Igcc and Ngcc

8/13/2019 Igcc and Ngcc

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Technology choice for power plants

under mean reversion in fuel prices

Luis M. AbadieBilbao Bizkaia Kutxa

Gran Va, 3048009 Bilbao, SpainTel +34-607408748

Fax +34-944017996E-mail: [email protected]

Jos M. ChamorroUniversity of the Basque Country

Dpt. Fundamentos del Anlisis Econmico IAv. Lehendakari Aguirre, 83

48015 Bilbao, SpainTel. +34-946013769Fax +34-946013891

E-mail: [email protected]

February 6th 2006

Abstract

In this paper we analyze the choice between two technologies forproducing electricity. In particular, the rm has to decide whetherand when to invest either in a Natural Gas Combined Cycle (NGCC)power plant or in an Integrated Gasication Combined Cycle (IGCC)power plant. Instead of assuming that fuel prices follow standard

geometric Brownian motions, here they are assumed to show meanreversion, specically an inhomogeneous geometric Brownian motion.

First we use the opportunity to invest in a NGCC power plant

Earlier versions of this paper have been presented under the title "Valuing exibility:The case of an Integrated Gasication Combined Cycle power plant". We would liketo thank seminar participants at the 9th Conference on Real Options (Paris 2005), XIIIForo Europeo de Finanzas (Madrid 2005) and the 1st Annual Congress of the SpanishAssociation for Energy Economics (Madrid 2006) for their helpful comments. We remainresponsible for any errors. Chamorro acknowledges nancial support from research grant9/UPV00I01.I01-14548/2002.

1


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2

as a base case, deriving the optimal investment rule as a function

of natural gas price and the remaining life of the right to invest.Additionally, the analytical solution for a perpetual option to investis obtained.

Then we turn to the IGCC power plant which may burn eithercoal or natural gas. We analyse the valuation of an operating plantwhen there are switching costs between modes of operation, and thechoice of the best operation mode. This serves as an input to computethe value of the option to delay an investment of this type.

Finally we derive the value of an opportunity to invest either ina NGCC or IGCC power plant, that is, to choose between aninexible and a exible technology, respectively. Depending on theopportunitys time to maturity, we derive the pairs of coal and gasprices for which it is optimal to invest in NGCC, in IGCC, or simplynot to invest.

Numerical computations involve the use of one- and two-dimensionalbinomial lattices that support a mean-reverting process for coal andgas prices. Basic parameter values are taken from an actual IGCCpower plant currently in operation.

Keywords: Real options, Power plants, Flexibility, Stochastic

Costs.JEL Codes: C6, E2, D8, G3.


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1 INTRODUCTION 3

1 Introduction

It is broadly accepted in the nancial literature that traditional valuationtechniques based on discounted cash ows are not the most appropriate toolfor evaluating uncertain investments, especially in presence of irreversibilityconsiderations, or a chance to defer investment, or when there is scope forexible management. In these cases, it is usually preferable to use themethods for pricing options, such as Contingent Claims Analysis or DynamicProgramming.

On the other hand, the energy sector is of paramount importance forthe development of any society. Besides, its specic weight both in the realand nancial sectors of the economy cannot be neglected. Therefore the use

of inadequate instruments in decision making may be particularly onerous.In addition, the high sums involved in the energy industry, its operatingexibility and environmental impact, the progressive liberalization of themarkets for its inputs and outputs along with many types of uncertainties,all of them render this kind of investments a suitable candidate to be valuedas real options.

The aim of this paper is to use the real options methodology to assessdecisions of investment in power plants. In particular, our rm is goingto decide simultaneously the time to invest and the choice of technology.Specically, we compare an inexible technology (Natural Gas CombinedCycle, or NGCC henceforth) with a exible one (Integrated Gasication

Combined Cycle, or IGCC). We derive the best mode of operation, thevalue of the investments and the optimal investment rule when there isan option to wait. We consider two mean-reverting stochastic processes forcoal and natural gas prices, both of the Inhomogeneous Geometric BrownianMotion (IGBM) type, and also switching costs between modes of operation.The output (electricity) price, though, follows a deterministic path. Basicparameter values in our computations refer to an actual IGCC power plantcurrently in operation.

Restricting ourselves to the energy industry, we would mention Paddock,Siegel and Smith [23] who apply real options theory to the valuationof undeveloped oil leases; they also provide empirical evidence of the

superiority of option values over those derived from standard Net PresentValue methods. Bjerksund and Ekern [3] value an oil eld when there isan option to defer operation and abandon the site. Lund and Oksendal[21] comprise several theoretical and empirical studies on the valuation ofoil investments and related issues. Kemma [15] presents some practicalcase applications, with a focus on the use of real options theory in capitalbudgeting decisions by an actual oil rm. More recently Ronn [26] providesan array of applications of the real options approach to the valuation ofdierent exibilities embedded in the operation of energy assets by theirowners.


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1 INTRODUCTION 4

On the other hand, Bhattacharya [4] investigates the accuracy of

traditional methods of valuation when cash ows follow a mean-revertingprocess, as opposed to the standard Geometric Brownian Motion, and theensuing biases in value. Laughton and Jacoby [19][20] and Hasset andMetcalf [11], among others, discuss several, possibly conicting, inuences ofmean reversion on investment decisions; as they show, neglecting this patternmay bias investment choices either way. First there is a risk-discountingeect: mean reversion reduces the long-run uncertainty, and this shouldresult in a lower risk premium or discount rate for cash ows far into thefuture. Thus reversion tends to increase the value of any claim to cashows that increase with long-term prices. At the same time, though, loweruncertainty also tends to reduce directly the value of long-term options of

any type because of the so-called variance eect; this in turn impingeson the trigger level to invest. But there is also a realized-price eectwhereby investment would be less likely due to the fact that a lower variancemeans a lower probability for the cash ows to reach that trigger level.Robel [25] develops some features of the IGBM process and analyses itsimplications on valuation. Insley [14] assumes that timber prices followan IGBM process and values forestry investments. Sarkar [27] adopts thissame stochastic process for analysing diverse investment decisions. Weir[32] deals with the valuation of petroleum lease contracts assuming also anIGBM process for crude oil prices.

Finally, several papers have analyzed dierent exibility options

concerning the types of inputs or outputs involved, the mode of operation,or the kind of technology to adopt. First we single out Herbelot [13], whostudies the fulllment of restrictions on SO2 emissions either through thepurchase of emission permits, or changing the fuel, or deleting pollutingagents in the factory itself.1 He and Pindyck [12] focus on the exibility toproduce either one of two dierent products; the model can also be appliedto the choice between distinct inputs. On the other hand, Kulatilaka [16]develops a dynamic model that allows to value the exibility in a exiblemanufacturing system with several modes of operation, using a matrix oftransition probabilities between states; in this paper, the importance ofexibility in the design of systems from the viewpoint of both engineers

and competitors is emphasized. Later on, Kulatilaka [18] analyzes thechoice between a exible technology (red by oil or gas) and two inexibletechnologies to generate electricity. Some numerical results appear inKulatilaka [17], where he normalizes the gas price in terms of the oil price.A similar problem with two stochastic processes is dealt with in Brekke andSchieldrop [5]. This is perhaps the closest work to ours. Specically, theyfocus on the choice between exible and inexible technologies when fuelprices are assumed to follow standard geometric Brownian motions; besides

1 A brief summary of this work appears in Dixit and Pindyck [8].


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2 THE NGCC AND IGCC TECHNOLOGIES 5

the rm has a perpetual option to invest, and the exible technology involves

no switching costs. In contrast, the rm in our paper faces the choice oftechnology assuming that fuel prices display mean reversion, changes in fuelinputs entail switching costs, and the investment opportunity is availableonly for a nite time span (this requires a numerical solution as opposed toan analytical one).

The paper is organized as follows. In Section 2 we briey introduce theIGCC and NGCC technologies. Section 3presents the stochastic model forfuel prices (IGBM) and its main features. Also in this section, the dierentialequations for the value of a power plant and that for an option to invest init are derived. In particular, rst we consider a perpetual option to investin an asset whose value depends on fuel price; we obtain a solution in terms

of Kummers conuent hypergeometric function. Then we consider the caseof a nite-lived option; hence the binomial lattices for one and two variablesthat we use later in the numerical applications are also shown. 2 The resultsfor the perpetual option later on serve as a benchmark to verify how thebinomial lattices behave. In Section 4 rst we report some basic parametervalues from an actual power plant that we use in our computations. Then wevalue an operating NGCC plant and the option to invest in it. We also valuean operating IGCC plant with switching costs between modes of operationand the right to invest in this exible technology. Finally we derive theoptimal investment rule when it is possible to choose between the NGCCand IGCC alternatives. A section with our main ndings concludes.

2 The NGCC and IGCC technologies

2.1 Some features of electric power plants

The production of electricity can be viewed as the exercise of a series ofnested real options to transform a type of energy (gas, oil, coal, or other)into electric energy. There are two sets of outstanding information. Therst one has to do with the characteristics of the energy inputs used in theproduction process. The second one comprises the operation features of theelectric power plants, among them: the net output, the rate of eciency

(operating heat rate), the costs to start and stop, the xed costs, thestarting and stopping periods, and the physical restrictions that preventinstantaneous changes between states. These factors determine the wedgebetween the prices of the energy consumed and produced. On this basis, ateach instant it is necessary to decide how much electricity to produce andhow, i.e. the possible state changes to realize.

2 The choice of the binomial lattice as a resolution method is due both to its suitabilityand acceptance by the industry. See Mun [22].


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2.2 Natural Gas Combined Cycle (NGCC) technology

It is based on the employment of two turbines, one of natural gas and anotherone of steam.3 The exhaust gases from the rst one are used to generate thesteam that is used in the second turbine. Thus it consists of a Gas-Air cycleand a Water-Steam cycle. This system allows for a higher net eciency, 4

close to 55%; future trends aim at reaching net eciencies of 60% in NGCCplants of 500 Mw.

The advantages of a NGCC Power Plant are:5

a) Lower emissions ofC O2, estimated about 350 g/Kwh, which allow aneasier fulllment of the Kyoto protocol;

b) Higher net eciency, between 50% and 60%;c) Low cost of the investment, about 500 AC/Kw installed;d) Less consumption of water and space requirements, which allow to

build in a shorter period of time and closer to consumer sites;e) Lower operation costs, with typical values of 0.35 cents AC/KWh.In addition, a NGCC power plant can be designed as a base plant or

as a peak plant; in the latter case, it only operates when electricity pricesare high enough, what usually happens during periods of strong growth indemand.

On the other hand, the disadvantages of a NGCC Power Plant are:a) The higher cost of the natural gas red in relation to coals;b) The insecurity concerning gas supplies, since reserves are more

unevenly distributed over the world;c) The strong rise in the demand for natural gas, which can cause a

consolidation of prices at higher than historical levels.

2.3 Integrated Gasication Combined Cycle (IGCC)technology

It is based on the transformation of coal into synthesis gas, which iscomposed principally of carbon monoxide (CO) and hydrogen (H2).

6 Thisgas keeps approximately 90% of the heating value of the original coal.However, the appearance of CO as a by-product demands a special carewith this technology, given the harmful impact of small quantities of this

gas on human health. Besides, the process of gasication increases the costsof the original investment and those of operating a plant of this type.

The synthesis gas obtained has several applications:a) Generation of electric energy in IGCC power plants;

3 Also known as Combined Cycle Gas Turbines or CCGT.4 The net eciency refers to the percentage of the heating value of the fuel that is

transformed into electric energy.5 Our reference is ELCOGAS [9]: Integrated gasication combined cycle technology:

IGCC and its actual application at the power plant in Puertollano (Spain).6 According to the following reactions: C+ CO2 ! 2 C O and C+ H2O ! CO +H2.


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b) Production of hydrogen for diverse uses, like fuel cells; note the

promising future of hydrogen as a fuel in general;c) As an input to chemical products, like ammonia, for manufacturing

fertilizers;d) As an input to produce sulphur and sulphuric acid.After the stage of gasication, it is time to clean the gas by means

of washing processes with water and absorption with dissolvents. Thismakes it possible that an IGCC plant has signicantly lower emissions ofSO2, N Ox and particles, in relation to those generated in standard coalpower plants. The emissions of CO2 per Kwh are also lower, but in thiscase the improvement derives mainly from the higher net eciency of thecycle (currently, typical values uctuate around 42%, whith a trend towards

approaching 50%, provided this methodology enhances its design, which isin its rst steps). Right now, the emissions of CO2 from an IGCC powerplant (about 725 g/Kwh ) are estimated at 20% lower than those from aclassic coal power plant 7. However, they are clearly higher than those fromNGCC (around 350 g/Kwh ).

In addition to the gasication plant, the operation of an IGCC powerplant requires another costly element, namely the air separation unit, forproducing oxygen as an oxidizing agent. This unit may represent between10% and 15% of total investment. Needless to say, the units for gasication,air separation, and other auxiliary systems raise the initial outlay and implyhigher operation costs.

From the viewpoint of real options, it is necessary to highlight thefollowing aspects of this technology:

a) The investment in an IGCC plant can be considered as a strategicinvestment in a new technology, whose ultimate results will depend on itsnal success or failure;

b) The IGCC power plant is a exible technology concerning the possiblefuels to use; apart from the synthesis gas, it may re oil coke, heavy reneryliquid fuels, natural gas, biomasss, and urban solid waste, among others. Inthis way, at each time it is possible to choose the best input combinationaccording to relative prices. The valuation of this operating exibilitybetween synthesis gas and natural gas constitutes one of the aims of this

paper.7 Coal power plants are classied in subcritical, supercritical and ultrasupercritical,

depending on the pressure and the temperature of the steam. The net eciency for coalpower plants rounds about 35% in the supercriticals, and increases with higher investmentcosts but also entails an increased hazard due to working at higher temperatures.


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3 THE STOCHASTIC MODEL FOR THE FUEL PRICE 8

3 The stochastic model for the fuel price

3.1 The Inhomogeneous Geometric Brownian Motion(IGBM)

In a model for long-term valuation of energy assets, it is convenient to keepin mind that prices tend to revert towards levels of equilibrium after anincidental change.8 Among the models which display mean reversion, wehave chosen the Inhomogeneous Geometric Brownian Motion (or IGBM)process: 9

dSt= k(Sm St)dt + StdZt; (1)where:

St: the price of fuel at time t.Sm: the level fuel price tends to in the long run.k: the speed of reversion towards the normal level. It can be computed

ask = log 2=t1=2, wheret1=2 is the expected half-life, that is the time for thegap between St and Sm to halve.

: the instantaneous volatility of fuel price, which determines thevariance ofSt att.

dZt: the increment to a standard Wiener process. It is normallydistributed with mean zero and variance dt.

Some of the reasons for our choice are:a) This model satises the following condition (which seems reasonable):

if the price of one unit of fuel reverts to some mean value, then the price oftwo units reverts to twice that same mean value.

b) The termStdZt in the dierential equation precludes, almost surely,the possibility of negative values.

c) This model admits as a particular solution dSt = Stdt+StdZtwhenSm = 0 and =k; therefore it includes Geometric Brownian Motion(GBM) as a particular case.

d) The expected value in the long run is: E(S1) =Sm; this is not true

in Schwartz [28], where E(S1) =Sm(e2

4k).

3.2 The fundamental pricing equation

When using binomial lattices below, we will make use of the risk-neutralvaluation principle. The change from an actual process to a risk-neutral oneis accomplished by replacing the drift in the price process (in the GBM case,

8 See, for instance, Dixit and Pindyck [8](pp. 77-78), Pilipovic [24](pp. 63), Ronn[26](pp. 36), Baker et al. [2](pp. 121), Cortazar and Schwartz [6](pp. 216).

9 This seems to be a well suited model for natural gas futures contracts fromNYMEX (Pilipovic [24] chapter 4). It is also known as Integrated GBM or geometricOrnstein-Uhlenbeck process. See Bhattacharya [4], Robel [25], Insley [14], Sarkar [27],Weir [32].


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) with the growth rate in a risk-neutral world, r , wherer is the risklessinterest rate and denotes the net convenience yield. Note though that theconvenience yield is not constant in a mean-reverting process.

In order to obtain the risk-neutral version of the IGBM process, wesimply discount a risk premium (which, according to the CAPM, is)

S (2)

to its actual rate of growth. In this expression, is the correlation betweenthe returns on the market portfolio and the fuel asset, and denotes themarket price of risk, which is dened as:

rM rM

; (3)

where rM is the expected return on the market portfolio and M denotesits volatility.

Let St denote the risk-neutral version ofSt; thus

dSt= [k(Sm St) St]dt + StdZt: (4)If certain "complete market" assumptions hold, it can be shown that the

value of an investmentV(S; t), which is a function of fuel price and calendartime, follows the dierential equation:

1

2

2S2@2V

@S2

+ [k(Sm

S)

S]

@V

@S

+@ V

@t rV + C(S; t) = 0; (5)

whereC(S; t)is the instantaneous cash ow generated by the investment.10

3.3 Valuation of the option to invest

Nex we want to derive the value Fof an opportunity to invest in a projectwhose valueV depends on an asset whose priceSfollows an IGBM process.

Starting from quation (5) above, in general F will depend on S and t.Since the option confers no cash ow to its owner, its value will satisfy:

1

22S2

@2F

@S2 + [k(Sm S) S]

@F

@S +

@ F

@t rF = 0: (6)

3.3.1 Analytical solution for the perpetual investment option

Let Fbe the value of a perpetual call option on an investment V which inturn depends on the value of an asset Sfollowing an IGBM process. In thiscase, the termFdisappears in (6), which now can be expressed as:

1

22S2F00+ [k(Sm S) S] F0 rF = 0: (7)

10 If we equate (r )St to the coecient ofdt in (4), the resulting expression for is afunction ofSt.


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This equation may be rewritten as:

S2F00+ (S+ )F0 F = 0; (8)

where the following notation has been adopted:

2(k+ )2

;

2kSm2

;

2r2

:

In order to nd a solution to this equation, we dene a function h(S1)by

F(S) =A0(S1)h(S1); (9)

whereA0 and are constants that will be chosen so as to make h()satisfya dierential equation with a known solution. As shown in Appendix A,this function turns out to be:

h(S1) =A1U(a;b;z) + A2M(a;b;z); (10)

where U(a;b;z) is Tricomis or second-order hypergeometric function, andM(a;b;z) is Kummers or rst-order hypergeometric function. Therefore,the general solution to (9) will be:

F(S) =A0(S1)(A1U(a;b;z) + A2M(a;b;z)): (11)

The boundary conditions will determine whether A1 or A2 in (11) arezero. In our case, Srefers to a fuel input so the rm faces stochastic costs.An upward evolution in Sentails a reduction in prots, so F(1) = 0 andz = 0, then A1 = 0 and the term in Kummers function remains.

11 Thesolution is:

F(S) =Am(S1)M(a;b;z); (12)

withAm

A0

A2

.12 The constantAm

and the critical value S

below whichit is optimal to invest must be jointly determined by the remaining twoboundary conditions:

a) Value-Matching: F(S) =V(S) I(S);b) Smooth-Pasting: F0(S) =V0(S) I0(S):

11 Due to the fact that U(a;b; 0) = 1 if b > 1; this will be shown to hold with ourparameter values below.

12 When a downward evolution of S entails a reduction in prots, so that F(0) = 0and z = 1, then A2 = 0 and the term in Tricomis function remains. The solution isAu(S

1)U(a; b; z)with Au A0A1, since M(a;b;1) = 1 and then it must be A2= 0.This would be the case of stochastic revenues.


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3.3.2 Numerical solution for the non-perpetual option to invest

In this case F satises the partial dierential equation (6) which must besolved by means of numerical procedures. Given the type of the optionsinvolved and the number of sources of uncertainty, the binomial latticeapproach is adopted.

The time horizon T is subdivided into n steps, each of size t = T =n.Starting from an initial value S0, at time i, after j positive increments, thevalue of the fuel asset is given by S0u

jdij, where d = 1=u.Consider an asset whose risk-neutral behaviour follows the stochastic

dierential equation (4):

dS= (k(Sm

S)

S)dt + SdZ:

As shown in Appendix B, adopting the transformation X = lnS, upward

movements must be sizeX=p

t; thereforeu = eptandd = e

pt.

The probability of an upward movement at node (i; j) is

pu(i; j) =1

2+

(i; j)p

t

2 ; (13)

where:

(i; j) k(Sm S(i; j))

S(i; j) 1

22: (14)

Now consider two assets whose prices are governed by the followingrisk-neutral processes:

dS1= (k1(Sm1 S1) 11S1)dt + 1SdZ1; (15)

dS2= (k2(Sm2 S2) 22S2)dt + 2SdZ2; (16)dZ1dZ2 = 12dt: (17)

Dening X1= lnS1, and X2= lnS2, Appendix B shows that

X1= 1p

t; (18)

X2= 2pt; (19)puu=

X1X2+ X21t + X12t + 12t

4X1X2; (20)

pud=X1X2+ X21tX12t 12t

4X1X2; (21)

pdu=X1X2 X21t + X12t 12t

4X1X2; (22)

pdd=X1X2 X21tX12t + 12t

4X1X2: (23)


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4 VALUATION OF ALTERNATIVE TECHNOLOGIES 12

In the above expressions, pud stands for the risk-neutral probability of a

downward movement in coal price and a simultaneous upward movement ingas price at a certain node; similarly for the probabilities puu, pdu and pdd.

The branches of the lattice have been forced to recombine by takingconstant increments X1 and X2 once the step size t has been chosen.Thus it is easier to implement the model in a computer. Nevertheless theprobabilities change from one node to another by depending on 1 and 2(see expression (14)). Besides, it is necessary that at any time the fourprobabilities take on values between zero and one.

4 Valuation of alternative technologies

4.1 Representative values used in the numerical applications

We have used the standard values that appear in Table 1 (ELCOGAS [9]).

Table 1. Basic parameters

Concepts Coal Plant IGCC NGCC Base

Plant Size Mw (P) 500 500 500Production Factor (%) (FP) 80% 80% 80%Net Eciency (%) (RDTO) 35% 41% 50.5%

Unit Investment Cost AC/Kw (i) 1,186 1,300 496O&M cts. AC/Kwh (CVAR) 0.68 0.71 0.32

The Production Factor is the percentage of the total capacity usedon average over the year. Using these data, the heat rate, the plantsconsumption of energy, and the total production of electricity can becomputed:

Heat Rate: HR= 3600=RDTO=1; 000; 000, in GJ/Kwh.13

Investment CostI= 1000 i P, in Euros.14Total annual production: A= 1000 P 365 24 F P, in Kwh.Fuel energy needs: B = 1000 P 365 24 F P HR, in GJ/year.Now with these formulae we estimate the parameters in Table 2.

Table 2. Resulting values

Concepts Coal Plant IGCC NGCC Base

Heat Rate GJ/Kwh 0.0103 0.0088 0.0071Total Investment (mill AC) (I) 593 650 248

Annual production (mill Kwh)(A) 3,504 3,504 3,504Fuel Energy (GJ/year) (B) 36,041,143 30,766,829 24,979,010

13 One Kwh amounts to 3,600 KJ, and a GJ (Gigajoules) is a million KJ (Kilojoules).14 Since power is measured in Mw.


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By assumption, the rm is assessing the opportunity to build a baseload

power plant, i.e. one that operates in almost any plausible scenario (asopposed to a higher-marginal-cost peaking generator, which only operateswhen electricity prices are high enough, typically in periods of strongdemand growth).

4.2 Value of an operating NGCC plant

Our rst purpose is to determine the value of a NGCC operating facility.The revenues are the receipts from the electricity generated, whereas thecosts refer to the initial investment, the average operation costs and thoseof the fuel consumed, in this case natural gas. We consider a deterministic

evolution of electricity prices and variable costs, both of which would growat a constant rate ra.15

The time-0 value of revenues over a nite number of periods is computedaccording to the following formula:

P V R= A:E:(1 e(rra))

r ra; (24)

and variable expenditures amount to:

P V Cvar =A:Cvar:(1 e(rra))

r ra; (25)

where:A= Annual production : 3,504 million Kwh.E= Current electricity price : 0.035 AC/Kwh.16

ra = rate of growth of electricity price and variable costs.r = riskless interest rate: 5%.Cvar = unit variable costs: 0.0032 AC/Kwh. = Estimated useful life of the power facility upon investment: up to

25 years.The cost of the initial investment Iis in our case 248 million euros. Thus:

P V R

P V Cvar

I= 1; 342; 055; 454 AC: (26)

As for the fuel costs, rst we compute the value (cost) of one fuel unitconsumed per year over the whole life-span of the plant (see expression (57)

15 As a long-lived investment, it is an estimate of the average electricity price during theassets life that really matters, instead of short-term swings.

16 We have undertaken some preliminary work on the monthly average price in theSpanish wholesale electricity market (OMEL). According to our results, this price seems tofollow an AR(1) model with strong mean reversion. Therefore we have used the estimatedequilibrium price as a realistic estimate of the deterministic mean electricity price (theseresults are available from the authors upon request). Other electricity markets displaysimilar features; see, for instance, Elliott et al. [10].


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in Appendix C). Then multiplying by the annual consumption we get the

present value of fuel costs:

P V C=B[kSm(1 er)

r(k+ ) +

S kSmk+r+ k+

(1 e(r+k+))]; (27)

whereB is the annual fuel energy needed in GJ and Sstands for the naturalgas price (AC/GJ). Assuming the additional values:17 Sm = 3:25 AC/GJ, = 0, = 0:40, k = 0:25, = 0:20 and substituting them in the aboveformula we obtain the average cost per unit consumed yearly during thefacilitys life: 35:5498 + 3:3315S. Now multiplying the annual unit cost byannual consumption (B in Table 2) we get 887; 999; 971+83; 217; 315S. Forthe specic value S= 5:45 AC/GJ, we nally compute:

N P V(NGCC) =P V R P V Cvar I P V C= 521; 120AC: (28)

4.3 Valuation of the opportunity to invest in a NGCC plant

4.3.1 The perpetual option

Consider the case of a NGCC plant described by the last column in Table 1.As already seen, the present value of revenues minus the investment outlayand the present value of variable costs amounts to1; 342; 055; 454 AC, whereasthe present value of fuel costs is 887; 999; 971+83; 217; 315S. Thus the valueof the investment if realized now is: V(S)I= 454; 055; 48383; 217; 315S.

In this case, the boundary conditions would be:a) Value-Matching: F(S) = Am((S)1)M(a;b;(S)1) =

454,055,483-83,217,315 S = V(S) I(S)b) Smooth-Pasting: F0(S) =Am((S

)1) (S)1M(a;b;(S)1) a

b(S)2M(a + 1; b +

1; (S)1)

= 83; 217; 315 =V0(S) I0(S)By computing =12:5, = 40:625, = 2:5, and = 0:1827, we

derive a and b: a= 0:1827, b = 14:8654.Thus we have a system of two equations that will allow us to determine

Amand S. Substituting the value Am((S)1), from the Value-Matching

condition, in the Smooth-Pasting condition, there results an equation in S

which has as its solution S= 2:7448.Next it is easy to determine thatAm= 94; 394; 000and, nally, that the

value of the option for a gas price S is:

F(S) = 94; 394; 000 ( 40:625S

)0:1827M(0:1827; 14:8654;40:625

S ):

17 The values for the parameters in the gas price process derive from a cursory look atNYMEX natural gas futures contracts on October 18th 2004. The price level expected inthe long run corresponds to the futures contract with the longest maturity. Conversely thespot price has been approximated by the contract with the nearest term. These futurescontract trade in US dollars per 10,000 million British thermal units.


15/41


In our case, with S = 5:45, initially the option value is 153; 870; 000 AC,

against a N P V = 521; 120 AC; thus it is optimal to wait.If there were no other option but to invest now or never, the hurdle

point would be V(S) = I(S), whence we get S = 5:4563. However,with a perpetual option it is preferable to wait, since in principle and inthe long run natural gas price is going to decrease and uctuate around alevel Sm = 3:25, and then keep on waiting until it reaches S

= 2:7448 orbelow this value, so that the option value be equal to the net value of theinvestment. When the option may be exercised only during a nite period,the threshold S will take on a value between 5.4563 and 2.7448.

Appendix D deals with the perpetual investment option in thedeterministic case. It also includes some sensitivity analysis with respect

to changes in , k and Sm.

4.3.2 The non-perpetual option

Now we want to determine the value of an investment option and theoptimal investment rule in a NGCC facility when the opportunity is availablebetween time 0 and T.

The costs of the initial investment are Ierbt, with 0 t T, where Iis the initial investment at time t = 0 and rb is its rate of growth (bothelectricity price and variable costs are now supposed to remain constant).

At the nal moment t = T, there is no other option but to invest right

then or not to invest. The decision to go ahead with the investment is takenif the present value of revenues exceeds that of costs:

N P V =P V(Revenues) P V(Expenditures)> 0:

A binomial lattice is arranged with the following values in the nal nodes:

W =max(N PV; 0):

At previous moments, that is, when 0 t < T, depending on the currentfuel price, we compute the present value of exercising the option to invest(N P V), and compare it with the value of keeping the option alive, choosing

the maximum between them:W =M ax(N P V ; ert(puW

+ +pdW)):

The lattice is solved backwards, which provides the time-0 value. Ifwe compare this value with that of an investment made at the outset, thedierence will be the value of the option to wait. Logically this optionsvalue will always be nonnegative.

By changing the initial value of the fuel unit cost it is possible todetermine the fuel price at which the option value switches from positive tozero. This will be the optimal exercise price at t = 0. Similarly, arranging


16/41


a binomial lattice for the investment opportunity with maturity t < T and

changing the fuel price, the optimal exercise price for intermediate momentsis determined.

At the nal date, since there is no chance for further delay, investmentis realized only if NP V > 0. The optimal point to invest will be foundby computing the gas price for which N P V = 0; in our case, this value isS0 = 5:4563. In the opposite case of an unlimited possibility to defer, whenra = rb = 0, it must be S

1 = 2:7448, which comes from the analytical

solution for the perpetual option. These results appear in Table 3. Theyshow a convergence towards those of the perpetual option as the maturityof the investment opportunity increases.18

Table 3. Trigger price S with nite time to maturityTerm ra= 0; rb = 0 ra= 0; rb = 0:025 ra= 0; rb = 0:05

0 5.4563 5.4563 5.456312 3.3268 3.5587 3.78231 3.2200 3.4417 3.65032 3.0864 3.3035 3.51073 3.0040 3.2250 3.43944 2.9480 3.1751 3.39825 2.9079 3.1413 3.37316 2.8782 3.1179 3.35757 2.8557 3.1012 3.3481

8 2.8386 3.0893 3.34239 2.8253 3.0808 3.3392

10 2.8151 3.0746 3.3378

1 2.7448 - -

The value of the option to invest increases with the options maturity. Inorder to be exercised optimally, a longer-lived option requires a lower criticalprice for the fuel resource. It can also be observed that a higherrb, ceterisparibus, quickens the time to invest: the higher the investment growth rate,the less stringent the hurdle to invest (as shown up by a higher S). Whenthe time to maturity is zero, though, there is no inuence from the rate of

growth of the initial investment.Next we analyze the optimal choice between investing or waiting, when

the investment opportunity is available for ve years, depending on theinitial fuel price.

18 Numerical results in Table 3 correspond to a setting with 1200 steps, irrespective ofthe options time to maturity. Take, for example, the option to invest in the next 5 years(then one month would comprise 20 time steps). When ra = rb = 0, it is optimal to investin a NGCC plant if gas price falls to 2.9079. With 150 steps (instead of 1,200), the triggerprice turns out to be 2.9394 AC/GJ, and with 10,000 time steps it is 2.8963 AC/GJ. Thusthe dierences in the critical price do not seem to be large (around 1% or less).


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Table 4. NPV and option values with T = 5 yearsS0 NPV(AC) Option value Max(NPV,Option) Optimal decision

5.45 521,120 119,170,000 119,170,000 Wait5.00 37,969,000 129,040,000 129,040,000 Wait4.50 79,578,000 141,640,000 141,640,000 Wait4.00 121,190,000 156,820,000 156,820,000 Wait3.50 162,790,000 176,420,000 176,420,000 Wait3.00 204,400,000 205,010,000 205,010,000 Wait

2.9079 212,070,000 212,070,000 212,070,000 Indierent2.50 246,010,000 245,900,000 246,010,000 Invest2.00 287,620,000 287,450,000 287,620,000 Invest

In principle, a lower fuel price ceteris paribus will render the powerplant more valuable; this in turn will increase the value of the option toinvest. Thus it is not obvious whether a cheaper natural gas will hasten theinvestment decision or not. As shown in Table 4, whenS0= 5:45, the NPVis very low and the option to invest is worth more than 119 million euros.As the value ofS0 decreases, the investment option increases in value butless than the NPV, with the equality being reached when S0 = 2:9079. Forlower gas prices, it is preferable to invest immediately.

4.4 Valuation of an operating IGCC plant

Now we must compute the value of an operating IGCC power station, bothright upon the initial outlay and at any moment along its useful life to deriveits remaining value. We accomplish this by means of two two-dimensionalbinomial lattices 19 which refer to initial states consuming either coal ornatural gas, respectively.

At the end of the plants useful life its value is zero, regardless of whetherthat instant has been reached consuming coal or natural gas:

W c= W g= 0;

whereW candW gstand for the plants (gross) value according to the nodes

in coal and gas lattices, respectively.At earlier times t we compute, for a time interval t, the prots by

mode of operation, which are determined as the dierence between electricityrevenues and the sum of variable plus fuel costs:

c= A:E:erat (1 et(rra))

r raBctSc A:Cvarc :erat

(1 et(rra))r ra

;

(29)

19 Given that the rm faces two stochastic fuel prices, there are two sources of risk andtwo-dimensional binomial lattices are needed.


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g =A:E:erat (1 et(rra))

r ra BgtSg

A:Cvarg :e

rat (1 et(rra))r ra

;

where:c: Net prots from operating with coal.g : Net prots from operating with natural gas.Sc: current coal price.Sg: current natural gas price.

A:E:erat (1et(rra))rra : value at time t of revenues from electricity over

the period t.

A:Cvar:erat (1et(rra))

rra : value at time t of variable costs incurred overt.

Bc: the coal energy needed per year in GJ.Bg: the natural gas energy needed per year in GJ.BctSc: Costs of coal consumed during t.BgtSg : Costs of natural gas consumed during t.I(c ! g) : Switching cost from coal to gas.I(g! c) : Switching cost from gas to coal.If initially the IGCC plant was consuming coal, the best of two options

is chosen:20

a) continue: the present value of the coal lattice is obtained, plus theprots from operating in coal-mode at that instant.

b) switch: the present value of the gas lattice is obtained, plus the prots

from operating in gas-mode at that instant, minus the costs to switchingfrom coal to gas, I(c ! g).Thus the binomial lattices will take on the following values:21

W c= M ax(c+ert(puuW c

+++pudW c+pduW c

++pddW c); gI(c ! g)+

+ert(puuW g++ +pudW g

+pduW g+ +pddW g

)): (30)

Similary, when the initial state corresponds to operating with naturalgas, we would compute:

W g= M ax(cI(g! c)+ert(puuW c+++pudW c+pduW c++pddW c);

g+ ert(puuW g++ +pudW g+

pduW g+ +pddW g

)): (31)

20 We have not considered the option not to operate, though it could be taken intoaccount easily. It could be included by a third lattice, corresponding to an idle initialstate. At every time we should maximize over three possible values, taking into accountthe switching costs between states. If we denote the idle state byp, in this case there couldbe a stopping cost: I(c ! p) or I(g ! p), and a restarting cost: I(p ! c) or I(p ! g). Ifrestarting costs were very high, stopping could amount to abandone.

21 We follow a similar procedure to Trigeorgis [31], pp. 177-184 (the case with switchingcosts).


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Finally, at time zero the optimal initial mode of operation is chosen by:

M ax(W c ; W g):

In this way, we have derived the value of a exible plant in operation.In this computation, for a plant operating for some time, the cost of the

initial investment plays no role, but it could be included at the outset inorder to compare this outlay with the present value of expected prots.

Table 5 shows the parameter values adopted in the base case.22 Some ofthem are taken from Table 1 but are repeated here for convenience.

Table 5. An IGCC power plant: base case

Concepts Coal Mode Gas Mode

Plant Size Mw (P) 500 500Production Factor(FP) 80% 80%

Net Eciency(%) (RDTO) 41.0% 50.5%Unit Investment cost AC/Kw (i) 1,300 1,300

Operation cost (ACcents/Kwh) (CVAR) 0.71 0.32Fuel price (AC/GJ) 1.90 5.45

Reversion value Sm(AC/GJ) 1.40 3.25Reversion speed (k) 0.125 0.25

Plants useful life (years)() 25 25Risk-free interest rate (r) 0.05 0.05Market price of risk () 0.40 0.40

Volatility () 0.05 0.20Correlation with the market (1; 2) 0 0

Correlation between fuels (12) 0.15 0.15Investment cost rate of growth (rb) 0 0Electricity price rate of growth (ra) 0 0

Switching costs (AC) 20,000 20,000

Making the computations with a lattice of 300 steps (one for each monthof useful life), Table 6 shows the ensuing results, as a function of switchingcosts.23

22

Parameter values for coal and gas price processes, as well as their correlationcoecient, have been computed from yearly average prices gathered by the US EnergyInformation Administration. Note that these prices are in US dollars per million Btu, sothey must be turned into AC/GJ.

23 As mentioned in the text, gures in Table 6 correspond to a setting with 300 steps,regardless of switching costs. As the number of timesteps is raised the scope for seizingthe palnts exibility increases. Take, for instance, switching costs of 20.000 AC; the plantsnet value amounts to 52,534,000 AC. If we consider 75 timesteps the net value falls to47,662,000 AC, whereas with 1200 steps this value becomes 53,693,000 AC. Nonetheless itdoes not seem very realistic to allow for very frequent changes in the mode of operation(say, daily or weekly). That is why we have chosen a monthly timestep.


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Table 6. Gross and net value (AC) of an IGCC Plant

Switching costs (A

C) Plants value Plants value - Initial investment0 702,662,000 52,662,000

10,000 702,598,000 52,598,00020,000 702,534,000 52,534,00050,000 702,345,000 52,345,000

100,000 702,129,000 52,129,0001,000,000 700,049,000 50,049,000

1 691,987,000 41,987,000

When there are no switching costs, the value of the plant exceeds theinitial investment (650,000,000 AC) by 52,662,000 AC. Thus it is worth an

8.10% more than the amount disbursed. As switching costs swell, the plantsvalue drops.

Comparing the value in each case with that of innite switching costs,the dierence shows the value of exibility; this amounts to 10,675,000 AC inthe absence of switching costs, just 1.5% of the initial investment. Note thatexibility in the IGCC plant may be valuable because of reasons dierentfrom harnessing at each instant the best fuel option. For instance, it maybe due to failures in elements necessary to operate in coal-mode, but thatdo not prevent the plant from operating in gas-mode and avert the totalstopping of the facility. The same could happen in the case of problemsconcerning supplies of a certain kind of fuel.

In the case of innite switching costs, in which operation turns out to beonly in coal-mode, the value of the plant changes with the number of stepsin the lattices. With 3,000 steps we reach a value (net of investment cost)of 43,353,000 AC, a gure which is very close to the 43,594,000 AC obtainedby following an analytical procedure akin to that for a natural gas plant.24

The plants value as a function of coal and natural gas prices appears inFigure 1. It can be observed that the investment value drops as the priceof either fuel rises. Also, for low gas prices, the value of the plant decreasessmoothly when coal price swells, since this is not the preferred fuel resource.However, for high gas prices, the plant burns mainly coal; if this gets dearer,an abrupt fall in value ensues.

Table 7 shows the (gross) value of the plant as a function of the remaininglife, for the base case with switching costs of 20,000 AC, without taking intoaccount the initial investment of 650,000,000 AC. For the sake of consistence,we have used one step per month again.25

24 With innite switching costs, there is no scope for exibility and the cheapest fuelresource is used exclusively. Thus the above formulas for the NGCC plant apply, but nowthey refer to coal.

25 Concerning the convergence of these results, for illustrative purposes let us consider aplant with 10 years to closure. With 1012=120 monthly timesteps, the gross value of theplant (ra = 0:00) is 361,090,000 AC. Again, the plants value increses with the number of


21/41


Figure 1: Gross value (AC) of an IGCC plant (a=0.00, =25).

Table 7. Value (AC) of an operating IGCC plant as a function of thegrowth rate of electricity price and variable cost (ra)

Remaining life Plant value ra = 0.00 Plant value ra= 0.03

25 years 702,534,000 1,245,700,00020 years 613,835,000 999,120,00015 years 501,490,000 742,186,00010 years 361,090,000 479,990,0005 years 191,020,000 223,980,0004 years 153,950,000 175,460,0003 years 116,150,000 128,500,0002 years 77,806,000 83,409,0001 year 39,044,000 40,477,0000 year 0 0

Whatever the rate of growth of the electricity output (and variable costs),ra, the plants value diminishes as its life-span comes to an end. Note alsothat at least two factors are at work here. For concreteness, consider theplants value with 1 and 2 years before closure. Clearly the reversion eectis stronger over 2 years than over 1 year; thus one would expect the plantsvalue with 2 years to be more than double that with 1 year to operate. Yetthe inuence of the discount rate r = 0.05 pushes against this trend. For

steps but the increase is not relatively large: with 1,500 timesteps it jumps to 362,050,000AC.


22/41


constant electricity prices (ra = 0), the eect of time discounting prevails

since 77,806,000 euros is less than 239,044,000 euros. This is not the case,though, with rising electricity prices (ra= 0.03), so now the reversion eectmore than osets the discount eect.

4.5 Valuation of the opportunity to invest in an IGCC plant

Let us assume that the investment opportunity is available from time 0 toT (the non-perpetual option). In this case, the procedure is very similarto that for the optimal timing to invest in a NGCC plant. Nonetheless,the value of an operating plant must be determined at each node of thetwo-dimensional lattice from another two-dimensional lattice which takes as

a starting point the prices at that node. 26

At the nal date, the choice must be made between investing then if theplants value is positive or not to invest:

W =max(N P Vigcc; 0).

With switching costs of 20,000 AC as in the base case, Table 8 showscombinations of coal and natural gas prices that imply a zero value (that is,the plant value matches initial outlay) at maturity.

Table 8. IGCC critical curve at t = T

Gas price (AC/GJ) Coal price (AC/GJ)

1 2.14105.4563 2.2325

5.45 2.23275.00 2.24464.50 2.26424.00 2.29743.50 2.36843.25 2.44853.00 2.58022.50 3.06822.00 4.06101.50 6.26461.00 18.9226

0.9897 126 Note that the rm must choose the optimal time to invest from time 0 to T. This

means that at every moment it must compare the continuation value (i.e. that of theoption to invest when kept unexercised) and the value of an immediate investment. Sincethe rm can decide to invest at any time given current fuel prices, the plants usefullife can start at any moment at the prevailing fuel prices. This implies that the abovetask of valuing an operating IGCC power plant now must be solved at every node of thetwo-dimensional binomial tree.


23/41


If the IGCC plant is to be worthy for a very high coal price, it is necessary

that natural gas prices remain below 0.9897 AC/GJ. This is due to the factthat the reversion process pushes gas prices towards 3.25 AC/GJ. However,if gas price were to grow arbitrarily high, the plants value would reach zerofor a coal price of 2.1410 AC/GJ, which is above its reversion value (Sm =1.40 AC/GJ). For lower coal prices, its value would even switch to positive.This seems reasonable since the IGCC plant is mainly designed to operatewith coal.

At previous moments, the best of the two options (to invest or tocontinue) has to be chosen:

W =M ax(N P Vigcc; ert(puuW

++ +pudW+pduW

+ +pddW)): (32)

At t = 0, the value obtained from the lattice is compared with that of aninvestment realized right then. The dierence is the value of the option towait.

By using a lattice with quarterly steps for the option to invest, and onewith monthly steps to value the plant, we get the valuations in Table 9 forthe base case, as a function of the time to maturity of the investment option:

Table 9. NPV and option values (IGCC plant)

Term (years) NPV (AC) Option value Option value - Plant NPV

5 52,534,000 76,398,000 23,864,0004 52,534,000 74,179,000 21,645,000

3 52,534,000 71,048,000 18,514,0002 52,534,000 66,651,000 14,117,0001 52,534,000 60,592,000 8,058,000

0.5 52,534,000 56,835,000 4,301,0000 52,534,000 52,534,000 0

The second column refers to the plant value under the assumption of anow or never investment in the exible technology. It includes the valueof the plants exibility.

The value of the option to wait increases with its maturity, given thestarting point of the prices (both above their reversion levels). The highest

yearly increase takes place in the rst period (8; 058; 000 AC); henceforth thatincrease is much lower since the eect of reversion is stronger in the initialperiods.27

The optimal investment rule in an IGCC technology can be derivedfollowing a procedure akin to that used for the NGCC technology in the basecase. Nonetheless, in this case and at each time, we will have to compute

27 With t1=2 = ln(2)

k one would guess that gas price will change from 5.45 AC/GJ to4.35 AC/GJ in 2.77 years time. On the other hand, the expected price for coal 5.55 yearsfrom now would be 1.65 AC/GJ. The value of t1=2 results from solving: S0 (

S0Sm2 ) =

Sm+ (S0 Sm)ekt1=2 .


24/41


the combinations of coal price and gas price for which the option to wait is

worthless.At anyt, a range of initial prices for coal is chosen; then, for each one of

them, we compute the natural gas price for which the option to wait turnsfrom positive to zero. For an option to wait up to two years, we get Table10. Again, the higher the price of one fuel, the cheaper the other one. Thatis, since both coal and gas are fuel resources, there must be a trade-o intheir prices for the options value to stabilise at a certain level (in this case,zero).

Table 10. Critical values with option to wait up to 2 years (IGCC)

Gas price Coal price

1 1.575.45 1.565.00 1.564.50 1.564.00 1.563.50 1.563.25 1.573.00 1.572.50 1.592.25 1.852.00 2.98

1.50 5.150.96 1

These results are shown in Figure 2. It can be observed that the newlocus has shifted downwards and to the left, in relation to the case in whichthere is no option to wait (Table 8). In other words, it makes sense to giveup (kill) the option to wait if fuel prices are now relatively lower thanbefore but not otherwise. The upper locus divides the price space into tworegions; above, the best decision is not to invest, and below it is optimal toinvest. Similarly, above the lower locus the rm should wait, but it shouldinvest immediately if fuel prices happened to fall below this locus.

Now let us consider for a moment the surface in Figure 1. Fixing naturalgas price at Sg = 5:00 AC/GJ, the resulting graphic would be a downwardsloping curve in the plants value / coal price plane. Subtracting from thesegross values the investment disbursement, and restricting coal prices to therange from 1.4 AC/GJ to 1.9 AC/GJ, the lower locus in Figure 3 arises. Forconvenience, in addition to the net value of the plant, the value of the optionto invest has also been drawn. It can be observed that optimal exercise ofthe option will take place as soon as coal price falls to 1.56 AC/GJ. If this ishigher, it will be better to wait.


25/41


Figure 2: Optimal divide between invest / not invest (upper locus) andinvest / wait (lower locus) in an IGCC power plant without and with anoption to wait.

Figure 3: Net value of the IGCC plant (forSg = 5:00 AC/GJ) and value ofthe option to invest with two years to maturity, both as a function of coalprice.


26/41


4.6 Valuation of the opportunity to invest in NGCC or

IGCCUp to now we have considered both the exible and the inexibletechnologies in isolation. When there is an opportunity to invest in eitherone of the two technologies, at each moment we face the choice:

a) to invest in the inexible technology (NGCC),b) to invest in the exible technology (IGCC),c) to wait and at maturity give up the investment.At the nal date, since there is no remaining option, the best alternative

among the three available ones is chosen:

W =max(N P Vigcc

; N P V ngcc

; 0);

where the valueN P Vigcc, at each point in the binomial lattice, is computedby means of a two-dimensional binomial lattice with the fuel prices at thatnode.

If, at time T, the only possibility is to invest in the NGCC technology,the investment is realized whenever the natural gas price is lower than 5.4563AC/GJ; see Table 3. Similarly, if, at time T, the only alternative is to investin the IGCC technology, the plant is built when the pairs of gas price andcoal price lay below the curve resulting from Table 8. Now Figure 4 showsboth decision rules, but it must be remembered that, in this case, it is notpossible to choose the best possibility. Thus, for natural gas pricesSg above

5.45 AC/GJ, it does not pay to invest in a NGCC plant. Concerning the IGCCfacility, it is optimal to invest for pairs of fuel prices below the decreasinglocus, but not otherwise. Note from Table 8 that this locus intersects thevertical border at a coal price Sc= 2.23 AC/GJ.

Now assume that at time T it is possible to choose between the twoalternatives (Figure 5). Clearly, if both fuel prices are relatively high, noneof the technologies will be adopted and there will be no new investmentin these power plants. There are also multiple pairs (Sg, Sc) such thatthe value of the investment is positive for the two plants; consequently theinvestment with the highest value will be chosen. Thus, given a high coalprice, ifSg drops enough the NGCC will become the technology of choice.

Alternatively, given a high gas price, ifScfalls enough the IGCC technologywill be preferred. This area will be divided into two by a line startingat the point (Sg = 5:4563, Sc = 2:2325) and pointing towards the origin,along which there is indierence between investing in IGCC and investingin NGCC.

At previous moments, the choice is:

W =max(N P Vigcc; N P V ngcc; ert(puuW

+++pudW+pduW

++pddW)):


27/41


Figure 4: Decision areas (at time T) when each technology is considered inisolation.

Figure 5: Investment / Continuation areas (at time T) when choosingbetween the two technologies.


28/41


Figure 6: Investment / Continuation areas when both technologies are onoer and there is an option to wait up to two years.

This computing procedure is iteratively followed until the initial value isobtained. At that instant, the NGCC technology will be adopted if W =N P Vngcc; similarly, the IGCC technology is chosen ifW =N P Vigcc. If thereis no investment at t = 0, this means that the best option is to wait.

The points over the optimal exercise curve are those for which the valueof the option to wait changes from positive to zero. In principle, there maybe two curves, one for the IGCC plant and another one for the NGCC plant.See Figure 6 for an option to invest in either technology with two years tomaturity. It can be observed that:

a) In order to invest, prices must be rather lower than those when thereis no option to wait. Thus the new locus for the NGCC moves leftwardsfrom the vertical line at Sg = 5.45 AC/GJ before; and the new locus for theIGCC moves downwards from the at line atSc= 2.23 AC/GJ. The reversioneect, given the initial prices, promotes this behaviour.

b) For a very high natural gas price, there will be investment in an IGCCplant if coal price falls below 1.57 AC/GJ; this is the same that we computedfor the IGCC investment option when this was the only technology available(Table 10).

c) Similarly, for a very high coal price, there will be investment in aNGCC plant if gas price is less than 3.17 AC/GJ. This result would applyas an option to waitt for two years in a NGCC plant with a lattice of 8quarterly steps, the same step size used in the option to invest with the two


29/41

5 CONCLUDING REMARKS 29

technologies on oer.

d) For fuel prices close to the iso-value line between immediateinvestment in NGCC and IGCC, the best choice is just to wait and seehow uncertainty unfolds. The waiting zone expands into the regions ofimmediate investment by driving a wedge along the indierence line (at T)between NGCC and IGCC.

The shape of the three zones in Figure 6 somehow resembles that inBrekke and Schieldrop [5], which is derived as the analytic result of aperpetual option to invest. In our case the solution is numerical becausethe option to invest is nite-lived.

5 Concluding remarksMany investments in the energy sector can be conveniently valued as realoptions. Frequently appearing features are the operating exibility and thepossibility to delay the investment or even not to invest at the nal moment.Special attention in the valuation must be paid to the nature of the stochasticprocesses that govern the underlying variables.

In this paper we have dealt with the choice between competingtechnologies for producing a certain output (electricity). In particular, wehave analyzed the valuation of the opportunity to invest in an IGCC powerplant, as opposed to the alternative of an NGCC power plant, according tothe current prices of natural gas and coal when these follow a mean-reverting

process, namely an IGBM. As a reference point we also have computed thevalue of a perpetual option to invest. The IGCC power plant has beenvalued with switching costs and choice of the optimal mode of operation;actual parameters from an operating plant have been used.

First we have computed the value of an operating NGCC plant. Thisis relatively easy since there is no special exibility in its usage. Then thevalue of a perpetual option to invest in it has been estimated. It serves asa limiting case or reference point for the nite-lived option. This has beenvalued by means of a binomial lattice for dierent maturities and growthrates of electricity price and investment outlay. The optimal investment rulefor a given term as a function of natural gas price has also been analyzed.

Second we have valued an IGCC plant in operation using twotwo-dimensional lattices, depending on whether initially the plant burnscoal or natural gas. The value of the plant has been computed as a functionof switching costs and for dierent useful life-spans with constant or growingelectricity prices. Next the non-perpetual option to invest in an IGCCplant has been considered, assuming again that this is the only technologyavailable. In this case, an optimal locus of fuel prices arises above which itis optimal to wait. As could be expected, the longer the options maturitythe closer the locus is to the origin in the prices space.


30/41

A ANALYTICAL SOLUTION OF THE PERPETUAL INVESTMENT OPTION30

Third we have assumed that both technologies are on oer. Our results

show the inuence of the mean-reverting process on the critical values whencurrent fuel prices are above their level expected in the long run. The valueof the operating exibility in the IGCC power plant seems to be low, becauseit is designed to function mainly with coal. We show that prior to optionsmaturity there is a small region in the prices space in which it is optimal towait whereas at maturity it was optimal to invest, since the values of bothtechnologies are very close; outside it, optimal decisions are clear-cut.

A Analytical solution of the perpetual investmentoption

In order to nd a solution to the ordinary dierential equation in F, wedene a function h(S1) by

F(S) =A0(S1)h(S1): (33)

whereA0and are constants that will be chosen so as to makeh()satisfy adierential equation with a known solution. The rst and second derivatives,divided by A0

, are:

F0(S)

A0

= ()S1h(S1) + S2h0(S1)();

F00(S)

A0

= (+ 1)S2h(S1) + ()S3h0(S1)() +

+S3( 2)()h0(S1) + S4h00(S1)2:

Substituting these expressions in (8) and simplifying we get:

Sh(S1) ((+ 1) ) + S1

S12h00

(S1) +

+h0(S1)( + (+ 2)

S12)

h(S1) = 0For this equality to hold, rst it must be:

(+ 1) = 2 + (1 ) = 0: (34)

This equation allows to determine the positive value of, since the remainingterms are known constants.

Once the value of has been obtained, the remainder of the equation is:

(S1)h00

(S1) + h0

(S1)(2+ 2 S1) h(S1) = 0:


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A ANALYTICAL SOLUTION OF THE PERPETUAL INVESTMENT OPTION31

This is Kummers Dierential Equation, 28 where: a = , b = 2+ 2 ,and z = (S1). The general solution to this equation has the form:

h(S1) =A1U(a;b;z) + A2M(a;b;z); (35)

where U(a;b;z) is Tricomis or second-order hypergeometric function, andM(a;b;z) is Kummers or rst-order hypergeometric function.

Therefore, the general solution to (33) will be:

F(S) =A0(S1)(A1U(a;b;z) + A2M(a;b;z)): (36)

The second-order hypergeometric function has the following representation:

U(a;b;z) = (1 b)(1 + a b) M(a;b;z) +

(b 1)(a)z(b1)

M(1 + a b; 2 b; z); (37)

where(:)is the gamma function and M(a,b,z) is Kummers function, whosevalue is given by:

M(a;b;z) = 1 +a

bz+

a(a + 1)

b(b + 1)

z2

2! +

a(a + 1)(a + 2)

b(b + 1)(b + 2)

z3

3! + : : : (38)

The derivatives of Kummers function have the following properties:

@M(a;b;z)

@z =

a

bM(a + 1; b + 1; z);

@2M(a;b;z)

@z2 =

a(a + 1)

b(b + 1)M(a + 2; b + 2; z):

The derivatives of Tricomis function satisfy:

@U(a;b;z)

@z = aU(a + 1; b + 1; z);

@2U(a;b;z)

@z2 =a(a + 1)U(a + 2; b + 2; z):

28 See Abramowitz and Stegun [1], equation 13.1.1.


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B BINOMIAL LATTICES FOR IGBM VARIABLES 32

B Binomial lattices for IGBM variables

B.1 Binomial lattice for a risk-neutral IGBM variable

The time horizonTis subdivided innsteps, each of size t= T =n. Startingfrom an initial value S0, at time i, after j positive increments, the value ofthe fuel asset is given by S0u

jdij, where d = 1=u.Consider an asset whose risk-neutral behaviour follows the dierential

equation:dS= (k(Sm S) S)dt + SdZ: (39)

This can also be written as:

dS= k(Sm S)S

! Sdt + SdZ=Sdt + SdZ: (40)Since it is usually easier to work with the processes for the natural logarithmsof asset prices, we carry out the following transformation: X= lnS. ThusXs= 1=S, Xss = 1=S2; Xt= 0, and by Itos Lemma:

dX= (k(Sm S)

S 1

22)dt + dZ= dt + dZ; (41)

where k(Sm S)=S 122 depends at each moment on the assetvalue S.

Following Euler-Maruyamas discretization, the probabilities of upward

and downward movements must satisfy three conditions:a)pu+pd= 1.

b)E(X) =puXpdX=

k(SmS)S 122

t= t. The

aim is to equate the rst moment of the binomial lattice (puXpdX)to the rst moment of the risk-neutral underlying variable ( t).

c) E(X2) = puX2 +pdX

2 = 2t+ 2(t)2. In this case theequality refers to the second moments. For small values of t, we haveE(X2) 2t.

From a) and b) we obtain the probabilities, which can be dierent ateach point of the lattice (because depends on S, which may vary fromnode to node):

pu= 12

+ t2X

: (42)

From c) there results X = p

t; therefore, u = ept. The

probability of an upward movement at point (i; j) is:

pu(i; j) =1

2+

(i; j)p

t

2 ; (43)

where:

(i; j) k(Sm S(i; j))

S(i; j) 1

22: (44)


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B BINOMIAL LATTICES FOR IGBM VARIABLES 33

B.2 Binomial lattice for two risk-neutral IGBM variables

Consider two assets whose prices are governed by the following risk-neutralprocesses:

dS1= (k1(Sm1 S1) 11S1)dt + 1SdZ1; (45)dS2= (k2(Sm2 S2) 22S2)dt + 2SdZ2; (46)

dZ1dZ2 = 12dt; (47)

where 1 and 2 denote the correlations of their respective returns withthose of the market portfolio. Adopting the transformations X1 = lnS1,X2= lnS2, and applying Itos Lemma:

dX1=

k1(Sm1 ^S1)

S1 11 12 21

!dt + 1dZ1= 1dt + 1dZ1; (48)

dX2=

k2(Sm2 S2)

S2 22

1

222

!dt + 2dZ2= 2dt + 2dZ2: (49)

Now it is necessary to solve a system of six equations:a)puu+pud+pdu+pdd= 1. The probabilities must sum to one.b) E(X1) = (puu+ pud)X1 (pdu+ pdd)X1 = 1t. This is the

expected value of the increment in X1.c)E(X21) = (puu +pud)X

21+ (pdu +pdd)X

21 =

21t + 1

2t2. This

is the expected value of the second non-central moment of the increment inX1.d) E(X2) = (puu+ pdu)X2 (pud+ pdd)X2 = 2t. This is the

expected value of the increment in X2.e)E(X22) = (puu +pdu)X

22+ (pud +pdd)X

22 =

22t + 2

2t2. Thisis the expected value of the second non-central moment of the increment inX2.

f)E(X1X2) = (puupudpdu +pdd)X1X2= 12t+ 12t2.This is the expected value of the product X1X2, which amounts tosatisfying the correlation condition.

The solution to this system of equations, ignoring the elements in t2,

is:29

X1= 1p

t; (50)

X2= 2p

t; (51)

puu=X1X2+ X21t + X12t + 12t

4X1X2; (52)

pud=X1X2+ X21tX12t 12t

4X1X2; (53)

29 Clewlow and Strickland [7] show a multidimensional lattice with two assets whichfollow correlated GBMs.


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C VALUATION OF AN ANNUITY IN PRESENCE OF MEAN REVERSION34

pdu=X1X2 X21t + X12t 12t

4X1X2; (54)

pdd=X1X2 X21tX12t + 12t

4X1X2: (55)

The branches of the lattice have been forced to recombine by taking constantincrementsX1 and X2 once the step size t has been chosen. Thus itis easier to implement the model in a computer program. However, theprobabilities change from one node to another by depending on 1 and 2.Besides, it is necessary that at any time the four probabilities take on valuesbetween zero and one.

C Valuation of an annuity in presence of meanreversion

Our aim is to value an asset V wich pays Xdt continuously over anite number of periods of remaining asset life, with X following amean-reverting process of the type:

dX=k(Xm X)dt + XdZt:

It can be shown that the asset V satises the dierential equation:

12

VXX2X2 + (k(Xm X) X)VX rV V = X; (56)

where it is assumed that the existing traded assets dynamically span theprice X. Let denote the correlation with the market portfolio, and themarket price of risk:

=(rM r)

M;

whererMstands for the expected return on the market portfolio and M itsstandard deviation. The solutionV(X; ) to the dierential equation mustsatisfy the following boundary conditions:

a) At = 0the value must be zero: V(X; 0) = 0.b) Bounded derivative as X! 1: VX(1; )< 1.c) Bounded derivative as X! 0: VX(0; )< 1.Using Laplace transforms we get:

1

2hXX

2X2 + (k(Xm X) X)hX h(r+ s) = X

s:

Rearranging:

1

2hXX

2X2 (k+ )XhX h(r+ s) = kXmhXX

s:


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C VALUATION OF AN ANNUITY IN PRESENCE OF MEAN REVERSION35

The general solution has the form:

h(X) =A1X1 + A2X

2 +X kXmk+

s(s + r+ k+ )+

kXm(k+ )s(s + r)

:

The derivative is bounded; thusA1= 0. Besides,h(0) = 0; thereforeA2= 0.The solution simplies to:

h(X) =X kXmk+

s(s + r+ k+ )+

kXm(k+ )s(s + r)

:

With the rst and second derivatives: hX = 1

s(s+r+k+) , hXX = 0, it ispossible to show, by substitution, that the dierential equation applies.

At this moment, the inverse Laplace transforms are taken. To do so weuse formula 29.3.12 in Abramowitz and Stegun [1], the nal result being:

V = kXm(1 er)

r(k+ ) +

X kXmk+r+ k+

(1 e(r+k+)): (57)

This formula may be useful to compute the present value of fuel costs overthe whole life of a plant with inexible technology, like an NGCC or coalplant.30

A series of particular, frequently used, cases may be derived from theabove general solution:

a) If = 0 or = 0, then the formula reduces to

V =Xm(1 er)

r +

XXmr+ k

(1 e(r+k)):

b) If! 1:

V = kXm

r(k+ )+

X kXmk+r+ k+

: (58)

In this case, it can be observed that the project value is the sum of twocomponents: one related to the reversion value and another one whichis a function of the initial dierence between the observed value and the

"normal" level ofX.c) When it is a perpetuity and also Xm= 0 and k + = , then:

V = X

r : (59)

d) When it is a perpetuity and also Xm= 0and k + = r + , then:

V =X

: (60)

30 See Bhattacharya [4](eq. 15) and Sarkar [27] (eq. 2-4).


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D THE PERPETUAL OPTION UNDER CERTAINTY 36

D The perpetual option under certainty

In this case = 0 and hence the fuel cost follows a deterministic path:St = Sm+ [S0 Sm]ekt (which corresponds exactly to the mathematicalexpectation of the stochastic case). If an investment is made at time T, itwill be worth: 454; 055; 483 83; 217; 315ST, which is the same value as inthe stochastic case when = 0. The only dierence is that this is a futurevalue and that ST = Sm+ [S0 Sm]ekT = 3:25 + 2:2e0:25T. Thereforethe present value at initial time of an investment to be made at time T isderived by discounting at the riskless interest rate:

[454; 055; 483 83; 217; 315[3:25 + 2:2e0:25T]]e0:05T:

Dierentiating with respect to T and setting the derivative equal to zero,the optimal investment time is T = 7:1557 years. In that moment thebenets from waiting for a sure drop in gas prices fail to balance the eectof discounting. Thus investment would take place when S = ST = 3:6177.This is the value S will approach to as ! 0.

Sensitivity analysisNext we analyze the sensitivity of the results with respect to changes in

, k and Sm.1. We assume several values for while the others remain the same

(k= 0:25, Sm= 3:25).

Table A1. Trigger price S as a function of 0.00 0.02 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

S 3.62 3.59 3.49 3.25 2.99 2.74 2.52 2.30 2.11 1.94As can be inferred from Table A1, the value of the option to wait

increases with volatility, which is reected in a lower threshold S.2. We consider several values for k while = 0:20 and Sm = 3:25.

First the special case in which k = 0 is computed;31 then the dierentialequation (7) has the form: 12F

002S2 rF = 0, whose solution is: F(S) =A1S

1 +A2S2 , where 1 > 0 and 2 < 0 are the roots of a quadratic

equation. Since F(1) = 0, it must be A1= 0. In our case, 2= 1:1583.The boundary conditions are:Value-Matching: A2(S

)2 = 1; 342; 055; 454

356; 448; 075S,Smooth-Pasting: A22(S

)21 = 356; 448; 075.They suce to determine the value ofS fork = 0, which is 2.0206.Computing the remaining parameters by the usual procedure, the results

are shown in Table A2.

31 The same results can be obtained by using equation (36) noting that if k = 0 then= 0, = 0, = 2:5, = 1:1583, A1 = 0 and M(a;b; 0) = 1. In this caseAm()

=A2.This shows that the solution using Kummers hypergeometric function includes othersimpler kinds of options as particular solutions.


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Table A2. Trigger price S as a function ofk

k V(S) I(S) Sk= 0:00 1; 342; 055; 454 356; 448; 075S 2.0206k= 0:05 928; 778; 968 229; 286; 079S 2.2594k= 0:10 712; 083; 008 162; 610; 399S 2.4276k= 0:20 507; 699; 469 99; 723; 156S 2.6608k= 0:25 454; 055; 483 83; 217; 315S 2.7448k= 0:30 415; 510; 405 71; 357; 291S 2.8143k= 0:40 364; 000; 824 55; 508; 189S 2.9225

According to equation (27), natural gas costs depend on k. Therefore,V(S) I(S) also depends on k. An increase in k pushes the trigger priceS upwards, since prices will drop faster. In the limit, with an innite

reversion speed, prices reach the equilibrium value instantaneously thusosetting the opportunity to wait until they fall. When k tends to zero,the result converges towards the values of the above case in which k = 0.

3. Finally, let us consider several values of Sm while = 0:20 andk= 0:25. First the special case in which Sm= 0is computed; although it isnot very realistic, it allows to analyze the convergence towards another typeof option. In this case, the dierential equation has the form: 12F

002S2 kS rF = 0, whose solution is: F(S) = A1S1 +A2S2 . Given thatF(1) = 0, it must be A1= 0. In our case 2 = 0:182712.

In this case the conditions:Value-Matching: A2(S

)2 = 1:342:055:454

83:217:315S,

Smooth-Pasting: A22(S)21 = 83:217:315,allow to determine the value ofS forSm= 0, which is 2:4914.After obtaining the remaining parameters by the usual procedure, Table

A3 shows the results.Table A3. Trigger price S as a function ofSm

Sm V(S) I(S) SSm= 0:00 1; 342; 055; 454 83; 217; 315S 2:4914Sm= 0:05 1; 328; 393; 916 83; 217; 315S 2:5016Sm= 0:50 1; 205; 440; 074 83; 217; 315S 2:5884Sm= 1:00 1; 068; 824; 964 83; 217; 315S 2:6737Sm= 2:00 795; 593; 934 83; 217; 315S 2:7925Sm= 3:00 522; 363; 173 83; 217; 315S 2:7848Sm= 3:25 454; 055; 483 83; 217; 315S 2:7448Sm= 3:50 385; 747; 793 83; 217; 315S 2:6769Sm= 3:75 317; 440; 103 83; 217; 315S 2:5645Sm= 4:00 249; 132; 413 83; 217; 315S 2:3687

It can be observed that changes in the trigger price S are relativelysmall in the range between Sm= 1:00and Sm= 3:50.

GivenS= 5:45, for very low values ofSmthe option value increases morethan the immediate investment value, with the equilibrium being reachedfor lower values ofS. For high values ofSm the option value is low, while


38/41


the net investment value takes on negative values which are oset if the

trigger price S falls more.


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REFERENCES 39

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