Valuing portfolios of interdependent real options under exogenous and endogenous uncertainties

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European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Stochastics and Statistics

Valuing portfolios of interdependent real options under exogenous and endogenous uncertainties ^R

Sebastian Maier

^a,b,∗

, Georg C. Pflug

^c,b

, John W. Polak

^d

aDepartment of Civil and Environmental Engineering, Imperial College London, London, UK

bRisk and Resilience Program, International Institute for Applied Systems Analysis, Laxenburg, Austria

cDepartment of Statistics and Operations Research, University of Vienna, Vienna, Austria

dCentre for Transport Studies and Urban Systems Laboratory, Imperial College London, London, UK

a rt i c l e i n f o

Article history:

Received 1 March 2018 Accepted 22 January 2019 Available online xxx Keywords:

Stochastic optimisation Stochastic processes Real options portfolio Endogenous uncertainty

Decision/state-dependent uncertainty

a b s t r a c t

Althoughthe valueofportfolios ofrealoptions is oftenaffectedby bothexogenous and endogenous sourcesofuncertainty,mostexistingvaluationapproachesconsideronlytheformerandneglectthelat- ter.Inthispaper, weintroducean approachfor valuingportfolios ofinterdependentrealoptions un- der bothtypesofuncertainty. Inparticular, westudy alarge portfolio ofoptions(deferment, staging, mothballing,abandonment)underconditionsoffourunderlyinguncertainties.Twooftheuncertainties, decision-dependentcosttocompletionandstate-dependentsalvagevalue,areendogenous,theothertwo, operatingrevenuesandtheirgrowthrate,areexogenous.Assumingthatendogenousuncertaintiescanbe exogenised,weformulatethevaluationproblemasadiscretestochasticdynamicprogram.Toapproxi- matethevalueofthisoptimisationproblem,weapplyasimulation-and-regression-basedapproachand presentanefficientvaluationalgorithm.Thekeyfeatureofouralgorithmisthatitexploitstheproblem structuretoexplicitlyaccountforreachability– thatisthe samplepaths inwhichresourcestatescan bereached.Theapplicabilityoftheapproachisillustratedbyvaluinganurbaninfrastructureinvestment.

Weconductareachabilityanalysisandshowthatthepresenceofthedecision-dependentuncertaintyhas adversecomputationaleffects asit increasesalgorithmiccomplexityand reducessimulation efficiency.

We investigatetheway inwhichthevalueofthe portfolioand its individualoptionsareaffectedby theinitialoperatingrevenues,andbythedegreesofexogenousandendogenousuncertainty.Theresults demonstratethatignoringendogenous,decision-andstate-dependentuncertaintycanleadtosubstantial over-andunder-valuation,respectively.

© 2019 The Authors. Published by Elsevier B.V.

ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

A fundamental issue in real options analysis and decision- making under uncertainty is how to account correctly and ade- quatelyforthe multiplesources ofuncertaintyoccurringin most practical real-life situations. Inthesesituations it isgenerally as- sumed thatthe effectivesourcesofuncertaintyare purely exoge- nousand, assuch, are independentof both the actions takenby thedecisionmakerandthestateoftheunderlyingsystemaffected by these decisions. For example, in the case of investment in a

R This paper is a significantly expanded version of a paper first presented at the 21st Annual International Real Options Conference in Boston, MA (USA), in June/July 2017.

∗ Corresponding author at: Department of Civil and Environmental Engineering, Imperial College London, London, UK.

E-mail address: sebastian.maier@imperial.ac.uk (S. Maier).

newwindfarm,the windfarm’s performancedependson factors suchaslocation,timeofdayandthewindturbines’characteristics;

however,parameterssuchasthewindspeed,andconsequentlythe amountofpowergenerated,are independentoftheinvestor’sde- cision ofwhetherto build thewind farm ornot. Likewise, ifthe amount of power generated by such a wind farm is sufficiently smalland/ortherelevantwholesaleelectricitymarkettowhichthe poweris soldis comparatively large,then the underlying whole- sale priceof electricity, andconsequently the investor’spotential revenuesarealsoindependentoftheinvestor’sdecision.

Thereare,however,manypracticalsituationsinwhichtherel- evantsourcesofuncertaintyareendogenous,i.e.dependentonthe decisionmaker’sactionsortheunderlyingsystem’sstate,orboth.

In the case of the wind farm example, if the above-mentioned conditions are violated, i.e. if the new wind farm is sufficiently large and/or the electricity market relatively small, then the introduction ofa newwind farm will affectthe wholesale price https://doi.org/10.1016/j.ejor.2019.01.055

0377-2217/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

Please cite thisarticleas: S.Maier, G.C. Pflugand J.W.Polak, Valuingportfolios of interdependent realoptions underexogenous and

ofelectricity and hence the investor’sfuture revenues. Similarly, although the “off-the-shelf” cost of new wind turbines may be knownandafeasibilitystudymayprovideaconstructioncostes- timate,the actual cost ofbuilding a new wind farm will not be known until the investor actually builds it. During the building process,theinvestorlearnsandrevealsthewindfarm’struecapi- talcost.Iftheinvestorwantstosellthewindfarmattheendofits lifetime,intheabsenceofasecondhandmarket,theresalevalue willdepend on its “state”, whichmay includesuch factors asits lifetime,assetvalue,wearandtear,anddecommissioningcost.

Despite the prevalence of exogenous andendogenous sources ofuncertainty in many real-life situations, there remains a need fora unified approach that accounts forboth when real options analysisisusedtoevaluatepracticalinvestmentproblems.Includ- ingbothtypesofuncertaintyinarealoptionsapproachhasrarely been studied in the related literature (Ahsan & Musteen, 2011).

Althoughportfolio ofreal options approacheshave been applied whenthere isonly exogenous uncertainty,there is aneed to in- cludebothtypesbecausethatenablesdecision-makerstomanage thetwo uncertainty types simultaneously(Otim& Grover, 2012).

Some authors have therefore suggested that future work should examinetheinteractionsbetweendifferentsources ofuncertainty and the portfolio’s individual options, e.g. see Tiwana, Keil, and Fichman (2006) and Li, James, Madhavan, and Mahoney (2007). Morerecently,acriticalreviewofTrigeorgisandReuer(2017)has suggestedfourextensions,threeofwhichareaddressedhere:port- folios of interdependent real options, multiple sources of uncer- tainty,andendogenousresolutionofuncertaintythroughlearning.

This paper introduces a valuation approach for portfolios of interdependent real options under exogenous and endogenous sources ofuncertainty. Studyingthe problemofa sequential and partiallyreversible investment project,we considera portfolio of options to: defer investment; stage investment; temporarily halt expansion; temporarily mothballthe operation; andpermanently abandontheprojectduringeitherconstructionoroperation.Inthe problemstudiedhere, theportfolio’s valueisaffectedbyfourun- derlying uncertainties. Ofthese, the project’sactual cost to com- pletion and its salvage value are decision- and state-dependent, respectively.Theseuncertaintiesevolveendogenously,whereasthe operatingrevenuesandtheirgrowthrateevolveexogenously.Sim- ilartoMaier,Polak,andGann(2018),weuseaninfluencediagram to graphically model the interdependencies between the portfo- lio’srealoptions andmathematicallytranslate theseintoaset of constraints.Theconstraintsandthestochasticprocessesdescribing theuncertainties’dynamicsarethenintegratedintoamulti-stage stochasticoptimisationproblemwhichisformulated asastochas- ticdynamicprogram.

Our decision model is a stochastic dynamic, discrete-time (Markovian) model: the transition of the state S_t of the under- lying system at time t to state S_t+ after a time increment is driven by our decisions as well as by the random processes describingthe uncertainties. Here we distinguish between exoge- nousandendogenoussources ofuncertainty.Modelledasstochas- ticMarkovian processes, the evolution of endogenous uncertain- tiesdependsonthedecisionmaker’sstrategyorthesystem’sstate, orboth,whilethose ofexogenous uncertaintiesare unaffectedby decisions and states. Compared to standard real option models, models with decision- or state-dependent random variables are much moredifficult to solve by simulation-and-regression meth- odssinceitisgenerallyimpossibletouserandomdeviateswhich have been sampled once at initialisation. However, as shown in SupplementaryMaterialC,itissometimespossiblebya reformu- lationtermed exogenisation touse the same fixed set ofrandom deviatesevenforendogenousuncertainty.Inthispaperitisthere- foreassumedthat endogenousuncertaintycan be exogenised. To approximate the value of this optimisation problem, we use an

extensionofstandardsimulation-and-regressionmethods(e.g.see Cortazar,Gravet,&Urzua,2008;Glasserman&Yu,2004;Longstaff

&Schwartz, 2001;Nadarajah,Margot,& Secomandi,2017; Tsitsik- lis&VanRoy, 2001)whosebasicstructureandprinciplesare de- scribedinSupplementaryMaterialD.

Themaincontributionsofthisworkareinthefollowingthree areas:

(1) Our model extends the standard models in several ways:

(i) standard models generally assume a single time step, meaning that the time evolves from t tot+1 after a de- cision has been madeat t, buthere the decisions at∈AS_t

imply the time delay h,i.e. the moment in time for the next decision ist+h. This makes our model moreflexi- bleasitallows usto addressproblemswithmultiple time steps;(ii)unlike standard problems,heretherandom vari- ables

ξ

^{t appearinginthetransitionfunctiondependonthe}

state ofthesystemSt.Toenablecomputationaltractability, however, we show how exogenised random factors

ε

t can be usedinsteadby assuming that the

ξ

^{t can be writtenin}

the form

ξ

^t(

ω

)=f(^St(

ω

),

ε

^t(

ω

)), where

ε

^t(

ω

^{) isindepen-}

dentofSt(

ω

^{) whenfollowingasamplepath}

ω

^{(see Supple-}

mentaryMaterialC);(iii)weexplainhowtheparametricre- gression modelcan be madedependent on the state St to account for the circumstance that some basis functions of theparametricmodelareimpossibleforsomestates.

(2) Wepresentanextended algorithmtoaccountforcomplex- ities induced by the extended model. First, compared to standard algorithms forproblemswithonly exogenousun- certainty, the incorporation of the decision-dependent un- certainty results in an additional path-dependency. We therefore propose a forward induction procedure in which the resourcestate space generation isinterleaved withthe MonteCarlo samplingsteps of theinformation state space generation.Secondly,we includereachabilityinourforward pass to account for the circumstance that some resource states may not be reachable, or only in a subset of sam- ple paths. This is a key feature as it enables us to design anefficientbackward approximationalgorithm thatconsid- ers only reachable resource states and the set of paths in which they can actually be reached. Thirdly, we describe how the structureof the problemto be solved can be ex- ploitedthroughdynamicallyandappropriately adaptingthe setofbasisfunctionsusedintheparametricmodelinorder toavoidnumericalinaccuraciesrelatedto1(iii).

(3) Wedemonstrate theapplicability ofourapproachandper- formaset ofdetailednumericalanalyses usinganillustra- tive exampleofan urban infrastructureinvestment inLon- don. We first conduct a reachability analysisto investigate the complexityof the problem in terms ofthe number of both resource states andsample paths, andshow that the presenceoftheendogenous,decision-dependentuncertainty generallyleadstoan increaseinalgorithmic costanda de- creaseinsimulationefficiency.Subsequently,weinvestigate thesensitivityofthevalueoftheportfolioanditsindividual optionstotheinitiallevelofannualrevenues, aswellasto the degreesofexogenous and endogenousuncertainty. We illustrate that the availability of realoptions is more valu- able for low values of initial revenues, and that the port- folioissubstantially morevaluable thanindividual options.

We also illustrate that theportfolio value increasesmono- tonically inboththeexogenous andtheendogenous, state- dependent uncertainty, but that there is a non-monotonic effect withrespect to the endogenous, decision-dependent uncertainty. More importantly, this work shows that ignoringdecision-andstate-dependentuncertaintycanlead Please citethisarticle as:S. Maier, G.C.Pflug andJ.W. Polak,Valuing portfolios ofinterdependent realoptions underexogenous and

to substantial over- and under-valuation, respectively, and alsoprovidesthereasonsforthis.

The rest of this article is organised as follows: Section 2 re- viewstherelevantliteraturewithanemphasisontheoperational research as well as on the finance and management literature.

Section3describestheinvestmentproblembyspecifyingboththe portfolioofinterdependentrealoptions(Section3.1)andthesetof uncertainties(Section3.2)consideredinthiswork.InSection4we present the modelling andvaluation approach together with the simulation-and-regression-basedvaluationalgorithm(Section4.3).

The approachandthealgorithm arethenapplied tothereal-case of a district heating network expansion investment in the Lon- don borough of Islington (Section 5). Results are presented and discussed in Section 5.4. Finally, some concluding remarks and suggestions for future research are provided in Section 6. Addi- tionalinformationregardingtheexogenisationofendogenousun- certainty,thebasicalgorithm,inputdataandillustrationofsample pathsisprovidedassupplementarymaterial.

2. Literaturereview

The classification ofuncertaintiesintoexogenous andendoge- nous hasreceived considerable attentionin differentbranchesof literature, andimportantly inthe operational research aswell as inthefinanceandmanagement literature.Withregard tothefor- mer, tothe bestofourknowledge, theworkof Jonsbråten,Wets, and Woodruff (1998)was the first to classify the formulationof stochasticprogramsinto“standard” formulationswithdecisionin- dependent random variables and “manageable” formulations, in which the distribution of the random variables is dependent on decisions.Callingtheformer“exogenous uncertainty” andthelat- ter“endogenousuncertainty” (Goel&Grossmann,2004),Goeland Grossmann(2006)specifiedthewayinwhichdecisionscanaffect thestochasticprocess– whichdescribestheevolutionofanuncer- tainparameter(Kirschenmann,Popova,Damien,&Hanson,2014)– by presenting two types of endogenous uncertainty. The first is whenthedecisionalterstheprobability distribution,whereasthe second relates tothe decisionaffectingthe timing ofuncertainty resolution,aprocessoftendescribedasinformationrevelation.

Considering the above specification of endogenous uncertain- ties, several relevant works have appeared in the operations re- search literature over the last few decades. As forthe first type ofendogenousuncertainty,Pflug(1990)wasthefirstto takeinto account decision-dependentprobabilities ina stochasticoptimisa- tionproblembyconsideringacontrolledMarkovchainwherethe transitionoperatordependsonthecontrol,i.e.thedecision.Other relevantarticlesrelatedtothistypeareinthecontextofstochas- tic network problems (Held & Woodruff, 2005; Peeta, Salman, Gunnec, & Viswanath,2010),globalclimate policy (Webster, San- ten, & Parpas, 2012) andnatural gas markets (Devine, Gabriel, &

Moryadee,2016).Bycontrast,thesecond typeofendogenousun- certainty has received considerable more attention in the litera- ture. The first work relatedto thistype was(Goel& Grossmann, 2004),whichpresentedastochasticprogrammingapproachforthe planningofaninvestmentintoagasfieldwithuncertainreserves representedthroughadecision-dependentscenariotree.Otherrel- evant works include the optimisation of R&D project portfolios (Solak, Clarke, Johnson, & Barnes,2010) andpharmaceuticalclin- icaltrialplanning(Colvin&Maravelias,2010;2011).

Moreover, several works have incorporated both the second type ofendogenousuncertaintyandexogenous uncertaintyinthe formulationofstochasticprogrammes.Forgenericproblemformu- lations andsolutionstrategiesseethe rathertheoreticalworksof Dupaˇcová (2006),GoelandGrossmann(2006),andTarhan, Gross- mann,andGoel(2013).Recentadvances andsummariesover ex-

istingcomputationalstrategieshavebeenpresentedbyGrossmann, Apap, Calfa, Garca-Herreros, and Zhang (2016) and Apap and Grossmann (2017). However, although almost all publications of thisbranchofliteraturerefertotheclassificationandspecification ofJonsbråtenetal.(1998)andGoelandGrossmann(2006),respec- tively,MercierandVanHentenryck(2011)arguedthatproblemsin which merely the observation of an uncertainty dependson the decisions,buttheactualunderlyinguncertainty isstillexogenous (=secondtypeofendogenousuncertainty)shouldbeclassifiedas

“stochasticoptimizationproblemswithexogenousuncertaintyand endogenousobservations”.

Unliketheoperationalresearchliterature,thefinanceandman- agement literature appears to be rather ambiguous, even some- what inconsistent when it comes the classification of uncertain- ties. Indeed, although the importance of taking this distinction into account has been widely recognised in this branch of lit- erature, especially in works related to the field of real options (Bowman& Hurry, 1993; Folta,1998; Li, 2007; Oriani& Sobrero, 2008),thereisnoclearandwidelyaccepteddefinition.Forexam- ple,Pindyck(1993)distinguishesbetweentechnicalandinputcost uncertaintywhilenotingtheirdifferenteffectsoninvestmentdeci- sionsastheseincentiviseinvestingandwaiting,respectively.Build- ing upon thisdistinction, McGrath(1997) calledfora third form ofuncertaintythatliesin-between.Furthermore,McGrath,Ferrier, and Mendelow (2004) refers to the exogenous and endogenous resolution ofuncertainty through the passing of time and learn- ing,respectively.Bycontrast,VanderHoekandElliott(2006)took noteofuncertaintiesthatare state-dependentratherthan depen- dentontheoptionholder’sdecisions.

Various researchers have applied real option approaches to valuation problems with both exogenous andendogenous uncer- tainty. Generalising the work of Roberts and Weitzman (1981), Pindyck(1993)evaluated a staged-investmentwithtechnical(en- dogenous) and input cost (exogenous) uncertainty using a finite difference method. Other relevant articles considered both types of uncertainty in the context of information technology invest- ment projects (Schwartz & Zozaya-Gorostiza, 2003), patents and R&Dprojects(Schwartz,2004; Schwartz&Moon,2000),pharma- ceuticalR&Dprojects (Hsu &Schwartz, 2008;Pennings &Sereno, 2011), product platform flexibility planning (Jiao, 2012), andnu- clearpowerplantinvestments (Zhu,2012).However, accordingto Miltersen and Schwartz (2007), the algorithms of Miltersen and Schwartz(2004),Schwartz(2004),Hsu andSchwartz(2008),and Zhu (2012), which are plain extensions of the basic algorithm of Longstaff and Schwartz (2001) for single American-style op- tions,“cannoteasilyhandletemporarysuspensionsofthe” invest- ment project nor isolate the options’ values. Also, these works considered only the abandonment option, rather than a real op- tionsportfolio.Withregardtostate-dependentuncertainty,Sbuelz andCaliari (2012) studied theinfluence ofstate-dependentcash- flow volatility on the investment decisions related to corporate growth options, whereas Palczewski, Poulsen, Schenk-Hopp, and Wang (2015) examined optimal portfolio strategies under stock pricedynamicswithstate-dependentdrift.

Nevertheless, these real option approaches are rather inflexi- ble and restricted in terms of the size of the real options port- folio, the number and types of uncertainties as well as the valuation method applied. This paper takes a fundamentally dif- ferent approach by introducing a framework for valuing portfo- lios of real options under exogenous and endogenous uncertain- ties. In particular, we study an investment problem with sev- eral types of real options (deferring, staging, mothballing, and abandoning),twoexogenousuncertainties(operatingrevenuesand their growth rate), and two endogenous uncertainties (decision- dependentcosttocompletionandstate-dependentsalvagevalue).

Using an illustrative example of a district heating network in Please cite thisarticleas: S.Maier, G.C. Pflugand J.W.Polak, Valuingportfolios of interdependent realoptions underexogenous and

London, we provide portfolio insights and find that the port- folio value increases monotonically in both the exogenous (rev- enue) and the endogenous, state-dependent (salvage value) un- certainty, but that the endogenous, decision-dependent (cost to completion)uncertaintyhasanon-monotoniceffect.Thiseffectis largelyduetotheavailabilityofabandonmentoptions,whoseval- ues– enabledbypartial reversibility– are directlyandindirectly drivenby state- anddecision-dependentuncertainty,respectively.

Mostnotably, we show that, in general, ignoring the former re- sultsinunder-valuation,whereasignoringthelatterleadstoover- valuation,thereby highlighting theimportance of accountingcor- rectlyforuncertainty.

3. Theinvestmentproblem

Inthissection,wepresenttheinvestmentproblemstudiedhere byspecifyingboththeportfolioofinterdependentrealoptionsand thesetofunderlyinguncertainties.

3.1.Portfolioofinterdependentrealoptions

We studythe problemof a decision makerwanting todeter- minethevalueofasequentialandpartiallyreversibleinvestment inaprojectwhosestage-wiseexpansion(development)canbede- ferred,temporarilyhaltedand/orabandonedaltogether, and,once operating,whosecashflowgeneratingassetcanbeuseduntilthe end of the asset’s project life in T₃^max time periods, temporarily mothballedand/orabandonedearly.

Representingthesetofflexibilitiesasaportfolioofinterdepen- dentrealoptions,theportfolio’ssingle,well-definedoptionsare:

(a) Optiontodeferinvestment:Insteadofstartingimmediately attime0,thedecisionmakermaychoosetodeferthestart ofthe expansionuntilthe expirationoftherightto under- takethisinvestmentinT₁^maxtimeperiods,withoutincurring anydirectcostsassociatedwithdeferring.

(b)Optiontostage investment:Asthedevelopmenttakestime to complete, the decision maker can invest at a rate of 0<Ct≤I^maxinperiodtaslongastheremaininginvestment cost Kt atthebeginning ofperiodt isgreater than0 – i.e.

while the construction is not yet completed –, where I^max andK₀ arethe maximumrateofinvestmentandtheinitial (expected)costofcompletion,respectively.

(i) Optionto temporarilyhalt expansion: Ifconditions turn out tobeunfavourable,thedecisionmakercanhalttheex- pansion(i.e.setCt=0)atacostofC^d,h,maintainthehalted expansionforatotalofT₂^max timeperiodsataperiodiccost ofC^h,and,ifdesirable,resumedevelopmentatacostofC^h,d. (ii) Option toabandon the projectduringconstruction (i.e.

whenKt>0):Whetherdevelopingorhalted,theprojectcan be permanentlyabandonedatanygivenpointintime tfor thesalvagevalueXt,whichisassumedtocontain anycosts thatabandonmentduringconstructioninvolves.

(c)Option to temporarily mothballthe operation: Ifoperation of the asset becomesuneconomic, the decisionmaker can mothballtheoperatingassetata costofC^o,m,maintainthe mothballedassetataperiodiccostofC^m,and,ifconditions become favourable again, reactivate the asset at a cost of C^m,o.

(d) Option to abandonthe project duringoperation (i.e.when Kt=0): Whether operating or mothballed, the decision makerretainstherighttopermanentlyabandontheproject at anytime tfor itssalvage value Xt,which isassumed to containallcostsrelatedtoabandoningduringoperation.

Theabovedescribedindividualrealoptionsarewell-knownand have beenwidely examined in the real options literature – such as “time to build” effects in Majd and Pindyck (1987) –, for an overviewseeTrigeorgis(1996).

3.2. Characterisationofuncertainties

This study considers four sources of uncertainty – also re- ferred to asstochastic factors orrandom variables – denoted by Kt,Vt,

μ

^{t andXt,}representingtheproject’sactual costtocomple- tion attime t, the revenues(net cash flow)generated by opera- tioninperiodt,thegrowthrateofrevenuesintandthesalvage value at time t,respectively. The first andthe fourthuncertainty are decision- and state-dependent, respectively. These uncertain- tiesevolveendogenously,whereasthedynamicsofthesecondand third factorare exogenous. While thechoice of stochastic factors obviously depends on the specific investment problem at hand, ourchoice, which issufficient forthe purposeof thiswork, cov- ersseveralrelevantandwidelyapplicable stochasticfactors, sois importantformanypracticalapplicationswherethesourcesofun- certainty areexogenous andendogenous.Unlikeprevious studies, whichhaveconsideredtheseuncertaintiesmostlyinisolation,here weconsiderthefouruncertaintiesjointlysincetheyarerelevantto mostprojects’ major phasesincludingconstruction (cost tocom- pletion),operation(revenues),anddecommissioning/disposal(sal- vagevalue).Notethattheconsiderationofastochasticgrowthrate allowsustomodelrandomvariationsinthegeneraleconomiccon- ditions andaddscomplexityto the problem, enablingusto both demonstratethecapabilityandtesttherobustnessofourproposed valuationapproach.

The fourstochastic factorsare describedby discrete-timeran- domwalkswithdrift,inageneralformby:

M_t+=

ϕ

^tMt+ft

(

^Mt,

θ

¹

)

⁺

σ

^t

(

^Mt,

θ

²

)

^√

ε

t^m+, (1) where

ϕ

^{t is a}discountingmultiplier,ft isthe driftfunction, ^is the time step,

σ

^{t isthe diffusion function, and}

ε

t^m+ is thedriv- ingzero-meanprocess.Notethatforendogenousstochasticfactors, theparameters

θ

1 or

θ

2,orbothdependonthedecisionorstate, orboth.The driving process

ε

^mt+ isalways Gaussian whitenoise (GWN),i.e.astandard normalrandom variablewhoseincrements are iid, but drivers for different stochastic factors may be corre- lated.¹ Table1summarisesthestochasticfactorsconsideredhere.

Thedynamicoftheproject’sactualcost tocompletion,Kt,de- pendsontherateofinvestment,0≤C_t≤I^max,chosenby thedeci- sionmaker,andisgivenby:

Kt+=Kt−Ct

+

σ

k

CtKt

ε

t^k+, (2)

where

σ

kisthe degreeoftechnicaluncertainty. Theabove equa- tion is a discrete approximation of the controlled diffusion pro- cessproposedbyPindyck(1993).AsanalyticallyshownbyPindyck (1993),Schwartz and Zozaya-Gorostiza (2003)and referred to as

“bang-bangpolicy” bySchwartz(2004),theoptimalrateofinvest- ment is either 0 or I^max, i.e. Ct∈{0, I^max}, because the processes (2)and(3)–(5)areuncorrelated.

Therevenuesreceivedattimetforoperationbetweentandt+ ,Vt,andtheir rateofgrowth,

μ

^t,evolve exogenouslyaccording to:

Vt+=e^−κvVt+

(

¹−e^−κv

)

^V0

(

¹+

μ

^tt

)

+

σ

v

1−e^−2κv 2

κ

v

ε

^vt+,

(3)

1This, of course, does not change the exogenous and endogenous dynamics of uncertainties.

Please citethisarticle as:S. Maier, G.C.Pflug andJ.W. Polak,Valuing portfolios ofinterdependent realoptions underexogenous and

Table 1

Summary of stochastic factors considered in this study.

Description Factor Defining eq. Dynamics Driving process ^a

Cost to completion K t (2) Decision-dep. GWN, independent of (3) –(5) Operating revenues V t (3) Exogenous GWN, correlated with (4) and (5) Growth rate μt (4) Exogenous GWN, correlated with (3) and (5) Salvage value X t (5) State-dep. GWN, correlated with (3) and (4)

a Gaussian white noise.

μ

t+=e^−κμ

μ

^t+

(

¹−e^−κμ

) μ

^¯ +

σ

μ

1−e^−2κμ

2

κ

μ

ε

^μt+, (4) where

σ

vand

σ

μ arethestandarddeviationsofchangesinVt and

μ

t, respectively, as well as

κ

v and

κ

μ are positive mean rever- sion coefficients that describe the rate atwhich the correspond- ing factors convergeto their lineartrend,V₀(¹+

μ

^tt), andlong- termaverage,

μ

¯,respectively.The nestedmodel(3)–(4)issimilar to the discrete versions of Schwartzand Moon (2001), who also used an Ornstein–Uhlenbeck process² to describe the evolution of

μ

^{t.FortheevolutionofVt,however,weapplyatrendingOrnstein–}

UhlenbeckmodelwithstochasticlineartrendadaptedfromLoand Wang (1995), which is more realistic than both the revenue dy- namicsinSchwartzandMoon(2001)andthegeometricmeanre- versionwithdeterministicexponentialtrend(i.e.V₀e^μ¯t)considered byMetcalfandHassett(1995).

Thestate-dependentsalvagevalueobtainedforabandoningthe project attime t, X_t, isa function ofthe expectedasset value at time t, Zt, which is a deterministic function of the state St (see (6)), andofa homoscedasticnoise term(i.e. errorindependentof the state), which is considered to be random. The salvage value processisdescribedby:

Xt+=Zt++

σ

^xZt+

ε

t^x+, (5) where

σ

^{x isthe standard deviationofXt. Unliketheexisting ap-}

proaches that allow forstochastic salvage (or abandonment) val- ues such as Myers and Majd (1990), Adkins and Paxson (2017), which assume these values evolve exogenously, we introduce a state-dependentsalvage value assuggested inVan der Hoek and Elliott (2006),thereby representone of themany practical situa- tions inwhich the salvage value depends on endogenousfactors (Trigeorgis,1993).Itisimportanttonote thatby “state” weactu- allymean its“resource” component (see Section4.1), ratherthan its “information” component, becausethe latter’sthree stochastic factorsgivenby(2)–(4)are,ofcourse,state-dependenttoobecause Markovian.

4. Methods

Thissectioncontainsthemodellingoftheinvestmentproblem as a multi-stage stochastic decision problem, the formulation of the valuation problemas a discrete stochastic dynamicprogram, andthedescriptionofthevaluationalgorithmapplied.Asummary ofthenotationusedispresentedinAppendixA.

4.1. Modelling

The flexibilities available to the decision maker when having the portfolio of interdependent real options of Section 3.1 are shownby the influencediagram in Fig.1. Itcontains nine nodes

2This simple mean-reverting process is more realistic than a geometric Brown- ian motion process in problems that involve natural gas and electricity price uncer- tainty (such as district heating networks) given that the underlying price processes in general exhibit mean reversion.

ofwhich fiveare decisionnodes andfourare terminal nodes,as wellas18transitionsthatlinkthesenodes.Thesetofnodes,de- cision nodes and transitions is given by N=

{

¹,2,...,9

}

, N^d=

{

¹,3,5,6,8

}

^andH=

{

¹,2,...,18

}

,respectively, andtheduration oftransitionh∈Hishtimeperiod(s).Tohelpunderstandthein- tuitionbehindFig.1seetheinfluencediagramforacomparatively simpleAmerican-styleoptioninMaieretal.(2018).

Thestateoftheinvestmentprojectattimetiswrittenas:

St=

(

t,

Nt

,Tt,Q

t Rt

,K

t,Vt

,

μ

^t,X

^t

It

)

, (6)

whereN_t∈N isthenodeattimet;T_t isthetimeleft att tode- fer investment/halt expansion/use the developed asset;Qt is the amountinvesteduptotimet;andKt,Vt,

μ

^{t andXtareasdefined}

inSection3.2.ThefirstfourvariablesofS_tarepartoftheresource stateRt,whereastheinformationstateIt ismadeup oftheprob- lem’sfourrandomvariables,twoofwhichareexogenousandtwo areendogenous.

Toeach decision node n∈N^d we associatebinary (0–1)vari- ables a_th in such away thata_th=1 indicates that transitionhis madeattime t and0otherwise. Itis clearthat the actionspace b^D(N_t),whichrepresentsthesetofoutgoingtransitionsofnodeN_t, isgivenby

b^D

(

^Nt

)

=

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

{

¹,2,3

}

, ifNt=1,

{

^4,5,6,7

}

^{, ifNt=3,}

{

^8,9,10

}

^{, ifNt=5,}

{

¹¹,12,13,14

}

, ifNt=6,

{

^15,16,17,18

}

^{, ifNt=8,}

{}

^{, otherwise.}

(7)

The decision variables a_t=(^ath)h∈b^D(N_t) must satisfy the feasible regionAS_t,whichdescribesthesetoffeasibletransitionsgivenSt

andisdefinedbythefollowingconstraints:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

h∈b^D(Nt)a_th=1,

∀

^Nt∈Nd,

(

⁸

)

at1T₁^max<Tt+T₁^max,

(

⁹

)

a_thTt=0,

∀

^h∈

{

^3,12,16

}

^,

(

¹⁰

)

at5Kt=0,

(

¹¹

)

(

¹−at5−at7

)

^K0<Kt+K0,

(

¹²

)

a_thT₂^max<Tt+T₂^max,

∀

^h∈

{

^6,9

}

^,

(

¹³

) (

¹−a_th

)

^T3^max<Tt+T₃^max,

∀

^h∈

{

^12,16

}

^,

(

¹⁴

)

wherea_th∈

{

⁰,1

}

,

∀

^h∈H.The meaningof theseconstraintsis as follows:(8) enforcesthat exactly one transitionismadeat ade- cision node;(9) and(13) ensure the investment can be deferred andtheexpansionhalted,respectively,onlyifthereisenoughtime left; (10) makes sure the development opportunity can only ex- pireat Tt=0 butnot before, and, together with(14), thesecon- straints make surethe developed project is completed at T_t=0; and,finally,(11)ensures thattheasset’soperationcanonlybegin ifKt=0,atwhichpointthedevelopedassethastobeabandoned dueto(12)ifnotoperated.

The transition function, which is generically written as S^M(^St,at,W_t+

h), describes the evolution of St from t to t+h

Please cite thisarticleas: S.Maier, G.C. Pflugand J.W.Polak, Valuingportfolios of interdependent realoptions underexogenous and

1 Undeveloped

2 Expired

Developing 3 6

Operating

4 Unfinished

9 Completed

5 Halted

7 Uncompleted 8 Mothballed Defer (1)

Continue (4) Continue (11)

Reactivate (15)

Mothball (13)

Maintain (17)

Abandon (18) Complete (16)

Develop (2)

Operate (5)

Resume (8)

Halt (6)

Maintain (9)

Abandon (10) Expire (3)

Abandon (7) Abandon (14)

Complete (12)

Fig. 1. Flexibilities provided by portfolio of interdependent real options.

after having made decision at with respect to AS_t and learned newinformationW_t+

h. Itiscomposedofthe resourcetransition functionS^R(·)^:Rt→R_t+

h andtheinformationtransitionfunction S^I(·)^:It→I_t+

h.With regard tothe former, thetransition oft is trivialasitsimplyevolvestot+h;thetransitionofN_tisimplic- itlygivenbytheadjacencymatrix(notshownhere)ofthedirected graph(N,H)^{underlyingtheinfluencediagram;thetransitionofT}t

isgivenby:

T_t+h=

⎧ ⎪

⎨

⎪ ⎩

max

{

^Tt−

h,0

}

^{, ifa}th=1,h∈H1, T₂^max, ifat2=1, T₃^max−

5, ifa_t5=1,

Tt, otherwise,

(15)

where T₀=T₁^max and H1=

{

¹,6,9,11,13,15,17

}

^{; and the transi-}

tionofQtisgivenby:

Q_t+h=

Qt+I^max

h, ifa_th=1,h∈

{

^2,4,8

}

^,

Qt, otherwise, (16)

where Q₀=0. In contrast to the deterministic transitions of the variables of Rt, the information state variables evolve generally stochasticallyaccordingto:

Kt+h=

⎧ ⎨

⎩

max

Kt−I^max

h

+

σ

k

I^maxKt

h

ε

_t+^k_h^,0

, ifa_th=1,h∈

{

^2,4,8

}

^,

Kt, otherwise,

(17)

V_t+h=e^−κvhVt+(¹−e^−κvh)^V0(¹+μtt)+σv

1−e^−2κvh

2κv ε^v _t+h, (18) μt+h=e^−κμhμt+(¹−e^−κμh)μ^¯+σμ

1−e^−2κμh

2κμ ε^μ_t+h, (19)

X_t+h=Z_t+h(^St+h)+σxZ_t+h(^St+h)ε_t+^x_h, (20) whereZt(St),theexpectedassetvalueattimet,isgivenby:

Z_t(^St)=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

−α^Imax, ifN_t=3,K_t>0, γ^Qt, ifNt=3,Kt=0,

−β^Imax, ifNt=5, γ^Qte^{−ζ (T}3^max−Tt) , ifN_t=6, δ^Qte^{−ζ (T3max−Tt)} , ifNt=8,

0, otherwise,

(21)

where α≥0 and β≥0 define the expected abandonment cost when Developing or Halted, respectively; γ≥0 and δ≥0 are pay-

outratiosdeterminingtheexpectedassetvalue whenOperatingor Mothballed,respectively;andζ istheperiodicdepreciationrate.

Lastly,thepay-off functionisrepresentedby:

t

(

^St,at

)

=−I^max

(

2at2+

4at4

)

+Vt

(

^at5+at11

)

+Xt

(

^at7+at10+at14+at18

)

+Xt

(

^at12+at16

)

−C^d,hat6−

(

^Ch,d+I^max

8

)

^at8−C^h

9at9

−C^o,mat13+

(

^Vt−C^m,o

)

^at15−C^m

17at17, (22) wherethefirsttwotermsontheright-handsiderepresentthecost fordeveloping andthe incomefromoperations, respectively; the secondline’stermsrepresentthenetincomefromabandoningand completing, respectively; the third line contains costs related to halting,maintaining andresuming (during development),respec- tively;andthelast line’stermsrepresentthecostofmothballing, thenetincomefromreactivatingandthemaintenance costwhen mothballed, respectively. Note that, for simplicity, it is assumed thatcompletingtheproject– bymakingeithertransition12(when Operating)ortransition16(whenMothballed)– resultsinapay-off ofthesalvagevalueXt,whichthusrepresentstheproject’sresidual value.

4.2. Valuationproblem

Thevalueoftheportfolioofinterdependentrealoptionsattime 0givenstateS₀,G₀(S₀),isobtainedbysolvingthefollowingmulti- stagestochasticoptimisationproblem:

G0

(

^S0

)

=max (^at)t∈T

E

t∈Te^−rt

t

(

^St,at

)

S0

, (23)

where S₀=(⁰,1,T₁^max,0,K₀,V₀,

μ

0,X₀)^{, at}=(^ath)h∈b^D(N_t), a_th∈{0,1}, at∈AS_t, T is the set of decision times, S_t+

h=

S^M(^St,at,W_t+

h),andristhediscountrate.

Applying Bellman’s well-known “principle of optimality”, the stochastic optimisationproblemin(23)canbe solved recursively, withthestochasticdynamicprogramming(SDP)recursionforcal- culatingtheoptimalvalueofbeinginstateSt givenby:

Gt

(

^St

)

=max

at

t

(

^St,at

)

+E

e^−rhGt+h

(

^St+h

)

St,at

(24)

s.t. a_th∈

{

^0,1

}

^,

∀

^h∈bD

(

^Nt

)

^{, (25)}

at∈ASt, (26)

Please citethisarticle as:S. Maier, G.C.Pflug andJ.W. Polak,Valuing portfolios ofinterdependent realoptions underexogenous and