Master Thesis
The value of electric energy storage including real
options: The case of Eemsdelta.
Jan Veijer, 1713280
January 2014
University of Groningen
Foreword
This thesis is written as part of an internship at Energy Valley focusing on a project concerning the production of hydrogen from electricity through electrolysis in the Northern Netherlands. My part of the internship considered the valuation of hydrogen production and storage.
I am grateful that Energy Valley provided me this opportunity to pursue a line of research that neatly fits my study profile. The field of Energy is not (yet) a prominent one at our faculty and as such finding a suitable thesis topic is not easy. I appreciate the learning opportunities that arose from this project and the involvement in todays’ hot topics in the field. Electricity storage is imminent and once it is implemented it will considerably change the electricity market as the property of storability is there.
Together with Ilco Kuipers, Orkhan Shukurov and Elham Khalili I was part of an interdisciplinary team able to focus on diverse aspects of the case. I enjoyed the cooperation and thank them for the provocative discussions and company. Further I thank my supervisors Catrinus Jepma and Peter Smid for providing me with useful comments and suggestions on draft versions of my thesis. I thank Steven Brakman for taking on the role as co-‐assessor. I also thank Dewi Eshuis for coordinating the team, and Koos Lok and Patrick Cnubben for their useful comments on our research during the meetings at Energy Valley. I further thank Fred Hage from Linde Gasses, Jeroen de Joode from ECN and Adriaan de Bakker from GasUnie for providing additional insights and information.
Abstract
The increasing penetration of renewable energy sources calls for energy storage techniques. This thesis investigates the economic feasibility of electric energy time-‐shift by producing hydrogen in the region of Eemsdelta in the Netherlands. Beside electric energy time-‐shift, the valuation includes a number of real options recognizing the output flexibility of hydrogen, the option to delay the investment, and the option to abandon the project. The results show that electric energy time-‐shift is not feasible whereas the option to sell hydrogen as feedstock considerably increases value. Nevertheless the overall value including real options is negative.
Keywords: Arbitrage, Energy storage, Hydrogen storage, Real option valuation,
Monte Carlo least squares.
Table of Contents
Chapter 1 ... 1
1.1 Introduction ... 1
1.2 Problem Statement and Research Questions ... 2
Chapter 2 Literature review ... 6
2.1 Electricity Prices and the Balancing Problem ... 6
2.2 The Holy Grail: Energy Storage ... 8
2.3 Power to Gas: Hydrogen energy storage ... 10
Chapter 3 Valuation Methodology ... 14
3.1 Value ... 14
3.1.1 Net Present Value ... 14
3.1.2 Real Options Valuation ... 15
3.2 Empirical application of Real Option Valuation ... 19
Chapter 4 Modelling of Price Series ... 21
4.1 Stochastic Processes ... 21
4.2 Seasonal Effects ... 23
4.3 Risk Neutral Valuation ... 24
Chapter 5 Eemsdelta Case Study ... 27
5.1 State Variables ... 27
5.1.1 Electricity Price data ... 27
5.1.2 Gas Prices ... 33
5.2 Operations Model and Parameters ... 36
5.3 Valuation ... 44
5.4 Real Options Valuation ... 46
5.5 Sensitivity Analysis ... 51
Chapter 6 Discussion ... 56
Chapter 7 Conclusion ... 59
References ... 60
Appendix A Storage Technologies ... 67
Appendix B Services and Benefits Provided by Energy Storage ... 69
Appendix C The Dutch Energy Market ... 72
Chapter 1
1.1 Introduction
Renewable energy sources (RES) are attracting attention nowadays, clearly mirrored in the European 20-‐20-‐20 target (European Commission, 2010), the recently agreed upon Dutch Energy Accord and the German Energiewende. Reasons for these political decisions are energy dependency, CO2 emission
reduction and the depletion of fossil energy sources in the foreseeable future. Following the Fukushima disaster in Japan, Germany drastically changed policies, proposed an Energiewende, and heavily invested in wind power production capacity. Although wind power is a clean alternative to conventional generators in terms of greenhouse gas emissions, the intermittent nature of wind power hampers the implementation of it in the electricity grid. Because electricity storage is not possible – or at least not at an efficient and sufficient scale – the supply of power must exactly match the demand of power. Therefore, in case supply of wind power exceeds demand of power, wind power production should be curtailed or the power must leave the grid in another way. In Leipzig, in Germany, the European Energy Exchange (EEX) allowed for negative price bids in 2008 to ensure that excess energy supply could leave the national grid in order to balance the grid.
These developments show that the power system is in need of storage capacity to integrate intermittent renewable energy sources and balance the system. Various solutions for implementation exist. For instance, excess energy supply from renewable sources can be stored centrally and used to balance the system in cases of undersupply. Another possibility is to allow market parties to store electricity and benefit from price differences. On a large scale, with sufficient storage operators and proper competition, this will also balance prices.
hydrogen applications exist: i) convert hydrogen back into electricity using fuel cells, or gas to power (GtP), ii) sell hydrogen to the chemical industry, iii) feed a small percentage volume of hydrogen into the gas grid, iv) produce methane by synthesizing hydrogen and carbon dioxide, and v) sell hydrogen to the automobile sector. The produced oxygen can also be sold as a feedstock for industrial applications.
Currently, a number of storage options exist including for instance: compressed air energy storage, pumped hydrogen storage (PHS), batteries, flywheel storage, and hydrogen storage. The reader is referred to Hall and Bain (2008), Beaudin et al. (2010) and Díaz-‐González et al. (2012) for a detailed discussion of available storage technologies. Appendix A provides a short description of the technologies. Although a variety of technologies exist, not all technologies are feasible due to geographical or power capacity limitations. For example, PHS applies only in regions with sufficient elevation differences, batteries are not yet available at a large scale and compressed air energy storage is still in a development phase. Hydrogen storage is a promising technique as it is versatile for its applications, geographically independent and applicable on a large scale.
In this thesis, the proposed location of the storage facility is the region of Eemsdelta in the northern part of the Netherlands. There is a nearby chemical industry and there are salt caverns optional for hydrogen storage. This study is one of the first attempts to value a hydrogen storage facility using real options valuation. Further it is the first study on electricity energy time-‐shift in the Netherlands. The outcomes of this study reveal further comprehension of practical operation strategies and the conditions under which it is optimal to invest in hydrogen storage.
1.2 Problem Statement and Research Questions
observed cross border trade of power between Western European countries is a manifestation of this balancing problem.1 The occurrence of negative prices on
the wholesale market is another manifestation of this problem.
Technically the imbalance is a considerable problem that needs technical solutions. Nonetheless, a price equilibrium can solve the technical problem of balancing, but comes at a cost for the supplier in case of oversupply and consequent prices decrease. Economically the imbalance offers an opportunity for arbitrage by means of storage. Arbitrage or time-‐shift is basically purchasing electricity at one time and selling it at a later time.2 Given that wind power
curtailment is unpopular from a European political perspective3, excess supply
of wind power tends to decrease the equilibrium price level. Through the merit order of electricity supply, the price of electricity is settled at the generator producing at the highest marginal costs. Because the marginal cost of producing wind is low, an high supply of wind power tends to reduce the equilibrium price of electricity.
Although it is theoretically possible to benefit from time-‐shift, in practice the value of arbitrage is depending on a number of parameters including: investment cost, operation and maintenance costs, equipment life cycle, real world energy price differences and others. The ultimate question is whether the benefits offset the costs. In this case study I investigate whether electricity time-‐shifting using hydrogen storage and production is economically feasible. The case is the region of Eemsdelta, which is in the Northern province of Groningen in the Netherlands. The research question leading this study is as follows:
What is the value of electric energy storage using hydrogen production and storage?
1 The trade primarily occurs in Germany that imported 7.3% and exported 11.1% of its
electricity consumption (Destatis, 2013).
2 From a finance perspective this sounds odd as the assumption of many theories is that
arbitrage opportunities do not exist. However, in electricity markets daily price variations exist, theoretically allowing for arbitrage if storage is available. Arbitrage and time-‐shift are alike and used interchangeably in this thesis.
3 Curtailing wind power production contradicts the implementation of the 20-‐20-‐20 program of
The general research question gives rise to four sub questions, each stipulating dimensions that are relevant given the nature of the project:
-‐ Why investigate the case of hydrogen storage? -‐ What is the context of Eemsdelta?
-‐ What factors drive the value of energy storage? -‐ What is the value of alternative output options?
The first two sub questions merely follow from the start of this research and serve as a motive or description of the context of this research. The last two sub questions deal with the value drivers of hydrogen production, storage and sale.
1 Why investigate the case of Power to Gas?
There exist a number of energy storage technologies. Each technology has its pros and cons in technical and economic terms. The main characteristics that should be considered are costs, efficiency and time-‐scale applicability (Hedegaard and Meibom, 2011). Power to Gas is a promising technique as it allows for large-‐scale storage, is not dependent on geography and allows for output flexibility.
2 What is the context of Eemsdelta?
The case study concerns the region of Eemsdelta in the Northern Netherland. In this region a number of chemical plants reside and there exist facilities for large-‐scale hydrogen storage in salt caverns. We use energy price data from the Netherlands and base the assumptions of the model on the location in the Eemsdelta.
3 What factors drive the value of energy storage?
The value of storage basically depends on revenues and costs. We identify the cost of investing in electrolyzers and fuel cells and model the revenues by their respective drivers: the prices of energy.
4 What is the value of alternative output options and strategic options?
the gas infrastructure, converted into methane, and can be converted back into electricity. Putting it simple, when one output becomes unprofitable we can switch to another output that is profitable. Switching from output along with profitability results in managerial flexibility that has a certain value under real options valuation (ROV). For this sub question I identify the relevant options in this project based on the options established in the ROV literature and subsequently value the recognized options.
Methodology
We assess the economics of hydrogen storage by standard net present value (NPV) analysis and ROV on top of that to capture the options that exist in the project. The point of departure is a daily output optimization using the prevailing prices.
Data
The data for this research comes from the Amsterdam Power Exchange (APX) that reports the prices of electricity on an hourly basis and the prices of gas on a daily basis. Data on future prices is attained from the ICE ENDEX. Price of hydrogen and oxygen are obtained from industrial gas companies. The parameters for the technology applied are primarily derived from DNV KEMA (2013).
Outline
This thesis proceeds as follows. Chapter two gives a literature review on the relevant aspects of this study including: energy storage technologies and the economics of energy storage. Chapter three provides the preliminaries for the valuation methodology. Chapter four presents the mathematics for modeling the state variables. Chapter five encompasses the Eemsdelta case study including the estimation of the state variables, operations model, valuation, and sensitivity analysis. Chapter six discusses the outcomes and chapter seven concludes.
Chapter 2 Literature review
The recent political decision-‐making that leads to the subsidized implementation of renewable energy sources has severe consequences for the energy system. I shortly discuss the technical consequences for the electricity system – the balancing problem – and elaborate more on the economic implications for energy prices and the energy business. Subsequently, I discuss the imminent solution of the balancing problem, which is energy storage. Then I reach at the point of departure for this research and continue with the methodology in the next chapter.
2.1 Electricity Prices and the Balancing Problem
Electricity Prices
The inherent characteristic of current deregulated markets is that the resulting prices tend to show a volatile pattern with price spikes from time to time. A number of factors attribute to this observed volatility. The prime reason is that with current technology it is not possible to store electricity in large volumes. Therefore the supply of power should exactly match the demand of power, which can be predicted on a daily basis. Program responsible parties, subjected to a fine otherwise, are there to ensure that supply continuously meets demand of electricity.4 Consequently, arbitrage opportunities are limited in practice and the
market-‐clearing price is equal to the marginal cost of production.
Further we can distinguish base load and peak load generators. Conventional baseload power plants operate steady and meet the baseload power demand. When the load increases, other generators start operating. Through the merit order, the baseload generators having the lowest marginal costs meet the baseload demand and when demand increases other generators start to operate in the order of lowest marginal cost of generation.
Based on this supply side and the demand pattern of electricity we can elicit a number of electricity price characteristics (Knittel and Roberts, 2005). These characteristics serve as theoretical rationale for the modeling of price series.
The first characteristic is the intraday variation in prices. As such, we can differentiate between base load periods – or off-‐peak periods – and peak load periods. In base load periods demand is relatively low and base load generators can meet the demand. Therefore, in a base load period generated volume is low and prices are also low. In peak load periods demand is higher and more generators need to be operated, generating power at a higher marginal costs along the demand curve. In such case, it might happen that demand peaks in a given hour giving rise to – sometimes extremely – high prices. Concerning the timing of baseload and peak load, in general base load occurs at night and peak load occurs during the day.5
The second characteristic is the seasonal variation in demand. Seasons influence the demand for electricity that is needed to heat or cool accommodations. For instance, on warm summer days electricity demand increases due to increased use of air-‐conditioning to cool accommodations. Winter days show a lower demand pattern as gas or coal meets the heating needs.6
The third characteristic is the limited distribution and transmission capacity. The generated electricity travels through the distribution and transmission lines to the destination of demand. The distribution lines have a maximum capacity which is the maximum of MW they can carry in an hour. Once the transmission exceeds the capacity, marginal costs of transmission becomes infinite.
These characteristics cause an inelastic supply of electricity in the short term, from hour to hour. Therefore in some cases small shifts in demand or supply of electricity can have a vast impact on the price charged per MWh. This is the main reason why huge spikes occur in the price pattern and the overall pattern is highly volatile (Knittel and Roberts, 2005).
Energy System
The increasing share of renewable energy sources in the energy mix has its impact on the energy system. Due to the intermittency of wind power generation there are large fluctuations in wind power supply depending on the availability
of wind on the one hand and the required power on the other hand. Further, Sensuß et al. (2008) observe that the increase in renewable energy sources, supported by feed-‐in tariffs, decreases the average electricity price thereby benefiting the demand side at the cost of the supply side.
Technically there is a balancing problem, but economically the price equilibrium can solve the balancing problem even if this leads to negative wholesale prices. Currently, price drops are primarily observed in the German wholesale market and in the cross border trade between Germany and its neighboring countries (Muche, 2009). However, negative prices are clearly not the producers’ interest while price spikes are not beneficial to the consumers.
2.2 The Holy Grail: Energy Storage
Although the property of storability is disregarded in the valuation of electricity derivatives and investments, the body of research on storage technologies grows. In particular due to the increasing penetration of renewable energy sources as photovoltaic and wind power, the necessity for storage increases in order to accommodate for the inherent intermittent supply of renewable energy sources.
Although a comprehensive review of available technologies is beyond the scope of this thesis, I briefly discuss the main storage technologies appearing in the literature in appendix A.7 The technologies discussed are Compressed Air
Energy Storage (CAES), Pumped Hydro Storage (PHS), Flywheel, batteries, and Power to Gas (PtG). For a further review the reader is referred to Hall and Bain (2008), Beaudin et al. (2010), and Díaz-‐Gonzales et al. (2012).
Economics of Energy Storage
Research on energy storage can broadly categorized on a regional economic system level as well as on a plant level. Research on the regional level is mainly concerned with the integration of large amounts of renewable energy sources into the system and the effect on system flexibility (Denholm and Hand, 2011) and the need for large-‐scale storage (Connolly et al., 2012; Diaf et al., 2008). Plant
7 Many of the available technologies are still in an R&D stage. The technologies discussed are also
level research considers the economics of a single plant with respect to storage. Various storage technologies have been assessed, i.e. compressed air energy storage (CAES) (Madlener and Latz, 2013), pumped hydrogen storage (PHS) (Muche, 2009), batteries (Walawalkar, 2007), flywheel (Walawalkar, 2007), and hydrogen storage (Floch et al., 2007).
The technologies mentioned in the previous section have different characteristics in terms of economics and technology. For the economic assessment, the most important characteristics are: power capacity measured in megawatt (MW), energy capacity measured in megawatt hour (MWh), and round trip efficiency. Power capacity is the maximum capacity that a storage device can realize in an hour. Energy capacity is the energy that is actually discharged by the storage device in a given time period. Round trip efficiency can be characterized as the share of charged electricity that is discharged into the grid after storage.
Using these characteristics along with financial data the feasibility of storage options can be assessed. Benefits of storage arise from a number of dimensions and applications. Eyer and Garth (2010) estimate a broad range of benefits – including the integration of renewable energy sources, load shifting, and ancillary services8 -‐ for small and large-‐scale electricity storage applications.
Appendix B shows the main benefits involved including a short description. The main benefits studied in the literature are electric energy time-‐shift (Muche, 2009; Sioshansi et al., 2009), avoidance of wind curtailment (Loisel et al., 2010) and, regulation services (Walawalkar et al., 2007). Further, a growing body of research considers the combination of storage with an existing power plant (Taljan et al., 2008) or wind park (Fertig and Apt, 2011).
Loisel et al. (2010) investigate the market potential of CAES and PHS in France and Germany focusing on the benefits of wind curtailment, price arbitrage and secondary and tertiary services. They found that storage is only viable when it receives a number of compensations for multiple services provided. They also stipulate that the accrued benefits from the services provided should be aligned because the investor captures not all benefits.
8 Following Eyer and Garth (2010): ‘ancillary services are necessary services that must be
Walawalkar et al. (2007) studied the feasibility of storage by sodium sulfur batteries and flywheel storage in the New York state electricity market. They show that there is a strong case for electricity storage in that market for applications such as energy arbitrage and regulation services. Another important benefit that occurs is the possibility to defer investments in transmission system upgrades.
Although different techniques have different applications and benefits, research shows that no single technology consistently outperforms other technologies on the various benefits in the system (Beaudin et al., 2010). Further there are a number of external factors such as mineral availability and geographical characteristics that might limit the applicability of certain storage devices.
2.3 Power to Gas: Hydrogen energy storage
Hydrogen energy storage is an electrochemical storage process in which, by means of electrolysis, electricity is used to convert water into hydrogen and oxygen. The distinct benefit of hydrogen storage is the ability to decouple production and storage of hydrogen. This makes it possible to store large amounts of energy, which is virtually impossible with most of the other storage techniques.
The basic process of storage is that electricity is used to split water into oxygen and hydrogen, given by the chemical equation
2 𝐻!𝑂 → 2 𝐻!+ 𝑂! , (2.1)
which implies that, through electrolysis, two molecules of water 𝐻!𝑂 convert into two molecules of hydrogen H2 and one molecule of oxygen O2.
industry, iii) infeed in the gas infrastructure, iv) methanation through reacting hydrogen and carbon dioxide, and v) delivery of hydrogen fuel to the automobile sector.
Conversion to electricity is the reverse of PtG; gas to power (GtP), a process that consequently releases electricity through the conversion of hydrogen and oxygen into water. These two processes, combined with the possibility to store hydrogen, are the essence of electricity storage through hydrogen.
Beside the basic option of electricity conversion, other hydrogen applications exist. One option is to sell hydrogen to the chemical industry. Hydrogen is valuable feedstock for the chemical industry. For example, the petrochemical industry uses hydrogen for crude oil refinements. Other applications in the chemical industry are water peroxide and methanol production. Another option is to feed hydrogen into the gas infrastructure in small proportion. In the Dutch gas grid the allowed volume equals 0.02 vol% (Donders et al., 2010), with an expected increase to 0.5 vol% in 2021 (EL&I, 2012). The fourth option is to convert hydrogen into methane by reacting hydrogen and carbon dioxide9.
Further the automobile sector increasingly uses hydrogen as fuel.
Natural Gas Prices
The natural gas markets are also deregulated in the 1980s and 1990s. Former monopolists do not exist any longer and the market settles prices. Where formerly the natural gas price was coupled to oil price, in recent years gas prices decoupled from oil prices in several gas markets (Erdős, 2012).
As for electricity, seasonality applies to gas prices. Gas is often used for heating in the winter periods and in the summer, as gas is often used as a mean to generate electricity, it is implicitly used for the electricity that is used for air-‐ conditioning. Therefore, the demand for natural gas depends on the weather, which in turn is related to seasons. In general gas prices are less volatile than electricity prices.
9 In the Netherlands explored natural gas consists approximately 80% of methane. Therefore,
Hydrogen Prices
In contrast to natural gas and electricity there does not exist an exchange for hydrogen. Hydrogen is often produced by companies for their own use or sold via contracts with gas producing companies. The hydrogen price depends on factors such as the form in which hydrogen is delivered and the transport distance from supplier to demander. Further, the prices are not publicly available for reasons of competition.
Research on Hydrogen Storage
Previous research on hydrogen storage mainly concentrates on the connection of a hydrogen storage facility with an existing power plant (Taljan et al., 2008; Floch et al., 2007) or wind farm (Olateju et al, 2014).
Taljan et al. (2008) show that hydrogen production storage is feasible for existent mixed wind-‐nuclear power plant, using the options of direct hydrogen sale and utilization of residual heat and oxygen. They conclude that the price of fuel cells does not justify the generation of electricity.
Floch et al. (2007) study the opportunities of producing hydrogen via electrolysis during off-‐peak periods. They conclude that in the French wholesale electricity market, with a substantial capacity of nuclear power plants, hydrogen production is not feasible with a percentage of use between 30% and 50%. The price of hydrogen does not cover the electrolyzer investments cost.
Olateju et al. (2014) study the production of hydrogen using a large-‐scale wind farm for serving the oil sands bitumen upgrading industry in Western Canada. They optimize the plant configuration and conclude that hydrogen production from wind energy is not competing with conventional hydrogen production based on fossil fuels.
Case Study
application in a certain context. Therefore, as a case study we choose the region of Eemsdelta in the Northern Netherlands where a number of companies reside including chemical companies, conventional electricity producers and wind parks. Furthermore, in the Northern Netherlands salt caverns allow for large-‐ scale hydrogen storage. The combination of a nearby chemical plant and large-‐ scale storage facilities is a unique combination. The chemical industry uses about 150 million Nm3 of hydrogen per year and therefore the Eemsdelta region suits
the case study. This study focuses primarily on the storage benefit of electricity time-‐shift and secondary on the allocation of alternative output options for hydrogen.
Although other benefits such as ancillary services might apply, I choose for electricity time-‐shift as research subject. Electricity time shift or price arbitrage is the main benefit studied in previous research and easily identifiable. Further, electricity time-‐shift is a predominant benefit due to the existent price difference between off-‐peak and peak hours.
Chapter 3 Valuation Methodology
3.1 Value
In any valuation methodology we apply, the main factor is the expected cash flow that is discounted to the present, accounting for the time value of money and the required – risk adjusted – return. The workhorse model in basic valuation exercises is the discounted cash flow calculation culminating in the net present value (NPV) after subtracting investment costs.
However, in case of energy storage, predicting cash flows can be a difficult task, as these series tend to follow a stochastic price pattern, induced by factors such as weather and technological developments but also (geo) political developments.10 Real options valuation (ROV) captures these uncertainties by
accommodating the NPV. As such, ROV does not resolve the inherent uncertainty but at least accounts for it by assigning a value.
Therefore, in this study we basically apply an NPV analysis with on top of that ROV to capture the project’s inherent flexibility.11 In this chapter I briefly discuss
the preliminaries of the NPV to continue with a comprehensive review of ROV.
3.1.1 Net Present Value
The NPV is a standardized algorithm and widely applied in empirical work due to its consistency, inclusion of the time value of money and corporate capital structure and, not unimportant, its ease to implement (Mun, 2006).
Basically the NPV is nothing more and nothing less then the difference between the required investment cost and the present value of the expected free cash flows (FCF) over the project lifetime. The NPV is given by:
𝑁𝑃𝑉 = !"!! (!!!)!− 𝐼 ! !!! , (3.1)
10 Developments in Germany clearly show that political decisions w.r.t. renewable energy
sources influence electricity price patterns.
11 In fact both traditional NPV and ROV are based on net present value. In this thesis NPV refers
where FCF represents the free cash flow from operations, r is the discount rate and I is the required investment.
Notwithstanding the advantages, the NPV faces severe shortcomings. The NPV assume a deterministic outcome regarding the project payoff whereas in reality the payoff is uncertain or stochastic. The arrival of new information cannot be captured in NPV whereas ROV does capture new information and as such offers managerial flexibility to alter the course of action.12
The characteristics of the hydrogen storage process, such as the inbuilt output flexibility and the inherent stochastic nature of the price series underlying the output options, challenges the standard discounted cash flow (DCF) analysis for valuation purposes. Therefore in this case, a real options valuation is a useful addition on top of the DCF analysis.
3.1.2 Real Options Valuation
Modern option valuation departs from the seminal papers by Black and Scholes (1973) and Merton (1971, 1973). The Black-‐Scholes-‐Merton (BSM) model has become a workhorse model for option valuation and is considered as the standard in many finance textbooks.13 Later Cox et al. (1979) develop the
simple binominal-‐option pricing based on the parameters of the BSM model. Myers (1977) has been one of the first to characterize the investment as a call option on the value of the project. Analogous to the notation of American plain vanilla call options we can define parameters for the option on the project. Exercising the option is equivalent to the decision to invest. Table 1 provides the comparison between plain vanilla options and options on the project.
The underlying asset is the project value and through paying the investment sum the option on the value of the project is exercised. The exercise price is equal to the investment cost and when the investment cost is higher than the value of the project it is not optimal to exercise the real option, i.e. the option to invest is out-‐of-‐the-‐money. The length of the project is the time to maturity. Option valuation is based on risk neutral valuation using replicating portfolios and as such the risk free rate is used as a discount rate. The riskiness of the
12 A full comparison of NPV and ROV is beyond the scope of this text but the reader may consult
Mun (2006) or Kodukula and Papudesu (2006) for a comprehensive comparison and discussion.
business, the volatility of cash flows, is captured by the volatility parameter 𝜎. Dividend yield decreases the stock value and similarly cash flows decrease the value of the project as long as the option remains unexercised. Delaying the investment implies sacrificing cash flows that subsequently reduce the value of the project.
Table 1
Similarity between financial options and real options
Financial Options Real Options Symbol
Stock price Value of the project S
Exercise price Investment cost K
Time to maturity Length of the project T
Risk free rate Discount rate r
Stock Volatility Riskiness of the business σ
Dividend yield Cash Flows q
Trigeorgis (1996) provides an in-‐depth discussion of ROV and categorizes
different options:
-‐ Option to defer regards the timing of the investment.
-‐ Time-‐to-‐built option allows for stages of investments, i.e. after a stage there exists an option to abandon or proceed.
-‐ Option to alter operating scale implies expansion or contraction of capacity.
-‐ Option to abandon is an option to abandon operations. The value depends on the salvage value of the equipment.
-‐ Option to switch provides an option to switch the output or input in operations.
-‐ Growth option implies that optional investments can lead to future growth options.
-‐ Multiple interacting option arises when several options exists that interact. In that case the sum of the parts might differ from the combination of options.
Solving Real Options
In practice the calculation of option values basically proceeds through analytical solutions or numerical solutions. The method based on analytical solutions proceeds by solving the differential equations of the underlying stochastic processes. The BSM model is a famous example of a closed form solution. For analytical solutions we can discern solutions based on dynamic programming and contingent claims analysis (Dixit and Pindyck, 1994). Contingent claims, as underlying the BSM model, uses a replicating portfolio and is based on risk neutral valuation. Risk neutral cash flows are discounted by the risk neutral discount rate. Dynamic programming uses the unadjusted cash flows and discounts the cash flows at a constant risk-‐adjusted discount rate, which is the main disadvantage of the method since it is questionable whether risk is constant through time. Dynamic programming is preferable over contingent claims when it is impossible to project risk-‐adjusted expected cash flows, provided that a proper discount rate does exist.
Nonetheless, analytical solutions rarely exist in reality, especially when the underlying process has more stochastic components or follows a path dependent process. In such case numerical solutions apply. Numerical solutions are approached via a number of available techniques including14:
-‐ lattice or tree methods, -‐ finite differences, and -‐ Monte Carlo simulation.
Numerous applications of lattices for option pricing exist. The best known is the binomial tree that includes an upward and downward movement at each node. In the limit the binomial tree solution equals the closed form solution in case of specific choices for the upward and downward probabilities (Cox et al., 1979).
Finite differences methods value a derivative by solving the applicable differential equation. In particular, the differential equation is transformed into a
14 The methods mentioned are by no account exhaustive. For a further explanation the reader is
set of difference equations. Subsequently the difference equations are solved iteratively.
Monte Carlo simulation basically simulates a number of possible price paths of the underlying asset. Then at each time period the payoff is calculated. The mean of the calculated payoffs for all simulated price paths is the expected payoff. The discounted payoffs provide an estimate for the derivative. Monte Carlo simulation can only be applied to European options. For American options one should apply Monte Carlo Least Squares (MCLS) simulation (Longstaff and Schwartz, 2001) in order to choose between the exercise value and continuation value.
The choice of the method depends on the nature of the underlying process. When the process is mean reverting or constant one should use trinomial trees or finite differences Hull (2012). In case of more than three underlying stochastic processes MCLS is applicable (Longstaff and Schwartz, 2001).
Monte Carlo Least Squares
In order to value an American style option it is necessary to choose between continuation and exercising at each exercise point before maturity. When the exercise value exceeds the continuation value it is optimal to exercise, otherwise it is optimal to wait. MCLS is an algorithm that facilitates this choice through finding the stopping times that maximize the value of the option (Longstaff and Schwartz, 2002). Table 2 summarizes the steps of the algorithm.15
The point of departure is the simulation space X with n trials and i paths. The algorithm starts on the last date and works backwards to the origin at i=1. At each point the algorithm indicates whether the value of continuation is higher than the value of exercising. In fact, the continuation value is estimated based on the conditional expectation of the payoff that results from keeping the option alive.
We calculate the conditional expectation by regressing the ex post realized payoffs from continuation on functions of the value of the state variables. The fitted value of the regression offers an estimate of the conditional expectation function. At each step the exercise strategy is optimized, resulting in one optimal
