financial and real options: a unifying framework with a
TRANSCRIPT
Financial and Real Options: A Unifying Framework
with a Market Completeness Assumption
Andrew J. Jack
Department of Mathematics
King’s College LondonThe Strand
London WC2R 2LS, UK
Mihail Zervos
Department of Mathematics
King’s College LondonThe Strand
London WC2R 2LS, UK
November 22, 2006
Abstract
We formulate an abstract mathematical model for investments in real assets from
the perspective of the real options approach. We then derive an analytic expression for
its fair price under a market completeness assumption. This expression is the solution
of a stochastic optimisation problem. Also, we consider certain associated control
theoretic aspects and we establish the dynamic programming equation. Our model is
so general that it can as well provide a framework for the study of financial options.
We illustrate the theory developed by means of examples drawn from both finance and
economics.
Keywords and phrases: real options, investment pricing, contingent claims, mar-
tingales, optimal stochastic control, dynamic programming
Mathematics Subject Classification (1991): 90A09, 60G44, 93E03, 93E20
1 Introduction
The traditional discounted cash flow approaches to the problem of asset valuation such asthe net present value approach attribute prices to investments in a static way. A standardobservation in the economics literature is that they often result in prices which are signifi-cantly lower than actual prices because they ignore the value of managerial flexibility. Thepast two decades have seen the emergence of new techniques which aim at correcting this
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discrepancy by incorporating a price for managerial decisions. The new research area hasbeen termed “real options” and has already been documented in numerous references, in-cluding the books by Dixit and Pindyck [DP] and Trigeorgis [T]. These books also containextensive discussions about the economics issues arising in the comparison of the real optionsapproach with the traditional discounted cash flow approaches.
Up to now, the real options approach has been developed within the framework of specificinvestment models. Such models typically involve a small number of managerial decisionssuch as the decision to activate a project and/or the decision to abandon it at discretionarytimes. Overall, apart from offering a generic framework of a qualitative nature, the newtheory appears to be rather fragmented. The motivation for this paper has been to developand analyse a general model that can provide a unifying quantitative framework for thetheory of real options.
Adopting an abstract point of view, we can state that the constituent elements of everyinvestment consist of a set of managerial decision flows and an uncertain flow of profits andlosses associated with each managerial decision strategy. This is the model that we study.More specifically, we identify every investment with a set of finite variation processes mod-elling cumulative profits and losses parametrised by an abstract set of managerial decisionflows.
This model is so general that it can as well be used as a framework for the study offinancial options. For instance, a European option involves the decision to exercise or not,and each of the two possibilities is associated with a different payoff. American optionsprovide more interesting examples because they involve a richer set of decisions, namely theset of all exercise times. Other examples include compound options such as American onAmerican options and passport options.
With regard to the economics assumptions of our analysis, we are going to adopt theidealistic point of view according to which the uncertainty faced by any investment is ex-actly the same as the uncertainty driving the prices of a finite number of actively tradedassets. This corresponds to the market completeness assumption often made in finance. Ofcourse, as far as actual investments are concerned, it is unrealistic to assume that everyinvestment is perfectly correlated with a given set of traded assets. However, such an as-sumption holds reasonably well for a number of important cases which include investmentsassociated with commodities for which there exist liquid futures markets. Furthermore, asdiscussed in Trigeorgis [T], the assumption that, given any investment, there exists a “twin”traded security which has identical risk characteristics is commonly made by the traditionaldiscounted cash flow approaches. Undoubtedly, the relaxation of this market completenessassumption presents a very important and challenging issue. It can conceivably be achievedby deploying existing ideas and techniques of mathematical finance which address problemsof asset pricing in incomplete markets. However, such a relaxation goes beyond the scope ofthis article.
The paper is organised as follows. In Section 2, we introduce the notation used in thepaper, and we consider the standard stochastic calculus model for a frictionless market of
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continuous trading (see e.g. the books by Elliott and Kopp [EK], Karatzas [K3], Karatzasand Shreve [KS2], Musiela and Rutkowski [MR] and Shiryaev [SA]). Since investments inreal assets often do not have a prespecified lifetime, we have to consider perpetual trading.This situation gives rise to several measure theoretic subtleties because the risk neutralprobability measure is not equivalent to the original one, but only locally equivalent. In thecontext of pricing perpetual financial options, the resulting theoretical complications havebeen addressed by Karatzas [K1] and Zervos [Z].
In Section 3, we formulate the abstract model for an investment that we are going toanalyse. A precursor of this model can be found in Knudsen, Meister and Zervos [KMZ2]whose analysis is developed within a Markovian framework with strong assumptions. Toillustrate the various notions, we draw examples from finance. In particular, we considerEuropean and American options as well as passport options (with regard to the latter ones,see Hyer, Lipton-Lifschitz and Pugachevsky [HLP] or Henderson and Hobson [HH]).
In Section 4, we introduce and discuss the pricing of an investment conforming with ourmodel. The arguments here are fairly standard. However, keeping in mind that we workunder a market completeness assumption, it is of interest to observe that we have to considerhedging portfolios which are non-trivial functionals of the decision strategy adopted by theinvestment’s management. This is an observation already made in Knudsen, Meister andZervos [KMZ2]. With regard to financial options such as passport options, this implies thatthe writer cannot hedge themselves unless they have full knowledge of the holder’s associateddecisions at every time.
Section 5 is concerned with deriving an analytic expression for the fair price of an in-vestment. This expression is the solution of a stochastic optimisation problem. The analysisitself differs from existing ones because it has to account for the facts that, contrary to fi-nancial derivatives, investments in real assets can present their owners with losses as well aswith profits, they can be traded several times during their lifetime, and they are genericlyof perpetual nature.
In Section 6, we consider certain control theoretic aspects of the problem derived inSection 5, and we establish Bellman’s principle of optimality under very general assumptions.With regard to applications, this is a very important result, e.g. because it enables the useof the theory of viscosity solutions when studying the solvability of the associated dynamicprogramming partial differential equations. We also derive the well known expressions forthe fair prices of European and American options as well as an expression for the fair priceof passport options.
To illustrate the theory, in Section 7, we formulate a realistic model for investments inreal assets. This generalises a number of models that have been studied in the economicsliterature by Brennan and Schwartz [BS], McDonald and Siegel [McS], Paddock, Siegel andSmith [PSS], Dixit [D] (see also several cases in the books by Dixit and Pindyck [DP] andTrigeorgis [T]), as well as in the mathematics literature by Brekke and Øksendal [BØ1, BØ2],Knudsen, Meister and Zervos [KMZ1], Shirakawa [SH], Duckworth and Zervos [DZ1, DZ2],and a number of references therein.
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For reasons of completeness, we prove a martingale representation result which has beenestablished by Zervos [Z] and is used by our analysis in the Appendix.
2 The market
Let W be a n-dimensional Brownian motion defined on a probability space (Ω,F, P ), let(F
t ) be the filtration obtained by rendering right-continuous the natural filtration of W , andassume that F = F
∞. Given a probability measure Q on (Ω,F), we denote by FQ and(FQ
t ) the completion of F under Q and the usual augmentation of (Ft ) by the Q-negligible
sets in F, respectively.We denote by P the set of all probability measures Q on (Ω,F), which are locally
equivalent to the measure P , i.e. the restriction of Q to Ft is equivalent to the restriction
of P to Ft , for all t ≥ 0. Furthermore, we define the filtered measurable space (Ω,F ,Ft) by
F =⋂
Q∈P
FQ, Ft =⋂
Q∈P
FQt , t ≥ 0.
The reason behind this definition is that we are going to consider probability measures whichare locally equivalent but not equivalent to the original probability measure P . The followingsimple observations will be important in our analysis.
Remark 1 Since Ft ⊆ Ft ⊆ FP
t , for all t ≥ 0, W is an (Ft, P )-Brownian motion. Moreover,if we denote by (FP,T
t )t∈[0,T ] the usual augmentation of (Ft )t∈[0,T ] by the P negligible sets
in FT , then FT,Pt ⊆ Ft, for all t ∈ [0, T ], for all T ≥ 0. As a consequence, by appealing to
a simple induction argument, we can assume that all of the processes that we are going toconsider have “nice” path regularity. In particular, we can assume that stochastic integralswith respect to W have continuous sample paths and are (Ft)-adapted. In effect, we canessentially work under the assumption that our filtration satisfies the “usual conditions”,even though this is not actually true.
We denote by S the set of all (Ft)-stopping times, and by C the set of all (Ft)-adapted,finite variation, cadlag processes C with values in R. Given a process C ∈ C, we denoteby C its variation process. Also, given a process Z, we make the usual assumption thatZ0− = 0, and we denote by Z+ and Z− its positive and its negative part, respectively,namely Z+ = Z ∨ 0 and Z− = − (Z ∧ 0).
We consider a market where n + 1 assets are traded continuously. One of them (an“ideal money market account”) is locally riskless, and has a price S0
t , at time t, that evolvesaccording to the differential equation
dS0t = rtS
0t dt, S0
0 = s0 > 0. (1)
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For notational simplicity, we introduce the discount factor defined by
Rt = exp
(
−
∫ t
0
rs ds
)
,
so S0t = s0/Rt. The remaining n assets are risky, and their prices are modelled by the
stochastic differential equations
dSit = bi
tSit dt + Si
t
n∑
j=1
σijt dPW j
t , Si0 = si > 0, i = 1, . . . , n. (2)
Here, dP denotes stochastic integration on the stochastic basis (Ω,F ,Ft, P ).We assume that the interest rate process r, the instantaneous rates of return b =
(b1, . . . , bn)′, and the dispersion matrix σ = (σij) satisfy the following assumption.
Assumption 1 r, b, σ are (Ft)-progressively measurable processes, and, for every T > 0,there exists a constant K(T ) such that
n∑
i=1
(
|rt| + |bit|)
+
n∑
i,j=1
|σijt | ≤ K(T ), ∀t ≥ 0, P -a.s., (3)
ξ′σtσ′tξ ≥ K−2(T )|ξ|2, ∀ξ ∈ R
n, ∀t ≥ 0, P -a.s.. (4)
Let us now consider a “small investor”, i.e. an agent whose actions cannot influence theprices, who starts with an initial capital x and invests in the n + 1 market assets. Thisinvestor selects a portfolio process π and a cumulative consumption process C ∈ C. Morespecifically, at every instant t, the investor decides how much money πi
t to invest in the i-thrisky asset and what the cumulative consumption Ct should be. By assuming that C is afinite variation process, and not simply an increasing process, we allow for the possibilitythat the “consumption” Ct+∆t −Ct in the time interval ]t, t + ∆t] can be negative, in whichcase an amount −Ct+∆t + Ct is actually infused into the portfolio. Following Karatzas [K2],we see that if Xt is the investor’s wealth at time t, then X satisfies
Xt = x +
∫ t
0
rsXs ds −
∫
[0,t[
dCs +
∫ t
0
(σ′sπs) ·
[
dPWs + σ−1s (bs − rs1n) ds
]
,
where 1n denotes the n-dimensional vector with each component unity. In view of theassumptions that we have made, note that X is a caglad process such that X0 = x. ThisSDE has a unique strong solution on (Ω,F ,Ft, P ) if we impose the following additionalassumption.
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Assumption 2 The portfolio process π = (π1, . . . , πn)′is (Ft)-progressively measurable and
satisfies∫ t
0|πs|
2 ds < ∞, ∀t ≥ 0, P -a.s..
Under this assumption, the solution of this SDE is
Xt = R−1t
(
x −
∫
[0,t[
Rs dCs +
∫ t
0
Rs (σ′sπs) · d
PWs
)
, (5)
where the (Ft, P )-semimartingale W is defined by
Wt = Wt +
∫ t
0
σ−1s (bs − rs1n) ds. (6)
Now, define the process L by
Lt = exp
(
−
∫ t
0
θs · dPWs −
1
2
∫ t
0
|θs|2 ds
)
, (7)
where the relative risk process θ is defined by θt = σ−1t (bt − rt1). Assumption 1 and Prob-
lem 5.8.1 in Karatzas and Shreve [KS1] imply |θt|2 ≤ K2(t)|bt − rt1|
2 ≤ nK4(t), so,
E
[
exp
(
1
2
∫ t
0
|θs|2 ds
)]
< ∞, ∀t ≥ 0.
As a consequence, Novikov’s criterion implies that L is a martingale (see Karatzas and Shreve[KS1, Corollary 3.5.15] or Revuz and Yor [RY, Corollary VIII.1.16]). Therefore, we can definea unique probability measure P on (Ω,F) such that, for every t ≥ 0, the restrictions of themeasures P and P on (Ω,F
t ) are equivalent, with corresponding Radon-Nikodym derivative(
dP/dP)
∣
∣
∣
F
t
= Lt (see the discussion on p. 192 of Karatzas and Shreve [KS1], or Revuz and
Yor [RY, Proposition VIII.1.13]). Furthermore, Girsanov’s Theorem implies that the processW defined by (6) is an (Ft, P )-Brownian motion, and
∫ t
0
Rs (σ′sπs) · d
PWs =
∫ t
0
Rs (σ′sπs) · d
PWs, P, P -a.s., (8)
where dP denotes stochastic integration on the stochastic basis (Ω,F ,Ft, P ), and both in-tegrals are (Ft)-adapted and have continuous sample paths (see Karatzas and Shreve [KS1,Corollary 3.5.2] or Revuz and Yor [RY, Theorem VIII.1.12, Proposition VIII.1.5]; see alsoRemark 1). As a consequence, the process X given by (5) satisfies
Xt = R−1t
(
x −
∫
[0,t[
Rs dCs +
∫ t
0
Rs (σ′sπs) · d
PWs
)
. (9)
At this point, observe that, under the assumptions that we have made, L is not in generala uniformly integrable martingale, and therefore, the measure P is not in general absolutelycontinuous with respect to P on (Ω,F). The following example illustrates this issue; it hasbeen inspired from Example 1.7.6 in Karatzas and Shreve [KS2].
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Example 1 Suppose that n = 1, and r, b, σ are all constants such that r 6= b. Sincelimt→∞ t−1Wt = 0, P -a.s., and limt→∞ t−1Wt = 0, P -a.s., if we define the events A =limt→∞ t−1Wt = 0 and B = limt→∞ t−1Wt = (r − b)/σ, then P (A) = 1, P (A) = 0 andP (B) = 0, P (B) = 1.
To obtain a viable analysis, we have to impose some additional assumptions on the tradingstrategies. More specifically, motivated by Karatzas and Shreve [KS2, Definition 1.2.4], weintroduce the following family of portfolios.
Definition 1 A portfolio π is tame if it satisfies Assumption 2 and, if M is the (FQt , P )-local
martingale defined by
Mt =
∫ t
0
Rs (σ′sπs) · d
PWs,
then the process M− is of class (D) with respect to P . We denote by A the set of all tameportfolios.
The assumption that the process M− is of class (D) is weaker than corresponding assump-tions in the literature; e.g. Karatzas and Shreve [KS2, Definition 1.2.4] assume that Mis uniformly bounded from below by a constant depending only on π. In the absence ofsuch an assumption, we can construct trading strategies which yield arbitrage opportunities.Karatzas and Shreve [KS2, Example 1.2.3] construct such an example if the time horizon isfinite. In the context of infinite time horizon, if the assumption above does not hold, then,by waiting sufficiently long, elementary strategies can yield a risk free profit.
Example 2 Suppose that n = 1, r ≡ 0 and b ≡ σ ≡ 1, and consider the constant portfolioπt = 1. In the case,
Mt = Wt = t + Wt.
Observe that there is an assymetry in the definition of a tame portfolio. Specifically, weassume that M− is of class (D) with respect to P , but not with respect to P . In fact, wecan easily construct tame portfolios such that M− is of class (D) with respect to P , but notwith respect to P . However, our assumptions are sufficient to ensure that the market doesnot present arbitrage opportunities in the following sense.
Definition 2 ...
The claim that tame portfolios exclude arbitrage opportunities from the market followsimmediately from the following result.
Lemma 1 Suppose that τ is an (Ft)-stopping time.
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Remark 2 If M is an (FQt , P )-local martingale such that M− is of class (D), then M is
an (FQt , P )-supermartingale (see Revuz and Yor [RY, Exercise IV.1.46]) which converges,
P -a.s., to an integrable random variable M∞ which closes M . To see this, consider a lo-calising sequence (τn) such that M τn is a uniformly integrable martingale. Given any timess < t, Fatou’s lemma implies E
[
M+t | FQ
s
]
≤ lim infn→∞ E[
M+t∧τn
| FQs
]
. On the otherhand, the uniform integrability of the family (M−
t∧τn) implies that the sequence (M−
t∧τn) con-
verges in L1 to M−t (Dellacherie and Meyer [DM1, II.21]), which implies E
[
M−t | FQ
s
]
=limn→∞ E
[
M−t∧τn
| FQs
]
. However, these observations imply E[
Mt | FQs
]
≤ Ms, which provesthat M is a supermartingale. The statement regarding the convergence of M follows fromDellacherie and Meyer [DM2, VI.6].
3 A general model for real options
We now introduce the general model for real options that we are going to study. To illustratethe various notions, we use examples drawn from finance. Surprisingly, European optionsrequire the most complicated analysis when addressed from within the framework that wedevelop. Later in Section 7, we are going to consider a further example for investments inreal assets.
Definition 3 A real option C∂, ∂ ∈ D is an economic entity that (a) offers its owner aset of decision strategies D, and (b) presents its owner with a cumulative payoff C∂ ∈ C ifthey adopt decision strategy ∂ ∈ D.
We are not going to impose any specific structure on the decision space D itself. Indeed,as we are going to see shortly, D can be a σ-algebra, a set of stopping times, or a set of tradingstrategies. Therefore, this is an abstract model that can be used to address problems thatinvolve no decisions, as well as problems that involve a continuum of decisions. Of course,as far as the modelling of actual problems is concerned, every decision strategy should benon-anticipative. This requirement is captured by the general assumption that C∂ is (Ft)-adapted, for every ∂ ∈ D. We can now consider the following examples.
Example 3 Given i = 1, . . . , n, a European call option on asset i with maturity T andstrike price K is the special case of the above definition which arises if we define D = FT ,and, given ∂ ∈ D,
C∂t =
(
SiT − K
)
1∂1T≤t, t ≥ 0.
According to these definitions, if the holder follows decision strategy ∂ ∈ D, then theyexercise the option if the event ∂ ∈ FT occurs, and only then.
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Example 4 Given i = 1, . . . , n, an American call option on asset i with maturity T andstrike price K is the special case which arises if we define D = S, and, given any stoppingtime ∂ ∈ D,
C∂t =
(
Si∂ − K
)
1∂≤T∧t, t ≥ 0.
Observe that, given a choice of an exercise time ∂ ∈ D, the option is exercised on the event∂ ≤ T and expires without being exercised on the event ∂ > T.
Example 5 Let any non-empty J ⊆ 1, . . . , n. A passport option on the trading accountcomposed by the assets in J with initial wealth z and maturity T is the special case whicharises if we define
D =
π ∈ A |∣
∣πjt /S
jt
∣
∣ ≤ Kj , πit = 0, ∀j ∈ J, ∀i /∈ J, ∀t ≥ 0, P -a.s.
,
and, given a choice of a trading strategy ∂ ∈ D,
C∂t = Z+
T 1T≤t, t ≥ 0,
where Zt is the total value of the trading account at time t, and is given by an equation like(5) or (9) with X = Z, x = z, C = 0 and π = ∂. Note that the bounds Kj > 0, j ∈ J , onthe allowable number of shares of each risky asset in J are specified by the contract.
We want our analysis to result in a price for a real option not only at time 0, but at anyfinite (Ft)-stopping time τ . For this reason, we introduce the following idea.
Definition 4 Given a decision strategy ∂ ∈ D and a finite stopping time τ ∈ S, we denoteby D(∂, τ) the random set of all decision strategies emanating from ∂ at time τ . Eachdecision strategy in D(∂, τ) “coincides” with ∂ on the stochastic interval [0, τ [, and so, if∂1, ∂2 ∈ D(∂, τ), then
C∂1
t 1t<τ = C∂2
t 1t<τ, ∀t ≥ 0, P -a.s.. (10)
In other words, D(∂, τ) is the random set of all decision strategies which are availableat time τ to the real option’s owner given that decision strategy ∂ has been adopted up totime τ . The idea behind this definition is the following. Suppose that we want to price thereal option at time τ , and its owner has followed decision strategy ∂ up to τ . At time τ ,the available decision strategies are the elements of the set D(∂, τ), and the valuation of thereal option has to be based on the fact that, after τ , the real option can yield payoffs/costsonly according to a cumulative payoff process which belongs to C∂′
, ∂′ ∈ D(∂, τ). At thispoint, it is worth noting that we have formulated the definition so that, given any ∂ ∈ D,D(∂, 0) = D.
Having imposed no structure on the decision space D, we cannot express this definitionmore formally. Clearly, (10) on its own is not sufficient to describe D(∂, τ). With regard tothe examples considered above, this definition gives rise to the following cases.
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Example 6 Consider a European call option as described in Example 3. Given a finitestopping time τ ∈ S, and a decision strategy ∂ ∈ D = FT , we can see that
D(∂, τ) =
∂ ∈ FT | ∃B ∈ FT : P(
∂ 4 [∂ ∩ τ > T ∪ B ∩ τ ≤ T])
= 0
,
satisfies the requirements of Definition 4. The definition of this set looks rather involved.To illustrate it, consider the cases where τ ≤ T , P -a.s., and τ > T , P -a.s.. In the first case,D(∂, τ) = D, which corresponds to the idea that, prior to or exactly at maturity, the decisionto exercise the option or not is available to the holder. In the second case, D(∂, τ) consistsof all sets in FT which are equal to ∂, P -a.s., which reflects the idea that, after maturity,the holder has exercised or not exercised the option irreversibly.
Example 7 Consider an American option as in Example 4. Given a finite stopping timeτ ∈ S, and an exercise time ∂ ∈ D = S,
D(∂, τ) =
∂′ ∈ S | ∂′1∂<τ = ∂1∂<τ, P -a.s.
,
i.e. D(∂, τ) is the set of all stopping times which are equal to ∂ on the event ∂ < τ.
Example 8 Consider a passport option as described in Example 5. Given a finite stoppingtime τ ∈ S and a trading strategy ∂ ∈ D,
D(∂, τ) =
∂′ ∈ D | ∂′1t<τ = ∂1t<τ, P -a.s.
,
i.e. D(∂, τ) is the set of all trading strategies in D which coincide with ∂ on the stochasticinterval [0, τ [.
Now, to avoid pathological situations, we have to assume that the set D(∂, τ) is not“sensitive” to sets of measure 0.
Assumption 3 Given any finite τ, τ ′ ∈ S such that τ = τ ′, P -a.s., D(∂, τ) = D(∂, τ ′).
Note that, since P , P are equivalent on Fτ , for every finite τ ∈ S, it would makeno difference if we had expressed this assumption in terms of P instead of P . Also, it isstraightforward to verify that this assumption holds in the cases of the examples consideredabove.
Observe that we have formulated Definitions 3 and 4 in terms of processes which are (Ft)-adapted. Since the measures P , Q are not, in general, equivalent to the original measureP involved in the description of the market model (2), and they arise by the needs ofthe analysis, this is the sensible choice from the perspective of modelling. However, fortechnical reasons, we need to work on the stochastic basis (Ω,FQ,FQ
t , P ). Now, since
(Ft) ⊆ (FQt ) ⊆ (F P
t ) and (Ft) is right continuous, given any stopping time τ ∈ SQ, thereexists a stopping time τ ′ ∈ S such that τ ′ = τ , P -a.s. (Dellacherie and Meyer [DM1, IV.59]).It is worth noting that τ ′ = τ , P -a.s., as well, because P , P are equivalent on FQ
τ . In viewof these observations, we introduce the following definition.
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Definition 5 Given a decision strategy ∂ ∈ D and a finite stopping time τ ∈ SQ, we denoteby D(∂, τ) the set D(∂, τ ′), where τ ′ ∈ S is such that τ ′ = τ , P -a.s..
Note that, in view of Assumption 3, this definition is well posed.
4 The pricing of a real option
We now define the fair price of a real option by comparing the profits and losses resultingfrom its ownership with the management of appropriate portfolios in the traded assets asmodelled in Section 2. Given a ∂ ∈ D and a finite τ ∈ S, we define the set of the seller’s
prices at time τ given decision strategy ∂ by
Vs(∂, τ) =
Z ∈ L1(Ω,Fτ , P ) | ∀∂′ ∈ D(∂, τ), ∃x ∈ R, ∃π ∈ A :
X(x,π,C∂′
)τ ≤ Z and lim inf
t→∞RtX
(x,π,C∂′
)t ≥ 0, P -a.s.
and the set of a buyer’s prices at time τ given decision strategy ∂ by
Vb(∂, τ) =
Z ∈ L1(Ω,Fτ , P ) | ∃∂′ ∈ D(∂, τ), ∃x ∈ R, ∃π ∈ A :
X(x,π,−C∂′
)τ ≤ −Z and lim inf
t→∞RtX
(x,π,−C∂′
)t ≥ 0, P -a.s.
Clearly, as long as they are not empty, Vs(∂, τ) is a stochastic interval which is a neighbour-hood of ∞, whereas Vb(∂, τ) is a stochastic interval which is a neighbourhood of −∞.
Definition 6 Given a ∂ ∈ D and a finite τ ∈ S, the lowest seller’s price at time τ given
decision strategy ∂ of the real option under consideration is
V s(∂, τ) = P - ess inf Vs(∂, τ),
whereas the highest buyer’s price at time τ given decision strategy ∂ of the real option is
V b(∂, τ) = P - ess supVb(∂, τ).
Moreover, if there exists an Fτ -measurable random variable Z such that V s(∂, τ) ≤ Z ≤V b(∂, τ), then Z is a rational or fair price of the real option at time τ given decision strategy
∂ .
Remark 3 We have formulated these definitions in terms of the measure P . However, thisdoes not contain any degree of arbitrariness because, given any finite τ ∈ S, the measuresP and P are equivalent on (Ft).
Also, observe that we have defined the real option’s prices in terms of Fτ -measurablerandom variables. This does not present any inconsistency with the fact that we have
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defined real options in terms of processes which are adapted to the uncompleted filtration(Ft). The reason is that, given any τ ∈ S and any Fτ -measurable random variable Z, thereexists τ ′ ∈ S and an Fτ ′-measurable random variable Z ′ such that τ ′ = τ and Z ′ = Z, P -a.s.(Dellacherie and Meyer [DM1, IV.59]).
Similar remarks pertain to our subsequent analysis.
The financial economics arguments behind the preceding definition are the following.Suppose first that, at time τ , the price of the real option is set at Z ≥ V s(∂, τ). In thiscase, the decision of the owner to sell the real option is rational because the owner can finda trading strategy that is at least as profitable as any decision strategy. Indeed, considerany ∂′ ∈ D(∂, τ), and let x ∈ R, π ∈ A be such that
X(x,π,C∂′
)τ ≤ Z and lim inf
t→∞RtX
(x,π,C∂′
)t ≥ 0, P -a.s.. (11)
At time τ , if the owner sells the real option for Z and invests X(x,π,C∂
′
)τ to maintain the
portfolio-consumption pair (π, C∂′
) after τ , then the owner makes a riskless profit equal to
Z − X(x,π,C∂
′
)τ at time τ and the portfolio’s value is eventually nonnegative. Obviously, this
is a situation which is preferable to maintaining the real option’s ownership and followingdecision strategy ∂′.
On the other hand, suppose that, at time τ , the price of the real option is set at Z ≤V b(∂, τ), and let ∂′ ∈ D(∂, τ), x ∈ R, π ∈ A be such that
X(x,π,−C∂′
)τ ≤ −Z and lim inf
t→∞RtX
(x,π,−C∂′
)t ≥ 0, P -a.s.. (12)
In this case, consider the strategy consisting of:(i) investing an amount Z to buy the real option and adopt the decision strategy ∂′,
(ii) investing the amount X(x,π,−C∂
′
)τ to maintain the portfolio-consumption pair (π,−C∂′
)after τ , and
(iii) infusing the profits resulting from the investment into the portfolio, and vice versa.
Clearly, this strategy, which yields a riskless profit equal to −Z − X(x,π,−C∂
′
)τ at time τ and
a total wealth that is eventually nonnegative, is at least as good as doing nothing. Notethat, since Z might be negative, “investing an amount Z to buy the real option” may referto a situation where an investor actually receives an amount Z to take ownership of andresponsibility for a non-profitable real option and its liabilities.
Now, suppose that there exists an Fτ -measurable random variable Z such that V s(∂, τ) ≤Z ≤ V b(∂, τ). In this case, both parties would be keen to accept a deal at the price Z. Thus,Z being a price at which the deal is satisfactory for both parties, it represents a fair pricefor the real option.
It is worth observing that the situation of the two parties involved in a deal is not sym-metric, which is reminiscent of the asymmetry occurring in the case of American contingent
12
claims. The seller already owns the investment, and thus needs to take into account the fore-gone income if it is sold. For this reason, the seller has to consider every available decisionstrategy. On the other hand, the buyer is not bound by the ownership of the investment,and, therefore, only has to consider a particular decision strategy.
Remark 4 In the above definitions, the portfolios considered may depend on the decisionstrategy adopted by the owner. With regard to investments having the structure of a Eu-ropean or an American option, this is not necessary, i.e. we can choose the correspondingportfolios in a way which is independent of the owner’s decisions ragarding whether to ex-ercise and its timing. However, this cannot be the case with other financial options or withmost investments in real assets. In examples like passport options, this implies that thereis no possibility for the writer to hedge unless they have full information of the holder’sassociated decisions at all times.
5 The fair price of a real option
We now solve the problem of pricing a given real option C∂, ∂ ∈ D as described in theprevious section under the assumptions on the market made in Section 2. To proceed, weimpose the following integrability condition.
Assumption 4 sup∂∈D E∫
[0,∞[Rs dC∂
s < ∞.
This assumption will ensure that the price of a real option is bounded away from infinity.
Theorem 2 Consider a real option C∂, ∂ ∈ D, and suppose that Assumptions 1, 3 and
4 hold. Given any decision strategy ∂ ∈ D and any finite (Ft)-stopping time τ ,
V s(∂, τ) = V b(∂, τ) = P - ess sup∂′∈D(∂,τ)
R−1τ E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
=: V (∂, τ). (13)
Moreover, if there exists a ∂∗ ∈ D(∂, τ) such that
V (∂, τ) = R−1τ E
[∫
[τ,∞[
Rs dC∂∗
s | Fτ
]
, (14)
then V (∂, τ) is the unique fair price of the real option at time τ given decision strategy ∂.
Proof. Throughout the proof, we fix a decision strategy ∂ ∈ D and a finite (Ft)-stoppingtime τ . Observe that Assumption 4 implies V (∂, τ) ∈ L1(Ω,Fτ , P ).
Let any Z ∈ L1(Ω,Fτ , P ). Given ∂′ ∈ D(∂, τ), suppose that there exist x ∈ R and π ∈ Asuch that (11) hold. With reference to the wealth equation (9), we define
Mt = RtX(x,π,C∂
′
)t +
∫
[0,t[
Rs dC∂′
s = x +
∫ t
0
Rs (σ′sπs) · d
PWs. (15)
13
By the definition of a tame portfolio in Definition 1, the process M− is of class (D) withrespect to P . In view of Remark 2, this implies that M is an (Ft, P )-supermartingale whichconverges, P -a.s., to an integrable random variable M∞ that closes M . As a consequence,
RτX(x,π,C∂
′
)τ = −
∫
[0,τ [
Rs dC∂′
s + Mτ
≥ −
∫
[0,τ [
Rs dC∂′
s + E [M∞ | Fτ ]
≥ −
∫
[0,τ [
Rs dC∂′
s + E
[∫
[0,∞[
Rs dC∂′
s | Fτ
]
,
the second inequality following from the second assertion of (11). However, this and the firstassertion of (11) imply
Z ≥ R−1τ E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
,
which proves that V s(∂, τ) ≥ V (∂, τ).On the other hand, given any ∂′ ∈ D(∂, τ), suppose that there exist x ∈ R and π ∈ A
such that (12) hold. By following exactly the same arguments as above with −C∂′
in placeof C∂′
, and (12) in place of (11), we can see that
Z ≤ R−1τ E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
,
which proves that V b(∂, τ) ≤ V (∂, τ).To prove the reverse inequalities, consider any ∂′ ∈ D(∂, τ), and let
x = E
∫
[0,∞[
Rs dC∂′
s . (16)
With reference to Corollary 5 in the Appendix, there exists a portfolio π satisfying Assump-tion 2 such that
E
[∫
[0,∞[
Rs dC∂′
s | Ft
]
− E
[∫
[0,∞[
Rs dC∂′
s
]
=
∫ t
0
Rs (σ′sπs) · d
P Ws. (17)
Since the left hand side of this equation defines a uniformly integrable martingale, both πand −π are tame portfolios. On the other hand, in view of (16) this equation is equivalentto
R−1t E
[∫
[t,∞[
Rs dC∂′
s | Ft
]
= R−1t
(
x −
∫
[0,t[
Rs dC∂′
s +
∫ t
0
Rs (σ′sπs) · d
PWs
)
.
14
If we compare this with (9), we can see that
X(x,π,C∂
′
)t = R−1
t E
[∫
[t,∞[
Rs dC∂′
s | Ft
]
, (18)
and
X(−x,−π,−C∂
′
)t = −R−1
t E
[∫
[t,∞[
Rs dC∂′
s | Ft
]
. (19)
Furthermore, Levy’s “upward” theorem implies
limt→∞
E
[∫
[t,∞[
Rs dC∂′
s | Ft
]
= limt→∞
(
E
[∫
[0,∞[
Rs dC∂′
s | Ft
]
−
∫
[0,t[
Rs dC∂′
s
)
= 0,
which, combined with (18) and (19), yields
limt→∞
RtX(x,π,C∂
′
)t = lim
t→∞RtX
(−x,−π,−C∂′
)t = 0. (20)
Now, in view of (13) and (18), for every ∂′ ∈ D(∂, τ),
V (∂, τ) ≥ R−1τ E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
= X(x,π,C∂′
)τ ,
where x and π ∈ A are as in (16), (17). Combining this with (20), we can see that X(x,π,C∂′
)
satisfies (11) with Z = V (∂, τ). However, this observation and the inequality V s(∂, τ) ≥V (∂, τ) imply Vs(∂, τ) = [V (∂, τ),∞[.
On the other hand, for every ∂′ ∈ D(∂, τ), (19) and (20) imply
X(−x,−π,−C∂′
)τ = −R−1
τ E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
and limt→∞
RtX(−x,−π,−C∂
′
)t = 0, (21)
where x and −π ∈ A are as in (16), (17). As a consequence,
R−1τ E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
∈ Vb(∂, τ), ∀∂′ ∈ D(∂, τ),
and so, V (∂, τ) ≤ V b(∂, τ), which, combined with the inequality V (∂, τ) ≥ V b(∂, τ), impliesV (∂, τ) = V b(∂, τ).
Finally, suppose that there exists ∂∗ such that (14) holds. (21) with ∂′ = ∂∗ and theinequality V (∂, τ) ≥ V b(∂, τ) imply Vb(∂, τ) = ]−∞, V (∂, τ)]. However, combining this withthe fact that Vs(∂, τ) = [V (∂, τ),∞[, we can see that V (∂, τ) is the unique fair price of thereal option at time τ given decision strategy ∂.
15
6 Optimality notions and the Bellman equation
In the previous section, we proved that the problem of pricing a real option C∂, ∂ ∈ Dreduces to solving the stochastic optimisation problems defined by (13). With reference tostandard theory of stochastic optimal control (see e.g. El Karoui [ElK]), this observationmotivates the following definition which generalises Definition 2 of Knudsen, Meister andZervos [KMZ2].
Definition 7 We say that a decision strategy ∂∗ ∈ D is 0-optimal if it satisfies (14) withτ = 0. Given a ∂ ∈ D and a finite τ ∈ S, we say that a decision strategy ∂∗ ∈ D(∂, τ) is(∂, τ)-optimal if it satisfies (14). Moreover, we say that a decision strategy ∂∗ is conditionally
optimal if ∂∗ is (∂∗, τ)-optimal for every finite τ ∈ S.
The idea behind this definition is that, if the owner follows any non-optimal decisionstrategy ∂, then the real option is depreciated in the sense that it would be better to sellthe real option and maintain an appropriate portfolio π with consumption C∂ and initialendowment equal to the amount received by the transaction. On the other hand, as long asthe owner follows an optimal decision strategy, there is no rational financial reason to prefera portfolio to the real option’s ownership or vice versa.
The optimality notions introduced by the preceding definition give rise to the followingrobustness issue. Suppose that a real option is traded at time 0 and its new owner follows anoptimal strategy ∂∗ up to time τ when it is sold. At that time, the real option’s price is givenby Theorem 2, and assuming that it exists, there is an associated (∂∗, τ)-optimal strategy ∂.Since the owner has been following an optimal strategy up to time τ , we should expect that∂ = ∂∗. Indeed, if this is not the case, it is straightforward to construct a trading strategythat yields arbitrage opportunities. To ensure that this is true, we impose the followingbifurcation assumption on our model.
Assumption 5 Given any ∂ ∈ D, any finite τ ∈ S, and any A ∈ Fτ , if ∂1, ∂2 ∈ D(∂, τ),then there exists ∂3 ∈ D(∂, τ) such that
C∂3
t = C∂1
t 1A + C∂2
t 1Ac , ∀t ≥ 0, P -a.s.. (22)
Observe that, following Definitions 3 and 4, we have formulated this definition in termsof the uncompleted filtration (Ft).
A further implication of this assumption is that the stochastic optimisation problemsdefined by (13) satisfy Bellman’s principle of optimality. The next result is concerned withthese issues.
16
Theorem 3 Suppose that Assumptions 1, 3–5 hold, and consider any decision strategy ∂ ∈D and any finite (Ft)-stopping times τ ≤ τ . The value of the real option satisfies the Bellman
equation
V (∂, τ) = P - ess sup∂′∈D(∂,τ)
R−1τ E
[∫
[τ,τ [
Rs dC∂′
s + RτV (∂′, τ) | Fτ
]
. (23)
Moreover, if ∂∗ is a (∂, τ)-optimal decision strategy, then ∂∗ is (∂∗, τ )-optimal. In particular,
if ∂∗ is 0-optimal, then it is conditionally optimal.
Proof. Given any ∂ ∈ D and any τ ∈ S, the family of random variables
H(∂, τ) =
E
[∫
[τ,∞[
Rs dC∂′
s | Fτ
]
| ∂′ ∈ D(∂, τ)
(24)
is directed upwards. To see this, let any ∂1, ∂2 ∈ D(∂, τ), and define
A =
E
[∫
[τ,∞[
Rs dC∂1
s | Fτ
]
≥ E
[∫
[τ,∞[
Rs dC∂2
s | Fτ
]
∈ Fτ .
Given τ ′ ∈ S such that τ ′ = τ , P -a.s., D(∂, τ) = D(∂, τ ′), by Definition 5. On the otherhand, there exists A′ ∈ Fτ ′ such that A′ = A, P -a.s. (see Remark 3). Now, in view ofAssumption 5, there exists ∂3 ∈ D(∂, τ) such that C∂3
t = C∂1
t 1A′ + C∂2
t 1(A′)c , for all t.However,
∫
[τ,∞[
Rs dC∂3
s = 1A
∫
[τ,∞[
Rs dC∂1
s + 1Ac
∫
[τ,∞[
Rs dC∂2
s , P -a.s.,
which implies
E
[∫
[τ,∞[
Rs dC∂3
s | Fτ
]
≥ E
[∫
[τ,∞[
Rs dC∂1
s | Fτ
]
∨ E
[∫
[τ,∞[
Rs dC∂2
s | Fτ
]
,
and the claim follows.Since D(∂, τ) = ∪∂′∈D(∂,τ)D(∂′, τ), and, with reference to Definitions 4, 5, C ∂
t 1t<τ =
C∂′
t 1t<τ, for every ∂ ∈ D(∂′, τ),
V (∂, τ) = P - ess sup∂′∈D(∂,τ)
P - ess sup∂∈D(∂′,τ)
R−1τ E
[∫
[τ,τ [
Rs dC∂′
s + E
[∫
[τ ,∞[
Rs dC ∂s | Fτ
]
| Fτ
]
= P - ess sup∂′∈D(∂,τ)
R−1τ E
[
∫
[τ,τ [
Rs dC∂′
s + P - ess sup∂∈D(∂′,τ)
E
[∫
[τ ,∞[
Rs dC ∂s | Fτ
]
| Fτ
]
= P - ess sup∂′∈D(∂,τ)
R−1τ E
[∫
[τ,τ [
Rs dC∂′
s + RτV (∂′, τ ) | Fτ
]
,
17
the second equality following from the fact that the family H(∂′, τ ) defined as in (24) isdirected upwards. However, this proves the Bellman equation (23).
Now, suppose that ∂∗ is a (∂, τ)-optimal decision strategy, and assume that it is not(∂∗, τ )-optimal, i.e.
V (∂, τ) = R−1τ E
[∫
[τ,∞[
Rs dC∂∗
s | Fτ
]
,
V (∂∗, τ) ≥ R−1τ E
[∫
[τ ,∞[
Rs dC∂∗
s | Fτ
]
,
with strict inequality on a set of positive probability. Under these assumptions,
V (∂, τ) ≤ R−1τ E
[∫
[τ,τ [
Rs dC∂∗
s + RτV (∂∗, τ) | Fτ
]
,
with strict inequality on a set of positive probability, which contradicts the Bellman equation(23).
We can now revisit European, American and passport options to see the implications ofour analysis.
Example 9 Consider a European call option as described in Examples 3, 6, and let any∂ ∈ D = FT , τ ∈ S, A ∈ Fτ and ∂1, ∂2 ∈ D(∂, τ). In view of the definition of D(∂, τ) inExample 6, there exist sets B1, B2 ∈ FT such that, for j = 1, 2,
P(
∂j 4 [∂ ∩ τ > T ∪ Bj ∩ τ ≤ T])
= 0.
If we define B3 = B1∩A∪B2∩Ac ∈ FT , then we can see that this formula with j = 3 definesa strategy ∂3 which satisfies (22). Now, given any ∂ ∈ D, since D(∂, T ) = FT , Theorem 2implies
V (∂, T ) = P - ess supA∈FT
(
SiT − K
)
1A =(
SiT − K
)+.
Furthermore, Theorem 3 implies that the fair price of the given European option written at
time 0 is E[
RT (SiT − K)
+]
, which is the well known formula.
Example 10 Consider an American call option as described in Examples 4, 7. Let any∂ ∈ D = S, any finite τ ∈ S, any A ∈ Fτ , and any ∂1, ∂2 ∈ D(∂, τ). Since ∂3 = ∂11A +∂21Ac ∈ D(∂, τ), we can easily verify Assumption 5. Furthermore, Theorem 2 implies thatthe fair price of the given American option written at time 0 is given by
supτ∈S
E[
Rτ
(
Siτ − K
)
1τ≤T
]
= supτ∈S
E[
Rτ
(
Siτ − K
)+]
,
which is the well known formula.
18
Example 11 Consider a passport option as described in Examples 5, 8. Given any ∂ ∈ D,any finite τ ∈ S, any A ∈ Fτ , and any ∂1, ∂2 ∈ D(∂, τ), we can easily verify that ∂3 =∂11A + ∂21Ac ∈ D(∂, τ), and so, Assumption 5 is satisfied. Also, Theorem 2 implies that thefair price of the given passport option written at time 0 is sup∂∈D E
[
RT Z+T
]
.
7 A model for investments in real assets
We now develop an investment model with special significance to the natural resource in-dustry. To simplify the presentation, we assume that the investment under considerationproduces a single commodity. Such an investment could be an oil field or a copper mine. Wemodel the commodity price on the stochastic basis (Ω,F ,Ft, P ) by the stochastic differentialequation
dYt = ct dt + et · dPWt, Y0 = y > 0, (25)
where c and e = (e1, . . . , en)′satisfy the following assumption.
Assumption 6 c, e are (Ft)-progressively measurable processes such that the SDE (25) hasa unique strong solution such that Yt > 0, for all t ≥ 0, P -a.s..
On the stochastic basis (Ω,F ,Ft, P ), the commodity price satisfies the SDE
dYt =[
ct − e′tσ−1t (bt − rt1n)
]
dt + et · dPWt, Y0 = y > 0. (26)
With reference to the market model described in Section 2, observe that the commodityprice is driven by the same uncertainty sources as the traded assets, and is adapted to thefiltration generated by W . In view of Assumption 1, this means that the commodity price isperfectly correlated with the traded assets. From the perspective of economic applications,such a case arises when there exists a futures market for the commodity under considerationand the process S provides a model for the associated prices. Of course, since we assumethat the investment produces a single commodity such as oil or copper, the process S whichmodels the associated futures prices should have a single component (i.e. n = 1).
We assume that the investment can have a maximal lifetime, denoted by T > 0. Thiscan be used to model lease expiration times, or it can be chosen to be infinity. Also, weassume that the investment can produce during its lifetime at most a given amount of thecommodity, say a > 0. With reference to the natural resource industry, this is the amountof the reserves that the investment has access to. In what follows, we assume for simplicitythat a is deterministic. However, the case where a is a random variable which is independentof W can easily be accomodated by enlarging the original probability space. Furthermore,a can as well be chosen to be equal to ∞.
The investment can operate in two modes, say “open” and “closed”. The transition fromone mode to the other one forms a sequence of decisions made by the owner. We model
19
these decisions by an (Ft)-adapted, finite variation, cadlag process I with values in 0, 1.We denote by I the set of all such processes which also satisfy
E
∫
[0,T [
Rs dIs < ∞. (27)
Given I ∈ I, It = 0 means that the investment is “closed”, whereas It = 1 means that theinvestment is “open”. Also, observe that assumption (27) puts a limit on how rapidly theowner can change the investment’s operating mode.
Given that the investment is “open”, the owner can decide on a production rate. Wemodel this by an (Ft)-progressively measurable process U with values in a compact subsetof the real line. We denote by U the set of all production rates.
Furthermore, we assume that the owner can totally abandon the investment at a timeof their choice. At abandonment, the investment is scrapped and subsequently presents nofurther profits or losses. An abandonment time τa is any (Ft)-stopping time τa ≤ T .
Now, the set of all decision strategies is
D = (I, U, τa) | I ∈ I, U ∈ U , τa ∈ S .
Given a decision strategy ∂ = (I, U, τa) ∈ D, and a finite (Ft)-stopping time τ ,
D(
(I, U, τa), τ)
=
(I ′, U ′, τ ′a) ∈ D | I ′
t1t<τ = It1t<τ, U ′t1t<τ = Ut1t<τ,
τ ′a ≥ τa ∧ τ, τ ′
a1τa<τ = τa1τa<τ, ∀t ≥ 0, P -a.s.
.
Clearly, Assumption 3 is satisfied.Given a decision strategy (I, U, τa), we denote by A the remaining amount of the com-
modity (resource) that can be produced. This satisfies the differential equation
dAt = −f(It, Ut) dt, A0 = a > 0,
where f > 0 is an appropriate function. Also, we define σ(I, U) = inf t ≥ 0 | At = 0,which is the time when the amount of the commodity (resource) that can be produced isexhausted.
Now, if decision strategy ∂ = (I, U, τa) ∈ D is adopted, the investment yields a flow ofpayoffs/costs according to the cumulative payoff process C(I,U,τa) defined by
C(I,U,τa)t =
∫ t∧τa
0
[
Ish(Ys, Us)1s≤σ(I,U) − IsKsto1σ(I,U)<s − (1 − Is)Kstc
]
ds
−∑
s≤t
[
KI(∆Is)+ + KO(∆Is)
−]
1t<τa − Ka1τa≤t,
where ∆It = It − It−. Here, h models the running payoffs/costs resulting from an “open”investment that can still produce positive quantities of the commodity. If in a time interval
20
[t, t+∆t] the investment is “open”, the commodity price is y and the production rate is set atu, then, depending on whether h(y, u) is positive or negative, the investment yields a profitor incurs a loss equal to h(y, u)∆t. The constant Ksto > 0 models the running costs resultingfrom an “open” investment after the amount of the commodity that can be produced has beenexhausted, whereas the constant Kstc > 0 models running “standby” costs resulting froma “closed” investment. The constants KI , KO > 0 are the costs resulting from “switching”the investment from a “closed” mode to an “open” one and vice versa, respectively, whereasKa > 0 is the abandonment cost.
In view of (27), observe that Assumption 4 is satisfied if
supU∈U
E
∫ T
0
Rs |h(Ys, Us)| ds < ∞.
Under this condition, Theorem 2 implies that the investment’s fair price at any finite (Ft)-stopping time τ given decision strategy (I, U, τa) ∈ D is given by
V(
(I, U, τa), τ)
= P - ess sup(I′,U ′,τ ′
a)∈D((I,U,τa),τ)
R−1τ E
[
1τ≤τ ′
a
(∫ τ ′
a
τ
Rs
[
I ′sh(Ys, U
′s)1s≤σ(I′,U ′) − I ′
sKsto1σ(I′,U ′)<s − (1 − I ′s)Kstc
]
ds
−∑
τ≤s<τ ′
a
Rs
[
KI(∆I ′s)
+ + KO(∆I ′s)
−]
− Rτ ′
aKa
)
| Fτ
]
.
With reference to Assumption 5, let any (I, U, τa) ∈ D, any finite τ ∈ S, and any A ∈ Fτ .Given any (I i, U i, τ i
a) ∈ D(
(I, U, τa), τ)
, i = 1, 2, if we define
I3t = It1t<τ + I1
t 1τ≤t∩A + I2t 1τ≤t∩Ac ,
U3t = Ut1t<τ + U1
t 1τ≤t∩A + U2t 1τ≤t∩Ac ,
τ 3a = τa1τa<τ + τ 1
a1τ≤τa∩A + τ 2a1τ≤τa∩Ac ,
we can verify that
(I3, U3, τ 3a ) ∈ D
(
(I, U, τa), τ)
and C(I3,U3,τ3a ) = C(I1,U1,τ1
a )1A + C(I2,U2,τ2a )1Ac .
As a consequence, Assumption 5 is satisfied, and the assertions of Theorem 3 follow.
8 Appendix: a martingale representation result
Since (Ft) is not the natural filtration of W , we cannot use directly the martingale rep-resentation theorem to express (Ft, P )-martingales as stochastic integrals with respect to
21
the Brownian motion W . We now prove that such an expression is actually true. The re-sult is taken from Zervos [Z] and is similar to the Fujisaki-Kallianpur-Kunita representationtheorem which is proved under more restrictive assumptions in Liptser and Shiryaev [LS,Theorem 5.20].
Proposition 4 Given a (F Pt , P )-local martingale M such that M0 = 0, there exists a locally
square integrable, (Ft)-progressively measurable process H such that
Mt =
∫ t
0
Hs · dPWs.
In particular, every (F Pt , P )-local martingale has a continuous, (Ft)-adapted version.
Proof. Since (F Pt ) satisfies the usual conditions, M has a version M ′ with cadlag sample
paths which, in view of Remark ??, is (Ft)-adapted. With reference to Proposition III.3.8in Jacod and Shiryaev [JS], LM ′ is an (Ft, P )-local martingale, and therefore, an (FP
t , P )-local martingale. Now, the martingale representation Theorem V.3.5 in Revuz and Yor [RY]asserts that there exists a locally square integrable, (Ft)-predictable process Y such that
LtM′t =
∫ t
0
Ys · dPWs, P -a.s., ∀t ≥ 0.
For this conclusion, we have also used the fact that every (FPt )-predictable process is P -
indistinguishable from an (Ft)-predictable process, which is proved in Dellacherie and Meyer[DM2, Appendice 1, Lemme 7].
Using the integration by parts formula and the fact that dPL−1t = L−1
t θt · dP Wt, wecalculate
L−1t
∫ t
0
Ys · dPWs =
∫ t
0
L−1s
[
Ys +
(∫ s
0
Yu · dP Wu
)
θs
]
· dP Ws.
By invoking Lemme 7 in Dellacherie and Meyer [DM2, Appendice 1] once again, we can seethat, since the processes L−1 and
∫ ·
0Ys·d
PWs are (FPt )-predictable (because they have contin-
uous sample paths), they are P -indistinguishable from (Ft)-predictable processes. (Note thatthe sample paths of these processes are not necessarily all continuous anymore.) As a con-sequence, since Y is (Ft)-predictable and, by the assumptions made, θ is (Ft)-progressivelymeasurable, the process H defined by
Ht = L−1t
[
Yt +
(∫ t
0
Ys · dP Ws
)
θt
]
,
is (Ft)-progressively measurable. Moreover, since all of the processes on the right hand sideare locally square integrable, the same is true for H .
22
Now, continuous, (Ft)-adapted versions of the processes∫ ·
0Hs · d
P Ws and∫ ·
0Hs · d
P Ws
are P -indistinguishable, and so,
M ′t =
∫ t
0
Hs · dPWs, P -a.s., ∀t ≥ 0.
However, P can be replaced by P here because the restrictions of the two measures on (Ω,Ft)are equivalent, for every t, and the proof is complete.
Remark 5 In view of the proof, it is clear that, as long as θ is (Ft)-predictable, H can bechosen to be (Ft)-predictable as well.
Our analysis has actually used the following result.
Corollary 5 Given a (F Pt , P )-local martingale M such that M0 = 0, there exists a portfolio
π satisfying Assumption 2 such that
Mt =
∫ t
0
Rs (σ′πs) · dPWs.
Proof. We choose π = R−1 (σ′)−1 H , where H is given by the preceding proposition.
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