasting optimality condition is embedded within the r% rule; they do not
play a separate role in the stopping algorithm.
The first term on the RHS of (30) is the cost of waiting. Given the intuition behind the r% rule under
certainty, why not stop the program once the expected rate of rise of Y falls to the discount rate ? This
unadjusted stopping rule,
E[dY ]
,
Y (Wˆ )dt0
(31)
has been called a myopic-look-ahead stopping rule (Clarke and Reed 1989, 1990a), an infinitesimal lookahead stopping rule (Ross 1970), and a certainty-equivalent stopping rule (Clarke and Reed 1990b). The
rule is correct under uncertainty if open-loop decision making is optimal. Open-loop decision making is
optimal when the stochastic process is monotone (Malliaris and Brock 1982, Brock et. al. 1989,
Boyarchenko 2004), or more generally when, once a stopping point is reached, the process cannot deviate
back into the continuation region (Ross 1970: 188-90). In these cases waiting indeed gains the discounted
expectation over possible values of Y next period, since the program will be stopped and Y will be gained
next period no matter what the outcome of the stochastic deviation. Trees with age-dependent growth are
an example where open-loop decisions are optimal (Reed and Clarke 1990),13 as is any stopping problem
11
The term is positive as a second order condition to the optimality of Ŵ : prior to stopping V > Y and V < Y,
which implies that V - Y > 0 at Ŵ .
12
The second-order conditions require that prior to stopping the expected rate of rise of NPV be greater than the
adjusted force of interest.
13
This is not to say that V is zero in these problems, only that the term is not relevant when deciding when to stop
the program.
15
Optimal Stopping
under certainty. In these cases, only pure postponement value need be taken into account in the stopping
algorithm.
In the case at hand, and for diffusion processes in general (Brock et al. 1989), there is a positive
probability that the process can drift back into the continuation region from Ŵ , at which point the
process is not stopped. The decision alternatives move from the open-loop choice of investing now or
committing now to invest next period to the closed-loop choice of investing now or conditionally
investing next period, the later clearly being superior to a commitment strategy when the stochastic
process can move disadvantageously. Previous analyses of stopping rules under these diffusion processes
(including jump processes) have specified only that
E[dY ]
  for stopping to be optimal in finite time
Y (Wˆ )dt0
(e.g., Brock et al. 1989, Mordecki 2002). The RHS of (30) is the exact rate to which the expected drift in
NPV must drop given (23). It is the intertemporal opportunity cost of delay, , less a credit for the pure
informational “quasi-option value” created through delay.
We now explain the functional form of the second-order adjustment term. In open-loop decision
making the global properties of V, as exhibited by V in equation (30), are irrelevant, while in closed-loop
decision making the program continues in the case of a downward movement in W (Reed and Clarke
1990). The quasi-option value of deferring the decision one more period is a function of the deviation of
V( Ŵ -h) from Y( Ŵ -h) in the event of a downward movement in W of magnitude h (see Figure 4),
where h is a function of 2. The larger V( Ŵ -h) - Y( Ŵ -h), the greater the incentive to wait for more
information prior to investing. In equation (30) this motivation to wait is engineered by the magnitude of
the V - Y term, which to second order is proportional to the deviation of V( Ŵ -h) from Y( Ŵ -h).
While equation (30) is not identical to the stopping rule under certainty, the “adjustment” to the
intertemporal portion of the r% rule to take into account quasi-option value (the value of information) is
not unprecedented. Clarke and Reed (1990b) discuss quasi-option adjustment factors applied to a
16
Optimal Stopping
boundary on value, Ŵ , calculated from stopping rule (31). Our approach applies an adjustment factor to
the (temporal) opportunity cost of waiting, which preserves the nature of the stopping rule as an r% rule.
The discussion thus far supports the following proposition, which holds for all values of 2  0.
Proposition 1:
At the optimal stopping point the expected rate of appreciation of the project NPV is equal to an
adjusted force of interest given in equation (30). That force of interest reflects both the
intertemporal costs and informational gains from deferring the decision to invest. Prior to the
stopping point project value is expected to rise at a rate greater than the adjusted force of interest.
This proposition implies the following:
Proposition 2:
In conjunction with knowledge of the properties of the investment itself, the (path of the) adjusted
force of interest is a sufficient datum for optimal stopping of the investment program.
We illustrate these propositions in the following examples.
Example 5: Arithmetic Brownian Motion
Let the forward value of the underlying asset, W (t0 ) , follow the Brownian motion process
dW  bdt0   dz ,
(32)
for constant positive values of b and σ. Such a process for birth value can be due to stochastic prices,
operating costs, closure costs, or any combination of these.
Typically, (32) represents the diffusion process for forward value only on an open set called the
continuation region. For investment or development timing problems, which are the focus of this paper,
17
Optimal Stopping
the upper bound to this continuation region is free, while at a lower bound Q there is some condition such
as V(Q) = 0 (Brock et al. 1989). To keep matters tractable we assume that Q = -.
Also, without loss of generality let there be no convenience yield or maintenance cost associated with
waiting to develop the asset, and let development be costless (C = 0) and Y = W. As before, let tˆ0 be the
optimal time to invest, though under uncertainty tˆ0 is not a deterministic parameter that can be decided
upon at time 0. Also let  be the constant risk-adjusted discount rate used to discount the investment
payoff W (tˆ0 )  Ŵ .
The market value of the opportunity or option to invest can be found by a combination of methods 2
and 4 above (e.g., Brock et al. 1989): the option value V (W ) satisfies the linear second-order linear
differential equation,
V (W )  bV (W )  12  2V (W )  0 ,
(33)
from which option value can be obtained given the appropriate boundary conditions. The solution to (31)
is of the form
V (W )  A1eW  A2e W
(34)
where  > 0 and  < 0 are roots of the characteristic equation
1
2
 2 2  b    0 .
(35)
The boundary conditions are then used to solve for Ŵ and the current value of the option. In this case
there are three boundary conditions. The first is that, in the absence of any holding costs and coerced
stopping at a finite lower bound Q, lim V (W )  0 . From this, A2 = 0. If the program is voluntarily
W 
ˆ  Wˆ and
stopped at Ŵ , we must have the value matching condition V (Wˆ )  Wˆ so that A1  We
ˆ   (Wˆ W ) .
V (W )  We
(36)
As was the case under certainty, V (W )  W  0 is the premium associated with the option to wait to
develop the project when W < Ŵ . The smooth pasting condition
18
Optimal Stopping
ˆ   (Wˆ Wˆ )  Wˆ  1
V (Wˆ )  We
produces the hitting boundary Ŵ   1 , which is the signal as to when to stop the program.
We now perform the stopping calculation using the r% rule, equation (30). The left hand side of (30)
equals
b
. Given equation (36) and Y   0 the second term on the right-hand side of (30) is
W
1
2
 2 2 ; the
program is stopped when
b
   12  2 2   b ,
W
(30)
the second equality following from (35). Therefore, Ŵ   1 > 0 as before. From (30) the value of the
option is:
ˆ
ˆ   (W W ) ,
V (W )  We
W  0;
ˆ   (Wˆ W ) ,
V (W )  We
W > 0 and
b
>   12  2 2 ;
W
V (W )  W ,
W > 0 and
b
   12  2 2 .
W
The r% rule stopping calculation has not made explicit use of a smooth pasting condition. The latter
is embodied within the r% rule, under which the asset owner measures project NPV in each period and
invests in the project once its expected rate of rise falls to  *    12  2 2 . Since all parameters of the
model are assumed to be known, the adjusted force of interest  * is a datum available to the investor and
is all that is needed to assess the stopping point.14 The investor need not understand or use smooth
pasting conditions nor solve for the value of the option in order to know when to invest; the investor
simply stops the program when the net opportunity cost of waiting exceeds the capital gains from waiting.
14
The discount rate is   r   (  r ) , where r is the risk-free discount rate and  is the required rate of return on
an asset with rate of return deviations dz (McDonald and Siegel 1986).
19
Optimal Stopping
The stopping point is now expressed in terms of the expected rate of increase in forward value as opposed
to a boundary on value.
Example 6: Geometric Brownian Motion
Consider an asset whose value follows a geometric Brownian motion with constant rate of drift b,
represented in differential form over a small time step dt0 as
dW  bWdt0   Wdz .
(37)
Let the required rate of return be represented by u and be constant and greater than b. Also let the riskfree rate be represented by r. With b constant the investment cost C must be positive to avoid bang-bang
now/never stopping solutions. Let the forward NPV be
Y W C .
(38)

Dixit et al. (1999) use method 2 to solve for the stopping point Wˆ 
C , where  >1 is the positive
 1
root of the fundamental quadratic equation
1
2
 2  (   1)  b    0
(39)
and   r   (u  r )  u (McDonald and Siegel 1986).
Here, we illustrate the r% stopping rule. From (37) and (38) the left-hand-side of (30) equals
bW
.
W C
Given that the option value is of the form (Dixit et al. 1999)

W 
V (W )   (W )Yˆ    Yˆ ,
 Wˆ 
(40)
the second term on the right-hand side of (30) is
1
2
 2  (   1) ,
where the variance in (30) is  2W 2 . At the stopping point Ŵ , therefore,
bW
   12  2  (   1)  b 
W C
(30)
20
Optimal Stopping
(with the second equality following from (39)). Given (30)
  12  2  (   1)  b 
bW
.
W C

Solving yields the stopping point, Wˆ 
C , as before.
 1
Using the r% rule stopping point the option value is

W  ˆ
 ˆ  Y,
W 
WC

W  ˆ
 ˆ  Y,
W 
W > C and
Y,
W > C and
bW
   12  2  (   1)
W C
bW
   12  2  (   1) .
W C
Once again, the investor need know nothing of smooth pasting conditions or the value of the option in
order to optimally time investment, though they must know the functional form for option value, equation
(40). The stopping point is fully calculable from (37), (38), and (40), given the parameters of the
problem. Figure 5 depicts this stopping problem for a series of specific parameter values. In this case
Wˆ  2 and Yˆ  Wˆ  C  1 . Note the similarity between the stopping rule in Figure 5, in the value domain,
and the same rule depicted under certainty in Figure 1, in the time domain.
III. More General Stochastic Processes
If W follows a combined geometric Brownian motion with an independent downward Poisson jump
of known percentage  and arrival rate  (Dixit and Pindyck 1994, pp. 167-173), the r% stopping rule
(equation 30) will hold as
(b   )W
   12  2  (   1) ,
W C
(30)
where  is now the positive solution satisfying
1
2
 2  (   1)  b  (    )   (1   )   0 .
21
Optimal Stopping
More general stochastic processes allow time to enter as a state variable, either via finite investment
horizons or through the stopping cost C(t) or a time-dependent discount rate (t), or via the addition of
time to the drift or variance terms in the continuous portion of the stochastic process for the underlying
variable,
dW  b(W (t0 ), t0 )dt0   (W (t0 ), t0 )dz .
(41)
In these cases, and using the same derivation as previously, the r% rule for an interior stopping point
0  tˆ0  T   is




1
Vt (Wˆ , tˆ0 )  Yt (Wˆ , tˆ0 )
 2 (Wˆ , tˆ0 ) VWW (Wˆ , tˆ0 )  YWW (Wˆ , tˆ0 )
2
E[dY ]
  * *   (tˆ0 ) 

.
Y (Wˆ , t )dt0
Y (Wˆ , tˆ0 )
Y (Wˆ , tˆ0 )
(42)
The partial derivative Vt is typically taken to be the depreciation of the option (Vt < 0) (Dixit and Pindyck
1994: 205-207), and Yt is any impact of inflating (Yt < 0) or deflating (Yt > 0) exercise costs. With time a
state variable, and for other processes such as mean reversion, there is no closed-form solution for the
option, and so the stopping rule in (42) is of more use intuitively than as an alternative to traditional
stopping calculations since the first and second partial derivatives cannot be calculated except under
simulation once the value surface has been solved. That intuition shows that at any interior stopping
point, even for finite-lived options, the asset’s expected rate of rise falls to the intertemporal opportunity
cost of waiting net of the value of information from further delay.
IV. Discussion
Under certainty the r% rule has the investor wait until the rate of rise of the NPV falls to a force of
interest before exercising an irreversible investment option. Optimal timing under uncertainty is
described as being “much more complicated” (Brealey and Myers 2003, p. 138), and indeed there
previously has appeared to be no solution other than by stochastic dynamic programming (Clarke and
Reed 1990b). We have shown that the same intuitive r% algorithm holds in the case of uncertainty; the
r% rule under uncertainty has the investor waiting until the expected rate of rise of the NPV falls to an
22
Optimal Stopping
adjusted force of interest. The adjustment adapts the intertemporal penalty of waiting to include a credit
for the quasi-option value of the information gained by delay. The rule is very general, and holds for
other stopping problems as well, such as optimal asset abandonment. In that case the asset manager waits
until the rate of rise of the intrinsic value of a shut-down falls to the adjusted force of interest.15
The intuitive attractiveness of the r% rule under certainty (equation (3)) has led practitioners in
mining and forestry to compare expected changes in asset value with the opportunity cost of capital when
deciding when to start a mine or harvest a stand of trees, as in equation (31) (e.g., Torries 1998: 44, 75).16
Recognizing this, academics have argued that the unadjusted r% rule (31) is useful, though inefficient, as
an approximate stopping rule under uncertainty (Brock et al. 1989, Clarke and Reed 1989, 1990a, 1990c,
Reed and Clarke 1990). The adjusted r% rule that we develop in equation (30) has the same motivation,
and yet is efficient. For simple stochastic optimization problems such as the examples in Section II, reallife decision makers need know nothing of smooth pasting conditions or of the option’s value; they need
only know the functional form of the option value, the opportunity cost of capital, and the parameters
associated with the stochastic process.
Seeing stopping under uncertainty as an r% rule clarifies three areas of confusion in the real options
literature. The first is the frequent claim that waiting to invest is only valuable under uncertainty.17 We
showed in Section I of this paper that waiting to invest has value even under certainty, that value being a
pure postponement value that has an exact counterpart under uncertainty The correct interpretation of
waiting to invest under uncertainty is that a delayed investment decision is only warranted under
15
In this vein one can use (30) to solve for the stopping point for the oil well abandonment problem in Clarke and
Reed (1990c), for example.
16
We have yet to learn of practitioners solving a stochastic dynamic programming problem for Ŵ .
17
Fisher (2000. p. 203), for example, states that “…the option to postpone the investment has value only because the
decision-maker is assumed to learn about future returns by waiting. If this were not the case, nothing would be
gained by postponing a decision to invest.”
23
Optimal Stopping
uncertainty. Delayed investment timing can be warranted under uncertainty and certainty, since pure
postponement value is common to both.
The second claim often made about investment under uncertainty is that the value of waiting is driven
by uncertainty (e.g., Amram and Kulatilaka 1999, p. 179). This acknowledges that there may be pure
postponement value under uncertainty, but that it is significantly less than the quasi-option value that
waiting under uncertainty creates. To examine this proposition, let the value boundary calculated from
the unadjusted r% rule, equation (31), be represented by WˆM . Option value is NPV plus pure
postponement value plus quasi-option value (Mensink and Requate 2005):
V(W) = Y(W) + PPV(W) + QOV(W).
(43)
The option premium, the value of waiting, is then
O(W)  V(W) - Y(W) = PPV(W) + QOV(W),
(44)
comprised of both pure postponement value and quasi-option value. Given a geometric Brownian motion
process and resultant option value (40), we can write (44) as
W 
 ˆ
W 


W
Wˆ  C  W  C    

  WˆM









 
 W 
W  ˆ
ˆ


WM  C  W  C     W  C  
WˆM  C
 Wˆ 
   Wˆ 
 M
 







.


(45)
The first term in the right-hand side is pure postponement value (which can be positive or negative), and
the second term is quasi-option value (which is always positive). Figure 6 shows both types of value for
the case of WˆM = 1.56 and Ŵ = 2.00. For W < Ŵ , the option premium is positive, and waiting is
optimal. For W < 1.56 pure postponement value is positive because the asset is rising at greater than .18
At the open-loop stopping point WˆM pure postponement value is zero (and hence, under an open-loop
strategy the program is stopped). For W > WˆM pure postponement value is negative (a cost) since the
asset is rising at less than . At the optimal stopping point the option premium is zero, and via (43) the
18
Traditionally, postponement value has been calculated by setting 2 = 0 (e.g., McDonald and Siegel 1986), which
may obfuscate the fact that there is postponement value under uncertainty.
24
Optimal Stopping
quasi-option value exactly offsets the negative pure postponement cost: QOV(W) = -PPV(W) and O(W) =
0.
Figure 6 shows that the value of waiting is driven by uncertainty, i.e., quasi-option value, only when
W is near Ŵ . For low values of W pure postponement value makes up the majority of the value from
waiting to invest, with the asset value is rising at a rate significantly greater than the rate of discount.
Figure 6 also shows the destruction of value by using the myopic stopping rule (31) instead of the
adjusted r% rule (30). Stopping at WˆM = 1.56 destroys only 8% of the investment’s value, its quasioption value,
1
V (WˆM | WˆM )
Y (WˆM )
Q(WˆM )
,
 1

V (WˆM | Wˆ )
Y (WˆM )  Q(WˆM ) Y (WˆM )  Q(WˆM )
which in this case is
  Wˆ

V (WˆM | WˆM )
1
 1
V (WˆM | Wˆ )
  1

  Wˆ  C 
W
WˆM
W
Wˆ
M

C
Wˆ
Wˆ
M


Wˆ  C 
Wˆ  C 
M
= 0.08.
This is consistent with the previously noted claims that the unadjusted r% rule, though inefficient, may
not destroy too much project value.
The third area of confusion is that uncertainty creates additional investment delay, beyond any pure
postponement value delay, by raising Ŵ . This is not the case. The r% rules under certainty and
uncertainty show that under certainty (2 = 0) the discount rate  falls to the riskless rate r and the
second-order term in (30) falls away. The adjusted force of interest under uncertainty, *, against which
asset appreciation is compared may be higher or lower than the force of interest under certainty, r, and
investment timing may be sped up or delayed. This result, generally unrecognized, has been
demonstrated by Sarkar (2000) and Davis (2002). For the example depicted in Figure 5, which is stopped
at Ŵ = 2 under uncertainty, the expected rate of growth of the asset only falls to the riskless discount
rate, 6%, when Ŵ = 6. For 2  W < 6 the program will be continued under certainty whereas it would be
25
Optimal Stopping
stopped immediately under uncertainty. Additional investment delay under uncertainty is only
guaranteed when  = r (comparing equations 3 and 30).
The r% rule has further implications for intuitive perceptions of equilibrium. For example, it
dramatically changes the traditional view of lowest-cost first in heterogeneous nonrenewable resource
extraction. Order of development is related to the rate of asset appreciation, rather than grade or cost
differentials: the mine whose expected value appreciation falls first to the equilibrium force of interest is
the first to come on line. In other cases discount rates can be different for similar assets. Ceteris paribus,
assets that are discounted more heavily, perhaps due to political risk, are brought on line sooner (cf.
Malliaris and Stefani 1994), in line with the economic intuition under certainty that higher discount rates
are a result of higher opportunity costs of waiting. The determination of investment timing under
uncertainty is thus the same as it is under certainty, the outcome of a complex sectorial equilibrium
involving price paths and interest rates that bring forth investment as needed to match demand.
The r% rule also supports the notion that this complex sectorial equilibrium involves project values
that must be in at least weak backwardation for new investment to be forthcoming under uncertainty
(Davis and Cairns 1999), since if the NPV of a project is always rising at or above the rate of discount the
program will, due to the delay incentive of quasi-option value, never be stopped. This in turn supports
evidence that commodities such as oil, which require irreversible lumpy investments to extract, have an
interior equilibrium that exhibits backwardation (Litzenberger and Rabinowitz 1995). This crucial
difference between equilibrium under certainty and uncertainty arises from the adjustment term in the r%
rule under uncertainty.
V. Conclusion
It is of some significance to observe that hitting boundaries and smooth pasting conditions apply
under conditions of certainty. Timing rules based on comparisons of mutually exclusive projects starting
at different times will lead to optimal timing decisions under certainty, just as they do under uncertainty.
26
Optimal Stopping
The substantive difference under uncertainty is one of method; problems are usually solved in the value
domain rather than the time domain.
By staying within the time domain we have shown that there is a consistent stopping algorithm, an r%
rule, that applies to problems under both certainty and uncertainty. In each case investment takes place
only once the NPV of the underlying project is expected to rise at a force of interest that represents the
opportunity cost of delay. Under certainty the opportunity cost of delay is clear – it is the time value of
money. Under uncertainty the opportunity cost of delay includes the time value of money and other
influences that decrease the opportunity cost of waiting. In a simple example we show that practitioners
need know nothing of smooth pasting or free boundaries to calculate the optimal stopping time under
uncertainty; they need only track the expected rate of appreciation of the NPV of their asset and invest
once it equals a force of interest that is fully calculable using the known parameters of the model.
Seeing the stopping problem under uncertainty as an r% rule that has close parallels to the case under
certainty reveals that the theory of investment under uncertainty is an incremental generalization of, not a
qualitative break from, the theory of investment under certainty; certainty is simply the limiting case of
uncertainty as volatility goes to zero.
27
Optimal Stopping
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Optimal Stopping
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Optimal Stopping
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30
Optimal Stopping
1.75
1.5
1.25
1
W
W
0.75
0.5
r = 0.20
0.25
5
6.25
10
15
20
time
Figure 1: Rate of appreciation of underlying asset as a hitting boundary problem for Example 1,
r = 0.20.
31
Optimal Stopping
Value
(Utils x 100)
1750
1500
1250
1218.25
1000
750

500
250
W
2
4
6
6.25
8
time
Figure 2: The stopping problem in Example 2 as a hitting boundary problem with W as the state variable,
and in Example 3 as a value matching and smooth pasting condition in the time domain.
32
Optimal Stopping
V(W, Ŵ )
(Utils x 100)
2000
1500
With
option to
wait
1000
500
No option
to wait
500
1000
Ŵ
1500
2000
W (utils x 100)
Figure 3: The stopping problem in Example 4 as a value matching and smooth pasting condition,
asset value domain.
•
Value
•
V(W)
Y(W)
h h
Ŵ
W
Figure 4: Possible deviations in W and program value given W = Ŵ .
33
Expected Rate of Growth
Optimal Stopping
0.35
0.3
W<C
W>C
0.25
0.2
0.15
0.1
Stopping
Continue
Region
0.05
Stop
0
0
0.5
1
1.5
2
2.5
Wˆ
W
Figure 5: The r% rule for geometric Brownian motion, comparing expected rate of growth of the
project forward NPV Y with the adjusted force of interest   12  2  (   1) for C = 1, r = 0.06, 
= 0.14, b = 0.05,  = 0.20.
1
Option Value = NPV + PPV + QOV
= NPV + Option Premium
0.8
Value
Option premium
Quasi-option
Value
0.6
0.4
Pure Postponement Value
NPV
0.2
1.2
1.4
1.6
1.8
2
W
Figure 6: Investment timing under geometric Brownian motion, showing option premium, pure
postponement value, and quasi-option value for C = 1, r = 0.06,  = 0.14, b = 0.05, and  = 0.20.
34