The Real Option Model – Evolution and Applications

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THE REAL OPTION MODEL EVOLUTIONAND APPLICATIONS

Jana Chvalkovsk

Zdenk Hrub

Institute of Economic Studies, Faculty of Social Sciences, Charles University, Prague, and Law Faculty, CharlesUniversity, Prague Institute of Economic Studies, Faculty of Social Sciences, Charles University, Prague


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1. INTRODUCTION

This paper presents an overview of the evolution, categories and application of the realoption model. The real option is a specific instrument for risk assessment developed on the

basis of a model created by Merton (1973) in his article The theory of rational optionpricing. This article follows the consequent development of this model into two evolutionbranchesone is represented by the simplified models based on the work of Box, Cox andRubinstein (1979), the second one expands the stochastic version of the Mertons model andcould be represented by the work of Dixit, Pindyck (1994).

Further on, the paper presents the general conditions for the application of the real option

model in investment valuation and summarizes the main categories of real options. This part

of the paper analyzes the major advantages of the utilization of the real option model as well

as its most important pitfalls.The introduction of the general principles of the real option model serves as a background

for the demonstration of the practical application of the real option models. Examined is both

the application of the simplified version of the model that found its main field of application

in management as well as the deployment of the more sophisticated, stochastic version of the

model. The overview of the real option applications is concluded by an example from the

energy sector that provides a practical manifestation of the utilization of both versions of the

model.

The aim of this paper is to analyze the application aspects of the real option theory in the

investment decision making. The base of the analysis is formed by the insight into the historyof the real option theory and of its two branches and thus provides a complex evaluation of

the real option context. The general description is accompanied by practical examples of the

real option use.


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2. OPTIONS VERSUS REAL OPTIONS

2.1 DEFINITION OF OPTIONS AND REAL OPTIONS

The real option model evolved and its valuation that is applied in praxis evolved from the

original modification theory option pricing that was developed by Merton (1973) based on

Black, Scholes (1972).

The currently most widely applied definition of option originates in finance and states that

option is a contract in which the writer (seller) promises that the contract buyer has theright, but not the obligation, to buy or sell a certain security at a certain price (the strike

price) on or before a certain expiration date, or exercise date. The asset in the contract is

referred to as the underlying asset, or simply the underlying. An option giving the buyer the

right to buy at a certain price is called a call, while one that gives him/her the right to sell is

called a put.1This definition basically corresponds to the one stated in the original work of Black,

Scholes (1972), who suggest that the option contract is a right to buy or sell another assetat given price within a specified period of time.

These definitions, however, are rather too much oriented on the perception of options as

mere financial market instrument. In order to understand the concept of real options, it is

necessary to understand the more general concept of option, which is provided in Merton

(1973)2, who understands the options as simple contingent-claim assets meaning that the

value of the assets depends on the occurrence of some specific state of the world.In financial terms, the Mertons definition can be understood as follows. The holder of theoption will exercise its right to buy the underlying asset in case of a favourable state of the

world. In case of a call option (see Farlex Financial Dictionary definition above), a

favourable state of the world is, when the market price of the underlying asset exceeds the

strike price established in the option. In such a case, the option becomes valuable, because it

enables its holder to purchase the underlying asset for a price that is lower than the market

price and consequently sell the underlying asset in the market, gaining thus the price

difference. On the other hand, in case of occurrence of the unfavourable state of the world,

1 Farlex Financial Dictionary (2009)2 pp. 141


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the value of the option becomes zero it will not be exercised. The financial option is acontingent-claim asset, because its value is conditioned by development of the price of its

underlying asset.

The real options are only another application of the contingent-claim asset concept.

According to Dixit, Pindyck (1994)a firm with an opportunity to invest is holding anoption analogous to a financial call optiona real option.3 In this case, for the firm theunderlying assets is the investment; the decision in front of the holder is basically to invest or

not to invest in the project, given the occurrence of conditions favourable for the project or

unfavourable, respectively. This concept is broader than the concept of the financial options.

The underlying asset of a real option can be practically anything including very complexprojects and investments. The favourable state of the world can be given by development of

prices of the inputs or outputs of the project, but also by more sophisticated conditions, such

as is approval of a specific legislation, etc.

2.2 TYPES OF REAL OPTIONS

Despite the fact that the definition of real option of Dixit, Pindyck (1994) cited in the

previous subchapter compares the real option of a firm to invest with a call option, this does

not mean that there are no real put options. The company may as well evaluate an

opportunity to divestand in this case the real option would be analogous to a put option.Similarly, the type of option structure deployed in the real option model is also not

restricted the models may deploy an American-style option as well as European-styleoption, Bermuda-style option or any type of exotic options depending on the structure of

the real-world evaluation in question. Nevertheless, the most frequently used structures ofreal options are the European- and America-style options due to their computational

feasibility.

There are also various uses for real option method as a project evaluation tool. BrealeyMeyers (2003) describe on page 617 following types:

the option to expand a project if the pilot investment is successful (option to keep a

project open)

the option to shrink or abandon a project

option to change inputs or outputs of the project (flexibility)

option to postpone the investment (timing)Again, the particular choice of real option model depends on the characteristic of the

underlying project or investment. In this paper there is a numerical and stochastic

demonstration based on European-style real option model of ex-ante project evaluation in the

energy sector.

3 pp. 4


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3. REAL OPTION EVOLUTION

The beginning of the option pricing model is connected with stock exchanges at the end of

the nineteenth century. In 1877, Charles Castelli published a book concerning the optionpricing and its relation with the price volatility of the underlying asset and the possibility to

use options in bond arbitrage between stock exchanges. The work of Charles Castelli was

rather descriptive and applied and contained only rather simple calculations (MacKenzie;

2003).

The mathematical principles of option pricing were explored by French mathematician

Louis Bachelier, who in 1900 published his dissertation thesis called Thorie de laSpculation. The key finding of the thesis was the application of arithmetic Brownianmotion on price movement modelingwhich is the basis of the contingent claims valuation.

Bacheliers pioneering work, however, did not achieve in his time wider recognition beingtoo much oriented on finance for mathematicians and too much sophisticated for financialeconomics (Courtault, et al., 2000, pp. 344). Thus, the continuous-time random walk model

of stock markets had to wait for nearly fifty years to be reintroduced by Paul Samuelson,

who, however, modified the model from arithmetic to geometric Brownian motion model4

(MacKenzie, 2003, pp. 17).

In mid-sixties of the last century, two economists with passion for earning money Kassouf, S.T. and Thorp, E.O.further developed the model depicturing in their book Beatthe Market (Kassouf, Thorp, 1967) graphically the relationship between the price of warrantand price of the underlying asset. The two economists later applied the theory in actual stock

trading for choosing delta-neutral hedge ratios of a stock versus its convertible bond andearned thus substantial amount of money Thorp (2003, pp. 36).

The seventies of the last century marked the real development of the continuous-time

random walk option pricing model. Merton (1973) and Black, Scholes (1972) developed the

theory of valuation of contingency claims based on Brownian motion. In this chapter, a

modified version of the Mertons (1973) model is deployed in order to demonstrate itsapplication in the energy sector. The method of option pricing developed by Merton, Black

4 The difference is that in arithmetic Brownian motion model the stock prices can be negative, whereas in geometricBrownian motion model Theky cannot (MacKenzie, 2003, pp. 17)


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and Scholes marked a new era in the pricing of financial market instruments and more than

thirty years later, won to Scholes and Merton5

the Nobel Prize for Economics (in 19976).

In the seventies and eighties of the last century the option pricing theory was blooming

and its applications were getting greater and crossed the borders of quantitative finance to

other fields of economics including management. The penetration of the option model tomanagement was facilitated by its simplification developed by Cox, Ross and Rubinstein

(1979) in their book Option Pricing: A Simplified Approach. The model of Cox, Ross andRubinstein (1979) model uses the binomial option pricing formula and applies it on discrete

time periods. A modification of the model is deployed in this paper for assessment of the

impact of carbon trading in the EU Emission Trading System on new power generation

projects.

Figure 1: Simplified scheme of the real option model evolution as summarized in this paper

The real option model evolved from the stock option valuation in the eighties and nineties

of the last century as a specific extension of the financial option models on real life situations

such as is investment planning, project evaluation, etc. (see more detailed description in thenext chapter). There are basically two modifications of the real option model that evolved

from the stock option valuation models (as suggests

Figure 1above).

The stochastic, mathematically more sophisticated real option models can be represented

by a work of Avinash Dixit and Robert Pindyck (1994) Investment under uncertainty,which provides a comprehensive overview of the modifications of the real option valuation

based on geometric Brownian motion models, mean reverting processes and Poisson jump

models. The book examines in depth the mathematical background of the various types of

real options as well as its applications in firms decisions, in industry evaluations, etc.

5

Fisher Black died in 1995, two years before the award (MacKenzie, 2003, pp. 51)6 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997 for a new method todetermine the value of derivatives (seehttp://nobelprize.org/nobel_prizes/economics/laureates/1997/).
http://nobelprize.org/nobel_prizes/economics/laureates/1997/http://nobelprize.org/nobel_prizes/economics/laureates/1997/http://nobelprize.org/nobel_prizes/economics/laureates/1997/http://nobelprize.org/nobel_prizes/economics/laureates/1997/


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As suggests Copeland and Tuffano (2004), the disadvantage of the real option models is

their complexity that frequently makes them impossible to apply on real world situations.

Therefore in the managerial decision making more widely used versions of the real option

model are those based on the simplified approach of Cox, Ross and Rubinstein (1979). These

models structure problems into binomial trees with assigned probability of upward or

downward movement of the value of the underlying project (or any other asset in question).

The mathematics behind this approach is rather less sophisticated than in case of continuous-

time random walk real option models, which makes them frequently less accurate, but on the

other hand less data demanding and more convenient to use (Copeland, Tuffano, 2004).

There are many books and articles describing in more or less details the methods of

application of managerial real option models. In this paper, I am deploying as an example a

model adopted and modified from Shockley (2006). Shockley (2006) in his book AnApplied Course in Real Options Valuation demonstrates several practical applications ofreal option modelling on valuation of existing projects from various industrial branches.

Similarly, Trigeorgis (1996) in his book Real Options: Managerial Flexibility and Strategy

in Resource Allocation also describes a wide range of managerial real option models andtheir applications.

As was already discussed, in this paper there are two examples of real option model

deployment on the investment decision making in the energy sector. One of the applications

is based on Deng, Johnson and Sogomonian (1999) and Merton (1973), the other represents

the managerial real option models and is based on Shockley (2006).


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4. APPLICATION OF REAL OPTION

4.1 GENERAL CONDITIONS OF REAL OPTION APPLICATION

The real option model is an alternative approach towards the investment valuation. The

model tries to capture in detail an important element of the investment decision making theuncertainty. The uncertainty could concern the future development of the markets and

especially the uncertainty about the future development of the regulatory framework. The

latter is often not fully captured in any of the traditional investment valuation model (such as

are the Discounted Cashflow (DCF) Valuation or the Relative Valuation approach7).

Laurikka, Piril (2005) discovered that in case of investments where the uncertainty is large8(such as is construction of new power plant) the application of the traditional valuation

methods may lead to systematic bias in the valuation.

The real option approach may have substantial advantages compared to the traditional

valuation methods, however, Scholleov (2008) suggests that on certain types of situations itcannot be applied. These situations can be summarized as follows:

decision making under certainty or zero risk; in this case the option value

disappears and the real option valuation equals to DCF

decision making that cannot be postponed or modified; the real option that

measures flexibility does not have sense when flexibility is not possible

twin options, when the option value would be assigned to more interdependent

projects; in such a case, the real option would over-valuate the flexibility low budget projects where the estimated option value would exceed the total

costs of the projects

Figure 2 adopted from Adner and Levinthal (2004) depicture graphically and simpler the

limits of use of the real option model. In case that the investment does not contain a

significant share of irreversible costs and/or is not tightly bound to uncertain factors the use

of real option is unnecessary. This definition is also described in Dixit, Pindyck (1994).

7 http://pages.stern.nyu.edu/~adamodar8

Harchaoui, Lasserre (2001) tested the efficiency of the real option model in evaluation of irreversible investmentsand their empirical results supported the hypothesis that in case of large, long-term investments (they were testinginvestments into mining and mineral extraction) the real option model is an accurate valuation tool.
http://pages.stern.nyu.edu/~adamodarhttp://pages.stern.nyu.edu/~adamodar


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Figure 2: Boundaries of Applicability for Net Present Value and Real Option (source: Adner, Levinthal,

2004)

4.2 AREAS OF REAL OPTION APPLICATION

There are multiple industries and sectors, where real option can be a more convenient

valuation tool than traditional valuation tools. The common characteristics of investment in

these sectors are that they are expensive, long term, affected by multiple risks (market risk,

regulatory risk, political and social risks, etc.) and are formed in large part from irreversible

costs. Thus standard valuation methods cannot fully capture their real value and result in

biased results.

Among the sectors, where real option valuation could be successfully deployed belong

above all the following:

Mining of minerals

Pharmaceutical industry

Research and development of hi-tech products (biotechnologies,

nanotechnologies, etc.)

Information and telecommunication technologies

Aeronautics

Energy production and transmissionExploitation of minerals is a very resource- intensive and expensive activity. Opening of

new mine takes preliminary works on geological exploration, negotiation of licenses and

contracts, investments in heavy machinery are large expenses that are from major part

irreversible. Development of a new mine or operation of an existing one is a long run project

that is subject to several types of uncertainty including the volatility of prices of the mined

commodity, changes of environmental, labour and other regulation, technological shocks,

etc.

Pharmaceutical industry and any sector manufacturing hi-tech products (such as

biotechnologies, nanotechnologies, etc.) are heavily dependent on research and innovation.

However, research is an expensive, long-term activity with uncertain results. Besides due to


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rapid development in these sectors, the product prices might be volatile and making revenues

from any project uncertain.

In information and telecommunication technologies, the rapid development of the industry

is a principle cause of the uncertainty of investments in this sector. Companies in this sector

have to be quick in developing new products in order to stay ahead of its competitors, but the

tough competition causes that the revenues from new products in the sector are hard to

predict, uncertain, respectively.

Aeronautics is a sector characterized by extremely long and extremely costly project

development. Design, construction and testing of a new type of airplane take years, the

revenues in the sector are volatile and the competition in air transport is tough. Thus the

conditions for application of real option are fulfilled in this sector.

4.3 APPLICATION OF REAL OPTION IN ENERGY SECTOR

The energy sector is marked by a high degree of uncertainty, which is caused by volatility of

prices of energies and energy commodities in the international markets as well as by the fact

that the investment projects in this sector are time-consuming, large-scale and extremely

expensive. In the following paragraphs, we will describe some examples from the literature

concerning the real option applications on cases of power plants projects.

4.3.1 Nuclear power plants

Nuclear power plants are source of electricity that vastly differs both from the renewable

sources as well as from the traditional fossil fuel power plants. From the climate changeperspectivecarbon dioxide emissions from nuclear power plants are close to zero and thusthe nuclear power plants seem to be appropriate solution for power generation in the carbon

constrained economy.

The boom of nuclear power plants is restrained by several sources of uncertainty (that

compose an important part of real option). Firstly, the nuclear power plants are using a

specific high-end technology that can be produced only by a limited number of companies9

that can thus influence the prices (volatile prices of production technology). Secondly, the

entire nuclear fuel production chain is submitted to sever regulation due to concerns aboutsecurity (potential abuse of the fuel for construction of nuclear weapons), environmental

safety (damages caused on environment by radiation especially from improper storage of

nuclear waste etc.) and about other risks related to human health, transportation of the

enriched fuel etc. (regulatory risk). Thirdly, construction of nuclear power plants takes long

time. Only the environmental impact assessment that is a necessary legal condition for

nuclear power plant construction lasts about 2- 3 years. The construction of the plant itself

lasts according to EG (2005) between 12 15 years, the life-cycle of the plant is then 40 50 years. Such a long duration of the project makes it vulnerable to changes in the political

9 Such as is the Russian Tvel Corporation (fuel production), AREVA NP (construction of reactors), UniStar NuclearEnergy, LLC (technology) or the US Westinghouse Electric Company


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and social preferences (political and social risks). Lastly, the value of the project is, of

course, sensitive to changes in market prices of electricity, uranium, etc. (market risk).

Taking into account the fact that the initial investment into a nuclear power plant exceeds

(according to EG; 2005) CZK 70 mil. per MW of installed capacity10 and is from majorpart irreversible, the nuclear power plants are a textbook example of an area of application of

real option model. Therefore, there is a substantial amount of literature describing the

application of real option in nuclear power plant projects.

For example, Roques et al. (2005) model how the value of leaving open the nuclear option

declines with the increasing prices of fossil fuels and of carbon. In their study they compare

the nuclear power plant with the Combined Cycle Gas Turbine and realize that high capital

cost, uncertain construction cost, potential construction and licensing delays, and economies

of scale are the main features that make nuclear power technology unattractive to private

investors in liberalized electricity markets. As the main driver for the current renewed

interest in the nuclear power was identified the security of supply concerns and the aim on

fuel mix diversification.

Graber, Rothwell (2005) examine the option value of a project focused on development ofopportunities for nuclear power plant construction. They realize that the high costs connected

with such commitment (the duration of this development project is estimated as approx. 56

months) are outweighed by the gained option to construct the nuclear power plant in the

future.

Kiriyama, Iwata (2005) analyze the value of an investment in power generation assets that

do not emit CO2 in the carbon constrained world. They compare the nuclear power plant with

other energy sources not dependent on fossil fuels with wind and photovoltaic powerplants. They discover that under all evaluated scenarios of the CO2 price developments the

nuclear power plant is a better option than the wind and photovoltaic power plants.

4.3.2 Combined Cycle Gas Turbine natural gas

The gas-fired combined cycle gas turbine power plants (CCGT) takes only 3 5 years toconstruct

11, has relatively low initial investment costs per MW

12and is flexible (could be

switched on and off according to peak-load hours). Its good efficiency and low minimum

technical capacity are its other positive features. In the carbon constrained world the CCGT

has also a comparative advantage compared to other fossil fuel plants for 1 MWh ofelectricity it produces only 0.43 tonnes

13of CO2 it is thus less vulnerable to the climate

change regulations than the coal- and lignite-fired power plants (that produce 0.9 tonnes of

CO2 per MWh, 1.25 respectively).

Compared to other power plant types, CCGT has relatively lower proportion of

irreversible costs, but is very vulnerable to changes of natural gas prices. It is also sensitive

10 And the optimal size of a nuclear power plant project is about 1000 MW of installed capacity.11 Still, the average estimated lifetime of a CCGT plant is about 25 30 years therefore it fits in the conditions

(principles) for application of the real option model.12 The optimal size of CCGT can range from very small projects up to large-scale ones.13 David (2006)


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on regulatory and political risks above all due to the fact that natural gas as fuel is quiterelated with the public preferences of environmental-friendly technologies.

Despite the fact that the sunk costs of CCGT are considerably lower than the costs of a

nuclear power plant, there is a considerable amount of literature on the real option

application on the CCGT.

Laurikka (2005) investigates the relationship between the value of flexibility of natural

gas power plants (option to alter the operation scale, to switch between products etc.) and the

climate policy. The findings in this study support the hypothesis that the climate policies

increase the option value of flexibility in power generation. As most flexible resulted the

cogeneration heat and power plant (CHP) and a multi-fired power plant (plant that can

combust more than one fuel).

Wickart, Madlener, Jakob (2004) develop a model that shall explain the investment

decision of large industrial companies, when they are deciding between construction of

private CHP plant or simple thermal unit and electricity from the grid. Focus of the analysis

is to examine the Swiss CHP regulation. The authors discovered that Swiss regulatory

framework in fact to some extent hinders the development of large CHP plants that couldpotentially endanger the Swiss emission reduction targets (given the large share of nuclear

and hydro power plants on the Swiss fuel mix).

Fleten, Nskkl (2003) analyzed the investment in gas-fired power plants understochastic development of natural gas and electricity prices. They discovered that by the time

of the investment decision the option to abandon the project does not have a significant value

unlike the timing and flexibility option.

Hirschl, Schlaak, Waterlander (2007) examined the investors choice between the CCGTand PCB-C. The CCGT results more economical only in case of low gas prices and relatively

high EUA price (NPV scenarios, IRR valuation methods). Teisberg (1993) shows with use of

the real option model why companies rather invest into smaller, shorter-lead plants (such as

are the small gas-fired plants) rather than in large projects. Alstad, Foss (2004) provide a

complex analysis of the investment decision to build a CCGT power plant in Tjeldergodden

in Norway. Analyzed is above all the option to postpone the investment given the future

development of natural gas and electricity prices.

4.3.3 Pressurized Coal Boiler - hard-coal (PCB-C) and lignite (PCB-L)

The PCB-L and PCB-C are the most traditional types of power plant. They are robustbase-load electricity sources usually built close to hard coal or lignite mine because of

transportation costs reduction. The fact, that both types of coal-fired plants usually use

domestically produced fuel could be considered as their major advantage, especially given

the EU concerns about the security of supplies and increasing dependency of the EU on

imported fuels. The initial investment costs of PCB-C and PCB-L lie somewhere in the

middle between the CCGT and nuclear power plant. EG (2005) estimated the investmentcosts as approx. CZK 38.5 mil. (PCB-C) and CZK 43 mil. (PCB-L) per MW of installed

capacity14

. The construction time is expected to be about 57 years.

14 The optimal size of the project is about 600 MW of installed capacity.


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There are several disadvantages connected to the PCB-C and PCB-L power plants. First of

all, they are extremely vulnerable to climate change policies. According to David (2006) the

PCB-C plants emits 0.9 tonnes of CO2 per MWh, the PCB-L emits 1.25 tonnes of CO2 per

MWh. They are also less efficient than the CCGT, based on EG (2005) the difference inefficiency between CCGT and PCB-C exceed 10%. The coal-fired power plants are also

subject to sever environmental regulation regarding the amounts of emitted.

Recentlyas a reaction on the carbon limitationsappeared several new coal combustiontechnologies. Reedman, Graham, Coombes (2006) compare the advanced coal technologies

deployed in Australia with use of the real option model. The compared technologies are:

Supercritical pulverized-fuel black-coal power plant

Black-coal-fuelled integrated gasification combined cycle power plant (IGCC)

Supercritical pulverized-fuel black-coal power plant fitted with post-combustion

capture of carbon dioxide

A black-coal-fuelled integrated gasification combined-cycle power plant fitted with

pre-combustion capture of carbon dioxide

The Australian study demonstrated that especially the carbon capture technologies have ahigh value of the option to wait because of the initial costs of the technology. Sekar et al.

(2005) also assesses the option to either postpone an investment into new coal combustion

technologyIGCCwith the standard pulverized coal technology in the US. The analysissimilarly to the Australiandiscovered that in most of the designed scenarios the pulverizedcoal technology stayed cheaper. The IGCC became more economical only under quite

extreme future carbon regulation.


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5. DEMONSTRATIONS

5.1 STOCHASTIC EXAMPLE

As was already described, the use of real option models goes beyond the borders of the

economic theory and the real option models are still more widely applied on evaluation of

investments in diverse industrial sectors. As was mentioned in the previous chapters of this

paper, there are several criteria under which the real option model can have better

explanatory power than any standard valuation tools.

Among the criteria belong the proportion of irreversible costs on the total value of the

investment, the length of the pay-back time of the investment and significant uncertainty

about the future development of factors that determine the costs or revenues of the

investment. A textbook example of such investment project is the construction of a new

power plant, which is both sizeable investment as well as an investment with a substantial

proportion of irreversible costs.

This chapter provides an example of a continuous-time stochastic real option model. This

model is often quoted in the literature in relation to the valuation of power generation assets

as a spark spread option model.

The aim of the spark spread option model is to evaluate the option of the investor in a

power generation plant to construct a gas-fired power plant. The model illustrates, how the

valuation of an unit of power generation capacity is driven by the difference between theelectricity prices and the value of the power plant input, natural gas. The risk is embedded

into the model through the natural gas and electricity prices that are inserted in a form ofcontinuous-time stochastic processes.

In this example, I will provide a summary of the spark spread option model as described

in Deng, Johnson, Sogomonian (1999) with an insight into the key steps of computation of

the PDE that I modified from Merton (1973). At the end of this subchapter, there will be a

brief discussion of other possible modification of the model and of their potential application

on energy sector and on risk evaluation of other long-term investment.


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5.1.1 Basic spark spread option model

This example corresponds to the model for the energy sector developed by Deng, Johnson,

Sogomonian (1999) and further evolved by the same authors of case of the exotic forms of

the real options (Deng, Johnson, Sogomonian, 2001). This development of the model will be

based on the initial model of Deng, Johnson, Sogomonian (1999) demonstrating itsderivation based on Merton (1973).

The spark spread option model is based on the idea that to power generators what matters

most is the spread between fuel costs (or costs in general) and the price of electricity. The

value of the spread might be denoted as follows:

= SEKHSF (5.1)where SE is the spot price of one MWh of electricity, SF is the spot prices of fuel (natural

gas) quantity that could be converted into one MWh of electricity and K H is the heat rate of

the power generation asset (a coefficient that stands for the technological parameters of the

boiler that is transforming the fuel into electricity).

The initial assumptions of the spark spread model are following: the markets, where electricity and the fuel are traded, work efficiently (no windfall

profits etc.) and are deregulated

there are no transaction costs or differential costs

trading takes place continuously

only assets with positive spark spread are operated (this assumption follows from

the previous one)

a complete set of financial instruments is traded in the market (in order to create the

replicating portfolio), there are no restrictions on buying and short-selling

the rate for borrowing and lending is the same the option to the project behaves as European-style option written on the spread

between price of electricity and of specific fuel (F) at a fixed heat rate KH

A European-style spark spread call option gives to its holder the right (not the obligation)

to pay KH times the unit price of the fuel F at the options maturity time (T) and receive theprice of one unit of electricity. The pay-off of the option at maturity date is:

C(ST

E, ST

F, T)=max(ST

EKHST

F, 0) (5.2)

Analogously, the European-style spark spread put option gives to its holder the right (not

the obligation) to pay for one MWh of electricity and receive KHtimes the price of the fuelat maturity time T.

P(STE, STF, T)=max(KHSTF - STE, 0) (5.3)The put- call parity can be expressed as:

CP= max(STEKHST

F, 0) - max(KHST

FST

E, 0) (5.4)

Assuming that

he risk-free rate (r) is constant

future spot price of electricity and the future spot prices of fuels at time t discounted

by the discount factor e-rt

equal to futures prices of electricity and futures prices of

fuels, FtE and F

tF, respectively

FtE = e-rt StE (5.5)

Ft

F = e-rt

St

F (5.6)C=e-rtmax(FtE - KH FtF) (5.7)


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P= e-rtmax(KH FtF - FtE) (5.8)The put-call parity can be expressed in a following way:

C1=P1+e-rt

(FtE - KH F

tF) (5.9)

From (5.4), (5.7) and (5.8) can be derived the lower and upper bounds of the European

spark spread call option. FtE and F

tG are the future prices of the electricity and of the fuel F,

respectively. The value of a spark spread call option C has then both lower and upper

boundary that are given as:

e-rt

max(FtE - KH F

tF, 0) C1 e

-rtF

tE (5.10)

In order to estimate the uncertainty about the future development of prices the variables

are in stochastic real option models described with use of continuous-time stochastic

processes (Dixit, Pindyck, 1994). The simplest continuous-time stochastic process is the

Brownian motion (Wiener) process, which I will use for description of the Spark spread

option model. Later in this chapter I will describe the potential use of other stochastic

processes.

The Brownian motion process B(t) has following properties (Dixit, Pindyck, 1994):

it is a Markov process the probability of distribution of all future values dependsonly on the current value and is unaffected by past values of the process

it has independent increments (dB(t))meaning that the probability distribution forthe change of the process in the time is independent of any other non-overlapping

time interval

changes in the process over a finite time interval are normally distributed (in case of

prices - and other variables that cannot fall below zero - we would assume that the

logarithms of the price are normally distributed) with a variance that increases

linearly with the time interval

dtdB t ; (dB)=0; var(dB)=dt; if t0 then B ~N(0, 1)(5.11)

Let us assume that:

the price processes of electricity and fuel futures (denoted as Fte and F

tf) follow the

geometric Brownian motion process

e

ee

e

e dBdtF

dF (5.12)

f

ff

f

fdBdt

F

dF (5.13)

where

[(dBe,f)2]=dt; [dBedBf]=dt (5.14)B

1and B

2are the Brownian motion processes (dB are the increments of the Wiener

process) with instantaneous correlation . In this model, the coefficients e, f (growthparameters), e, f (proportional variance parameters) are assumed to be constant (whereas inthe model with the mean-reverting process they will be functions of time).

The value of the spark spread call option, which matures at time T, is denoted as

V(x,y,t)Ce-rt(Fet,T

, Fft,T

, T-t) (5.15)

Fet,T

and Fgt,T

are the commodity futures prices at time t with maturity date T.

Based on Merton (1973) the following steps has to be made in order to obtain the partialdifferential equation (PDE) suggested in Deng, Johnson, Sogomonian (1999):


17/35

x

xx dBdtx

dx (equation 5.12 in simplified notation 5.15) (5.16)

y

yy dBdty

dy (5.17)

x,y are instantaneous expected returns and x,y are instantaneous variances of the expectedreturns.

assumed is no serial correlation

tstdBsdB

tstdBsdB

xy

yy

;0),(),(

;0),(),(

(5.18)

correlation between future prices with different maturity is possible

dtTtdBtdB T,),(),( (5.19)

instantaneous correlation between x and y),( yxcorr

(5.20)

We assume that the option price function V (5.15):

has second derivation in x, y and first derivation in tThen we can apply Itos Lemma (Fundamental Theorem of Stochastic Calculus15 and

derive the option price function V(5.15):

dxdyyx

Vdy

y

Vdx

x

Vdy

y

Vdx

x

Vdt

t

VdV

222

22

2

2

22(5.21)

from (5.16) and (5.17) we compute dx

2

and dy

2

and substitute into (5.21):2223

2

222 2 dtxdtdtxdx xxxx (5.22)

2223

2

222 2 dtydtdtydy yyyy (5.23)

if dt0 then dt2/3 and dt2 go to zero faster than dt and could be ignored16

)2(2/12222 dtyVxydtVdtxVdtVdyVdxVdV yyyxyyxxxxtyx (5.24)

when dx and dy from (5.16) and (5.17) is substituted to (5.24) we obtain:

)2(2/1)()(

2222

dtyVxyVdtxVdtVdBdtVdBdtVdV yyyxyyxxxxty

yyy

x

xxx (5.25)

This equation can be simplified if we replace some parts of the expression as follows:yx

VdBVdBVdtdV

(5.26)

where

15 Itos Lemma (Fundamental Theorem of Stochastic Calculus) expresses, how the value of an option reflects the

underlying stochastic processes (for more detail see Dixit, Pindyck, 1994, page 80)16 from now on I will use the simpler notation, where Vx is , Vxx is etc.


18/35

)2

1

2

1(

1 2222yyyxxyyyxyyxxxx VyVxVyVxyVxV

V

(5.27)

V

yVyy1

(5.28)

VxVxx

1

(5.29)

The elements of the replicating portfolio (used for the option-pricing) could be described

as a combination of W1, W2 and W3. Where W1 denotes the amount of money invested into

the electricity futures; W2 the amount of money invested in the option and W3 the amount of

money invested into the fuel futures. The R denotes the return of the portfolio, dR denotes

the instantaneous return to the portfolio and the condition of zero aggregate investment is

expressed as:

W1+W2+W3=0 (5.30)

y

dyW

V

dVW

x

dxWdR 321

(5.31)

Into (5.31) we substitute for dx/x, dV/V and dy/y

W3= -W1- W2 (5.32)

yy

x

xyyx dBWWWdBWWdtWWdR )()()( 2122121 (5.33)

The coefficients W1 and W2 can be set in such a way that the resulting dR will be non-

stochastic

0)(;0 '2'

1

'

2

'

1 WWWW yyx (5.34)together with the equilibrium condition

0)()( '2'

1 WW yyx (5.35)A non-trivial solution of the 3*2 linear system exists if and only if

y

y

xyx

y

(5.36)

From the non-trivial solution and from the definitions of , , follows:

yx

1

(5.37)

yx

xx

x VxV

1

11

(5.38)

y

yy

xV

yV

VxV

1

1

(5.39)

V

yV

VxV

y

x 11

(5.40)


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y

yx

y

)(

(5.41)

)()2

1

2

1(

1 2222yx

x

ytyyxxyyyxyyxxxxV

xVVyVxVyVxyVxV

V

(5.42)

02

1

2

1 2222 tyyyxyyxxxx VyVxyVxV

(5.43)

The equation (5.43) is the linear second-order partial differential equation of the parabolic

type used for European-style spark spread valuation.

The draft of crucial steps of solution of the PDE (5.43) is again based on Merton (1973)

and modified in order to obtain the solution in Deng, Johnson, Sogomonian (1999).The boundary conditions of a European call option with price V are:

V(0,y,t)=0 (5.44)

V(x,0,t)=x (5.45)

V(x,y,0)=max(x-y,0) (5.46)

We define an auxiliary variable z,

z(t)=x/y (5.47)

dz(t)/z is defined by (16) and (17) and if applied the same procedure with Itos Lemma,we obtain

yy

xxyxyyx BBdt

zdz )( 2 (5.48)

we define the instantaneous variance of return on z(t) as

yxyxz 2)var(22 (5.49)

and new variable v(z,t) independent from y and

v(z,t)=V(x,y)/y (5.50)

if v and var(z) are substituted into (5.43), (5.44), (5.45) and (5.46) and assuming the

homogeneity of v, we will get

yxxyxxyxyx vvzzvvz )var(21)2(

21 2222 (5.51)

The modified boundary conditions for v are

v(0,t)=0 (5.52)

v(x,0)=max(0, x-1) (5.53)

V is homogenous degree 0 in x,y (see Merton, 1973, 166) and =var(z)(T-t)and insert it into (5.51), we obtain

02

1 2 yxx vvz (5.54)


20/35

To solve the equation we further introduce the change-of-variable transformation:

Z=ln(z)+(T-t)/2 (5.55)

(Z,)=v/z (5.56)

the problem will transform to a form that could be according to Merton (1973, page 167)

solved by separation of variables or Fourier transformation and result in the solution of the

Black-Scholes PDE:V(x,y,)=x(v1)ye

-rt(v2) (5.57) is a standard normal cumulative distribution function

duu

xx

)2

exp(2

1)(

2

1,0 (5.58)and v1 and v2 equal to

))(2(

))(2(

2/))(2()/ln(

22

12

22

22

1

tTvv

tT

tTyxv

yyxx

yyxx

yyxx

(5.59)

If we substitute for V(x,y,t) the original C=e-rt

(Fet,T

, Fft,T

, T-t), the result will be the closed

form solution of the Black-Scholes partial differential equation the value of the sparkspread European call option (as suggested in Deng, Johnson, Sogomonian, 1999):

C(Fet,T

,Fft,T

,T-t)=e-r(T-t)

(Fet,T(v1)-KHFf

t,T (v2)) (5.60)where v1 and v2 equal to

))(2(

))(2(

2/))(2()/ln(

22

12

22

22,,

1

tTvv

tT

tTFKFv

ffee

ffee

ffee

Tt

fH

Tt

e

(5.61)

Deng, Johnson, Sogomonian (1999) uses the above spark spread call European option

value (58) for valuation of the power generation assets. In order to be able to do that they

impose on the model several assumptions:

considered is only an investment into new power plant

operational and maintenance costs are assumed to be stable over time

assumed is that the prices of fuel G and electricity follow the Wiener (mean-

reverting, mean-reverting with Poisson jumps) process no depreciation of the power plant

Then value V of one unit of capacity of power plant with useful life T has both lower and

upper bound (this follows from the definition of the spark spread European call option and

from the no arbitrage condition):

T T

t

e

rtt

fH

t

e

rt dtFeVdtFKFe0 0

)0,max(

(5.62)

The estimated value of the power generation asset on a competitive market can be

expressed as follows:


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T T

t

fH

t

e

rtdtvFKvFedttC

0 0

21 )()()(

(5.63)

The estimated value of one unit of power generation asset is thus expressed as a right to

operate a power plant that generates electricity from fuel F over certain time period.

5.1.2 Modifications of the spark spread option model

There are several modifications and extensions of the above derived model. Deng,

Johnson, Sogomonian (1999) use for modeling of the price of electricity and fuel a mean-

reverting process instead of the Brownian motion with a drift.

According to Dixit, Pindyck (1994) the simplest mean-reverting process is the so called

Ornstein-Uhlenbeck process, which would in this case look as follows:

dBdtFFdF )( (5.64)where is the speed of reversion and F is the price level to which the price tends to revert.

This is also the main difference between the Wiener process - which in the long run could

grow indefinitely and between the mean reverting process which is anchored to the Fprice level.

17The expected value and variance of the Ornstein-Uhlenbeck are according to

Dixit, Pindyck (1994):

tt eFFFF )( 0 (5.65)

)1(2)var(2

2t

t eFF

(5.66)Deng, Johnson, Sogomonian (1999) use a modification of the mean-reverting process in

following form:e

eeeeeee dBFtdtFFdF )()ln( (5.67)f

fffffff dBFtdtFFdF )()ln( (5.68)where is the long-term mean of the price, is the speed of reversion and the variance

parameters are time-dependent; dB are again two Wiener processes with instantaneous

correlation .The price processes in the mean-reverting form are again inputted to the partial

differential equation (5.43) and solved and solved analogously to the solution for Brownian

motion processes. The solution would be then again C1(Fet,T

,Fft,T

,T-t)=e-r(T-t)

(Fet,T(v1)-KHFf

t,T

(v2)), but the value of v1 and v2 would in this case be18

:

17Hence the mean-reverting process satisfies the Markov property, but does not have independent increments.

18 the mean-reversion parameters of the futures price of electricity and generating fuel do not enter the pricing

formula of a spark spread option since the futures contracts of electricity and generating fuels are traded securitiesand therefore the mean-reverting effects are eliminated through the construction of the replicating portfolio usingtraded future contracts; Deng, Johnson, Sogomonian (1999; page 4)


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tT

dssss

tTdv

tT

tTFKFv

T

tffee

Tt

gH

Tt

e

)2()()(2)(

)(

)(

2/)()/ln(

22

2

2

12

2

2,,

1

(5.69)

The main advantage of use of the mean reverting process in the formulation of the real

option value is that it provides more realistic approach towards modelling of the future

development of fuel and electricity prices. In the numerical model in this thesis I have

assumed that the fuel prices will follow a linear trend. This assumption is of course

simplifying, but so would be in fact the application of the Brownian motion process. The

Brownian motion process with a drift assumes that the fuel and electricity prices will grow

indefinitely. The mean-reverting process, on the other hand, assumes that there is certainfactor that anchors the prices in the long-run. Dixit, Pindyck (1994) suggest that in case of

raw commoditiessuch as are also the fossil fuels, their price might be related to the long-run marginal production costs. In such case, the mean-reverting process would be able to

better capture this element in the estimation of price development.

The above described spark spread option model could be used for assessment of the

option value of the power generation assets in relation to the prices of EU emission

allowances19

(this means in case analogous to the numerical case described below). Similarly

to fuel prices, EU emission allowances can be also bound to the volume of production of the

power plant and influence the variable costs of the power plant. The transformation factor in

case of EU emission allowances will again be the technology used in the power plant, butnow it would include the carbon intensity factor. The estimation of the real option based on

EU emission allowance prices will be, however, complicated by the regulatory uncertainty

influencing the EU emission allowance prices.

There is, however, a process that could describe the price development of EU emission

allowances including the price shifts related to changes in the regulatory framework. The

process uses a combination of a Brownian motion process and of Poisson-jump process. This

combination reflects that the variable follows prevailingly some continuous stochastic

process with occasional discrete jumps. According to Dixit, Pindyck (1994) the jumps can be

of fixed or random size and their arrival times follow a Poisson distribution. The common

notation is that stands for mean arrival rate during certain time interval. For aninfinitesimally short interval dt this means that dt represents the probability that an eventwill occur, 1- dt is the probability that the event will not occur. The jump process can thusbe described as:

)(1

)1(0

dtyprobabilit

dq

dtyprobabilit

(5.70)

combined with the geometric Brownian motion the process would be

19 Allowances for emissions of greenhouse gases within the framework of the EU Emission Trading System.


23/35

dx=f(x,t)dt+g(x,t)dq (5.71)

where f(x,t) and g(x,t) are some known non-random functions. In case of EU emission

allowances the infrequent jumps in the model could illustrate the destabilization of the

markets caused by negotiation of the rules of emission trading in the EU or by changes of the

allocation method. Thus the model would have to capture jumps that arrive in quite regular

time periods, but are of unknown size. This approach would, however, require further

research.20

5.2 NUMERICAL EXAMPLE

The numerical example demonstrates the application of the standard binomial option-pricing

model that was originally designed by Cox, Ross, Rubinstein (1979). The inspiration of the

setting of the problem was a managerial modification of Box, Ross, Rubinstein (1979)

published in Shockley (2006). Shockley (2006) uses the dynamic valuation of investment

decision of a Bowen power plant in Georgia under the price uncertainty within the US SO2

emission trading. The company had three main options either to buy the necessaryallowances or switch to low sulphur coal or install a scrubber. The real option model was

used in this case for design of an optimal dynamic strategy of the power plant in this

situation.

In this paper the real option is being applied on an investment-decision in a backgroundsimilar to the one solved by Shockley (2006). The evaluated real option in this model

example is the decision of an investor to build either coal-fired or gas-fired power plant.

Similarly to the stochastic example, the drivers of the investment decision are in this case

also the prices of inputs and output of the power plant (in this case the prices of natural gas

and electricity are accompanied by prices of hard-coal and of prices of EU emission

allowances). The difference from the previous example is that this type of real option model

does not utilize continuous time modelling and is deterministic. This model operates with

fairly strict assumptions, hence it serves more as a tool for managerial decisions rather than

as analytical tool. On the other hand, this model can operate even with quite limited amount

of data and can be thus used for instance as an instrument in preliminary evaluation of

projects under uncertainty (regulatory uncertainty, price developments uncertainty, etc.).

5.2.1 Case study: Impacts of EU Emission Trading System on investment decision making of power generators

20 Stochastic models with jumps and spikes (although not for EUA modeling) were applied e.g. by Deng (2000) orEthier (1999).


24/35

This example assesses the impacts of the regulatory uncertainty in the sector of power

production. The case evaluated is, as was already mentioned above, the choice of an investor

to construct a gas-fired or coal-fired power plant. The regulatory uncertainty stems from the

potential change of rules of the EU Emission Trading System (EU ETS) that affects the

operational costs of power plants.

The coal-fired power plants21

has, in general, lower operational costs that gas-fired power

plants, because of lower price of hard-coal relative to natural gas. On the other hand, coal-

fired power plants have relatively high emissions of greenhouse gases compared to gas-fired

power plants and therefore their operational costs under EU ETS can be high - in case that

for each tonne of greenhouse gas emissions it has to buy emission allowance (EUA).

Figure 3 illustrates a case, when the EU ETS will oblige power plants to purchase EUA to

cover some significant part or all their greenhouse gas emissions. In such a case, the variable

costs of the gas-fired power plants (CCGT technology) will be lower than those of coal-fired

power plants. In case that some significant part or all the EUA will be given to the power

plants for free, the coal-fired power plants will stay cheaper in terms of variable costs. The

reason, why the real option model is applied, is that currently, the investor does not know,whether the EUA will be granted for free or not. Therefore, due to this regulatory risk, the

investor may also take into account the possibility to postpone the decision what power plant

to build until the EU ETS structure is known. This possibility is evaluated by the real option

model. The investor has thus three options to build a coal-fired power plant, to build a gas-fired power plant or to postpone the decision (this option has opportunity costs of lost

revenues from sales of electricity).

Figure 3: Merit order curve of power plants in the Czech Republic and the potential impact of EU ETS (merit

order curve adopted from Svoboda (2007))

21 Combined Gas Cycle Turbine (CCGT)


25/35

The numerically assessed real option is structured as a European-style option that can be

exercised only once a year at certain date. Thus the option can make only one

upward/downward step per year. The EUA22

is assumed to only take a step up or down, but

not both simultaneously. Tested implied volatility of EUA prices in the coming phase of the

EU Emission Trading System and beyond are set as 20% (Shockley 2006), and 50%.The equations describing the steps are:

Upward step U=e*t, where is the volatility, t=1 year

Downward step D=1/U

The probability of the upward movement is as follows:

Probability of an upward step q=(exp(risk-free interest rate)-D)/(U-D); where

the risk free rate is 5% (discount rate)

Probability of a downward step 1-q=(U-exp(risk-free interest rate)/(U-D).

The model is structured as a basis for managerial decision fixed to a certain date23

.

The assessed problem is structured as a cost minimization problem that could be described

in the following notation:min C (initial investment, M&O

24, fuel, EUA)

s.t. power plant type

power plant operation mode

EU Emission Trading System: allocation structure for next phase

of trading

EUA price (2013 and beyond).

Power plant types assessed are the Combined Cycle Gas Turbine (CCGT) and the

Pressurized Hard Coal Boiler (PCB-C). The reason for this choice is the proximity of costs of

these two power plant types as suggested in the picture below.The CCGT and PCB-C do not operate in the same way. PCB-C is a typical base-load

source. This means that switching a PCB-C power plant on and off again according to the

momentary electricity demand is not possible from technological reasons25

. The CCGT is a

classic peak-power plant. It is operating only during hours with highest electricity demand

and the rest of the time it is switched off. Therefore, two potential operation modes of power

plant types are being assessed:

both power plants as base-load sources5 000 hours per year per MW installed(4 867 t.a.

26hours CCGT, 4 667 t.a. hours PCB-C), base-load electricity prices

PCB-C as base-load5 000 hours per year per MW installed (4 667 t.a. hours

resp.), base-load prices and CCGT as peak-load3 000 hours per year per MWinstalled (2 920, resp.), peak-load prices

22 EU emission allowance23 All data adopted from Carbon and Energy exchanges as well as the exchange rate adopted from the Czech

National Bank exchange rate are set as of July, 10, 2008 in order to avoid problems with calculation of long-termprice averages of commodities traded in foreign currencies under current appreciation of the Czech crown.24 Maintanence and operational costs25The coal-fired boiler has a very long start up time. Switching the coal-fired boiler on and off would thus be very

expensive and would further decrease the efficiency of the PCB-C plant.26Technically attainable amount of hours per year.


26/35

The future allocation structures illustrate the amount of EUA that the individual industrial

installations will have to purchase. due to auctioning or substantial emission reduction

targets. Based on recent statement of the EC (EurActive 2008a) I decided to use for this

model as benchmark amounts 50% EUA to be purchased and 100% EUA to be purchased

(no grandfathering alternative).

The model takes into account 2 scenarios concerning the per cent of EUA that will have to

be purchased by the power plants under the EU ETS. These scenarios were based on the

results of political discussions about the form of EU ETS after the year 2012. Based on

recent statement of the EC (EurActive 2008a,b) the benchmark scenario counts with 50%

EUA to be purchased and 100% EUA to be purchased.

To model the prices of inputs (hard coal, natural gas, EUA) and outputs (electricity), the

real option model will use the binomial option pricing model27

.

The EUA 2013 Future traded on ECX28

:

EUA13 =EUR 34.65

carbon intensity ratio 0.9 tonnes of CO2 per MWh (PCB-C) and 0.43 tonnes of

CO2 per MWh (CCGT) 29 exchange rate = 23.465 CZK/EUR

In the model it is assumed that compliance with the EU ETS will be always more

advantageous than non-compliance, which is in line with the EU ETS rules.

The investment parameters of the assessed power plants were adopted from the EG(2005) report and are summarized in the Table 1 below. Costs of both power plants are

computed for one MW of capacity installed.30

Type of unit CCGT PCB-C

Fuel Natural gas Hard coal

Investment costs (CZK/MW) 20,714,000 38,547,000

M&O costs 2,394,195 4,253,814

Table 1: Technical parameters of investment into new power plant (EG (2005))

The lifetime of both power plants is set as 30 years31

, the construction time as 4 years32

(the estimated period 20132043).The payment of the investment price is calculated as a 30 years (=T) credit with r=8%

interest rate with standard annuity (A):

1)1(

)1(

T

T

r

PVrr

A (5.72)

where PV are the investment costs as stated above.

27 That is based on, as was described above, Box, Ross, Rubinstein (1979) and on the managerial vision described in

Shockley (2006).28 see footnote 2329 David (2006)30 The usual size of CCGT is according to EG (2005) 300 MW of installed capacity, of PCB-C 600 MW ofinstalled capacity.31 EG suggests 25 years in case of CCGT, but for instance Roques et al. (2005) estimate the CCGT lifetime as 30

years.32 EG suggests 5 years construction time of PCB-C, but for purposes of the model 4 years for all three types ofpower plants fit better (besidesone year difference is a relatively insignificant for the model).


27/35

The inflation applied in the model is the long-term inflation target of the Czech National

Bank 3% (= ); the discount rate applied is i=5% (as in Shockley 2006) 33.The estimated present value of annual costs of PCB-C and CCGT without the EUA will

be computed as follows:

30

1 )1(

)_(*cos_

)1()1(

)1(*coscos

tt

t

tt

t

i

rategrowthtsFuel

i

Annuity

i

tsMOtsPV

(5.73)

Since the power plant construction time is approx. 4 years for both types in the model, it is

necessary to apply fuel, electricity and EUA prices as of the year 2013. All the fuel and

electricity prices were adopted from trading prices on ECX and EEX34

:

Natural Gas EGT35

Year Future 2013 = 40.99 EUR/MWh

Phelix Base-load Year Future 2013 =93.50 EUR/MWh

Phelix Peak-load Year Future 2013 =138.50 EUR/MWh

Exchange rate = 23.465 CZK/EUR

ARA36 Coal Year Future 2013 (10th July, 2008)=200 USD/tonne Average hard coal conversion rate

37= 27 GJ/tonne

Unit hard coal consumption38

= 8 GJ/MWh

Exchange rate = 14.938 CZK/USD

The hard coal conversion from USD/tonne is expressed as follows:

ARA Coal Year Future 2013 price*unit hard coal consumption*exchange rate/average

hard coal conversion rate=200*8*14.938/27=885.2148 CZK/MWh.

The electricity prices are used to compute an estimate of opportunity costs of the investors

stemming from postponement of power plant construction. The estimate is of the annual

opportunity costs per one MW of capacity installed is as hours per year per MW*electricity

price* efficiency of the power plant.

The EG (2005) estimates the CCGT netto efficiency as 54.2% and of PCB-C 45%.The model uses reference10%, 15% and 20% growths of prices of electricity, natural gas

and coal.

The model is focused on testing of changes in the EUA prices, which is modelled with use

of the binomial-pricing model.

The difference is given by the carbon intensity ratio of CCGT plant is 0.43 tonnes of CO 2

per MWh and 0.9 tonnes of CO2 per MWh for PCB-C (David (2006)).

The EUA costs are further merged with the M&O and fuel costs and with the annuity(5.72) and resulting cumulative costs are adjusted by the probabilities (q; 1-q) from the

binomial option-pricing model.

The result is a comparison table, where the probability-adjusted cumulative costs of each

option are compared. This is a cost-minimization problem, so selected is always the

33The depreciation and taxes were not taken into account.

34 see footnote 2335 E.ON Gas Trading36

Amsterdam-Rotterdam-Antwerp

37 BP Historical Data 1861 - 200738EG(2005)


28/35

minimum. An example of a final table depicturing Situation 1 (natural gas, electricity and

coal price growth 15%) is

Table 2. The investor can wait until the year 2012 in order to obtain more information

about the future development of EUA prices (the opportunity costs are lower than the actual

costs of the power plant).

Scenario 1

base-load operation mode for both power plants

no-grandfathering (100% EUA to be purchased)

20% volatility

Price growth Natural Gas 10% Natural Gas 15% Natural Gas 20%

Electricity 10%

Coal 10%CCGT PCB-C PCB-C

Electricity 15%

Coal 15%

CCGT

Option to wait; low

EUA prices-PCB-C

(2016/2017); high

EUA prices -CCGT

(2017)

PCB-C

Electricity 20%

Coal 20%

CCGT CCGT

Option to wait; low

EUA prices-PCB-C

(2022); high EUA

prices -CCGT (2024)

Electricity 20%

Coal 15%

CCGT

Option to wait; low

EUA prices-PCB-C

(2016); high EUA

prices -CCGT (2016)

PCB-C

Electricity 15%


Electricity 20%


Table 2: Resultsscenario 1

Conclusions

The results of Scenario 1 are summarized in the matrix of results in

Table 2 above. The option to postpone the investment decision appears only, when the

growth of prices of natural gas and of coal follow basically the same trend. Whereas CCGT

has larger share of fuel costs on the total present value of costs of the plant, PCB-C has larger

share of the EUA costs on the present value of total costs of the plant. If the relative gap

between the fuel prices does not change, the only factor that could lead to a switch in cost-

effectiveness between these two types of power plants are the costs of EUA. The growth in

the prices of both fuels has to be a substantial one; otherwise the CCGT will become due to


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its low initial costs less costly than the PCB-C regardless on the EUA costs (see natural gas

10%, coal 10%, and electricity 10% in Table 2). In case of the options, the higher is the

growth of electricity the higher gets the opportunity costs of the option (the costs of

postponement of the investment decision), even though the difference in case of option to

wait with 20% and 15% electricity growth (natural gas 15%, coal 15%) is not that substantial

(one year change).

The option to postpone the investment decision is, however, not present in cases when the

relative prices of coal and natural gas change. If natural gas becomes more expensive than

coal, the preferred investment choice will be the PCB-C. There are 8 cases in Scenario 1

under which the PCB-C is the preferred option, whereas there are only 7 cases, where CCGT

is the preferred alternative, from which three cases can be considered as highly improbable

situations (the shaded cells). It must be, however, stated that it is quite improbable that the

coal prices would grow more steeply than the natural gas prices. Completely implausible

combinations (such as e.g. when growth of electricity prices would be lower than the growth

of prices of both fuels39

, etc.) were therefore excluded from the matrix of results.

Scenario 2

base-load operation mode for both power plants

50%-grandfathering (50% EUA to be purchased)

20% volatility


Electricity 10%


Electricity 15%Coal 15%

CCGT PCB-C PCB-C

Electricity 20%

Coal 20%CCGT CCGT PCB-C

Electricity 20%


Electricity 15%



CCGT PCB-C PCB-C

Table 3: Scenario 2 results

Scenario 2 drafts the situation, when in the EU ETS would require the power plants to

purchase only 50% of the necessary EUA and the remaining 50% would be given to the

power plants for free. In such case there is no option to wait (the option ceases to exist when

39

The price of electricity is dominantly formed by prices of fossil fuels. Besides it includes margin and other costsrelated to the electricity generation, transmission, distribution etc. It is a common sense assumption that in long-runthe electricity cannot grow more slowly than the prices of fuels for its production.


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less than approx. 66%EUA has to be purchased40

; the lower is the non-grandfathered EUA

share, the more preferred are the PCB-C) and the PCB-C plant is preferred in 11 cases.

Again, the higher is the electricity price growth the faster increase the opportunity costs and

the sooner has the investor the incentives to build the power plant.

Scenario 3

peak-load operation mode for CCGT, base-load operation mode for PCB-C power

plant

no-grandfathering (100% EUA to be purchased)

20% volatility


Electricity 10%


Electricity 15%



CCGT CCGT CCGT

Electricity 20%


Electricity 15%


Electricity 20%


Table 4: Scenario 3results

The results of Scenario 3 confirm the general observation that CCGT is currently

advantageous only as a peak plant that is switched on only in case of high demand. The

introduction of more stringent targets on the emission reduction within the EU ETS will only

intensify this situation CCGT will be more preferred alternative in 10 cases, PCB-C in 8cases.

Scenario 4

peak-load operation mode for CCGT, base-load operation mode for PCB-C power

plant 50%-grandfathering (50% EUA to be purchased)

20% volatility

The Scenario 4 matrix of results does not differ from Scenario 3. The EU ETS does not

have any significant impact on the relative costs of CCGT and PCB-C under their standard

operation modes. The only important factor that could influence the investment decision in

Scenario 4 identified by this model is the change of the price gap (relative prices) of the

fuels.

40 Therefore the matrix of results would be the same for 80% grandfathering (20% EUA target).


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6. CONCLUSION

In this paper we examined the evolution, types and applications of real option model. We

have demonstrated that from the original option model developed by Merton (1973) werederived two fundamental forms of real option modela stochastic one and a managerial one.Further on, we identified the conditions for successful application of the real option model.

The most important ones are uncertainty about the future developments of variables that

determine the success or failure of an investment project and irreversibility or partial

irreversibility of costs of the project.

The real option use was demonstrated on examples from the field of energy. Firstly, we

have shown some examples from the literature concerning the real option application on

projects of nuclear, coal-fired and gas-fired power plants. Later on we proceeded to

numerical and stochastic examples of real option application, also from the energy sector.The numerical example demonstrated the use of the managerial real option valuation. It

assessed the impacts of the potential development of the EU Emission Trading Scheme on

the investment decision-making in the power generation namely on the trade-off betweenthe deployment of coal and natural gas.

The stochastic example demonstrated the real option valuation on case of a so called spark

spread option of natural gas-fired power plants. We have shown the evolution of the sprak

spread option valuation based on the model developed by Merton (1973) and modified by

Deng, Johnson, Sogomonian (1999) and illustrated also some potential further modification

of this model that might improve it in order to capture more risks namely the regulatory

risk.


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The Real Option Model – Evolution and Applications