master thesis u. appendix

8/11/2019 Master Thesis u. Appendix

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Abstract

The thesis has three objectives. The first objective is to introduce real option theory and

investigate the possibility of developing a real option valuation model to improve

valuation in real estate development projects. The second objective is to examine howflexible value is created by the real option analysis and evaluate different real option

methods in comparison of traditional valuation methodology and the practical usability.

The final objective in the thesis is to examine how the different input parameters affect

the value of the project. In order to reach the objectives a case study is applied with a

valuation of a large real estate development project.

The valuation of the real estate project is based on a compound real option methodology

applied by academics and finance practitioners in valuation of R&D projects. It is found

that the compound real option method can be applied to valuate real estate development

projects. A lattice model and a closed-form model are applied in order to estimate the

value of the real estate development project. The applied compound real option method

increased the payoff of the project by 15% created by the sequential decision strategy

incorporated into the valuation model. The optimal sale price/m2threshold increases as

well, compared to the static NPV threshold.

The two different valuation models estimated similar results. The closed-form model is

preferred if the compound is less than four folds, while the lattice model is preferred if

the number of folds increases.

The results obtained are consistent with empirical evidence found in the literature. The

theory states that greater price uncertainty should lead to delay in development and raise

the value of the option and thereby decreasing the probability of exercise.


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Contents

1. Introduction...............................................................................................................................4

1.1 Motivation...........................................................................................................................4

1.2 Problem statement...............................................................................................................4

1.2 Structure..............................................................................................................................6

1.3 Research methodology........................................................................................................7

1.4 Delimitation.........................................................................................................................8

2. Real option theory.....................................................................................................................8

2.1 Traditional valuation methodology.....................................................................................8

2.2 Introduction to real options.................................................................................................9

2.3 Real Option Methods and Approaches..............................................................................13

2.3.1 Methods......................................................................................................................13

2.3.2 Approaches.................................................................................................................20

2.4 Practical implementation and limitations of real options..................................................23

3. Real estate...............................................................................................................................25

3.1 Real options in real estate..................................................................................................25

3.2 Empirical evidence............................................................................................................27

3.3 Real options applied..........................................................................................................28

4. Models applied in real estate construction & planning...........................................................29

4.1 Compound options............................................................................................................29

4.1.1 n-fold compound option.............................................................................................32

5. Lighthouse* case.....................................................................................................................34

5.1 Project description.............................................................................................................35

5.2 Underlying drivers.............................................................................................................38

5.2.1 Cash inflows...............................................................................................................38

5.2.2 Cash outflows.............................................................................................................38

5.2.3 Cost of capital............................................................................................................39

5.3 NPV model........................................................................................................................42

5.4 Uncertainty........................................................................................................................43

5.4.1 Geometric Brownian Motion.....................................................................................44

5.4.2 Demand......................................................................................................................45

5.5 Volatility...........................................................................................................................45

5.6 Traditional valuation.........................................................................................................48

5.6.1 Expected NPV............................................................................................................48


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5.7 Real option valuation........................................................................................................49

5.7.1 Applied Compound option model..............................................................................51

5.8 Lattice method...................................................................................................................52

5.9 Closed-form solution.........................................................................................................54

5.10 Results.............................................................................................................................56

5.10.1 Method / Model........................................................................................................58

5.11 Sensitivity analysis..........................................................................................................59

5.11.1 NPV sensitivity analysis...........................................................................................59

5.11.2 Real option sensitivity analysis................................................................................60

6. Critique & considerations........................................................................................................64

6.1 Critique..............................................................................................................................64

6.2 Considerations...................................................................................................................65

7. Conclusion...............................................................................................................................65

8. References...............................................................................................................................68

9. Appendixes..............................................................................................................................73

List of tables:Table 2.1 Financial options versus Real options ....................................................................... ................. 10

Table 2.2 Types of real options ...................................................................................... ............................ 10

Table 2.3 Real option Approaches and Methods ................................................................. ....................... 22Table 5.1 Area and construction schedule .................................. ............................................................... 35

Table 5.2 Peer group .................................................................... .............................................................. 39

Table 5.3 Peer group ratios ................................................................... ..................................................... 41

Table 5.4 Industry ratios ............................................................................................ ................................ 42

Table 5.5 Input parameters ....................................................................................................... ................. 48

Table 5.6 Input variables for compound option model ...................................................................... ......... 52

Table 5.7 Valuation results........................................................... .............................................................. 56

Table 5.8 Sensitivity analysis for Rfand .................................................................................................. 59

Table 5.9 Sensitivity analysis for Voand ........................................................... ...................................... 61

List of figures:Figure 1.1 Thesis structure ....................................................................... .................................................... 6

Figure 2.1 Binomial lattice of underlying asset ......................................................................................... 19

Figure 2.2 Reasons for not using real options ........................................................................................... 24

Figure 4.1 Sequential compound strategy ................................................................... ............................... 30

Figure 5.1 Sale-price/m2...................................................... ............................................................... ....... 37

Figure 5.2 Sequential strategy ................................................................................................................... 50

Figure 5.3 Compound options structure .................................................................... ................................. 51

Figure 5.4 Binomial lattice model ..................................................................... ......................................... 53

Figure 5.5 Payoff structure for project .................................................................................... .................. 57

Figure 5.6 Sensitivity analysis of the real option value (V0and ) ............................................................ 62

Figure 5.7 Sensitivity analysis optimal thresholds ................................................................ ..................... 62


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1. Introduction

Traditional valuation methods like NPV are still the preferred valuation tool for many

practitioners, even if it has been criticized for decades, because its failure to incorporate

flexibility [Hayes & Abernathy (1980)]. Several authors have argued that real option

methodology should be applied instead, but the method can be quite complex, which

makes it difficult to adopt by practitioners. Within real estate several papers have been

published regarding the use real option theory as valuation tool, most of the literature

has focused on conversion from agricultural land to urban land [Capozza & Sick

(1994)]. Other topics in real estate are less developed and the literature is limited, this is

especially within planning and construction areas. Planning and construction of large

real estate projects are characterized by high sunk costs, capital intensive outflow that

are not immediately recovered and uncertainty in sale price/m2and demand. This makes

real estate project development a risky business if all the investment decisions are made

at once. Especially in relation to the situation on the Danish real estate market today,

during the last decade the real estate market in Denmark has changed dramatically.

From 1992-2007 the sale prices increased significantly1 and no one thought about

flexibility in project development by making sequential decisions, since this would lead

to increasing construction costs. In 2008-2009 when the prices decreased several large

real estate developers and real estate projects defaulted because of unrealistic

assumptions for future real estate prices based on simultaneous decisions and static

valuation mythology.

1.1 Motivation

The motivation for the thesis is found in the above situation, regarding construction of

large real estate project under uncertainty. As mentioned large real estate projects are

exposed to several uncertain factors, by making sequential investment decisions instead

of simultaneous decisions it should be possible to gain from the uncertainty according to

real option theory.

1.2 Problem statement

Main hypothesis:

Is it possible to value large real estate development projects by applying real option

theory and is the methodology applicable in practice?

1Association of Danish Mortgage Banks


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Part1

IntroductiontoRealoptiontheory

Literature review

Appliedmodelsandtheory

Part2

Lighthouse*casestudy

Traditionalvaluation

Realoptionvaluation

Part3

Sensitivity analysis

Evaluation

Conclusion

1.2 Structure

The thesis is divided into three main objectives these objectives are divided into three

parts. A brief outline in the structure of the thesis is presented below.

Figure 1.1Thesis structure

Section one in the thesis contains the problem statement and research questions, these

are briefly discussed in relation to the main hypothesis and the objectives. The structure

of the thesis, research methodology and delimitation are also stated in this section. In

section two an introduction of real option theory is given and compared with traditional

valuation theory. The section provides an analysis of the different real option methods

and approaches. In section three a literature review of previous studies and findings are

discussed in relation to real estate development. The literature review regarding real

options in real estate development is backed up by a review of empirical evidence and

applied real options in case studies. Section four introduces specific real option modelsregarding real estate development and the theory behind.

In section five the case project is introduced and a description of the data used in the

case study is given. The section contains the project analysis, where the theories and

models from section three and four are applied in order to develop a model for valuation

of the project. The results are analyzed and compared with findings from the literature

review. In order to verify the results a sensitivity analysis is performed to test the

parameters in the model and identify the most important value drivers. In section six a

critique of the modeling framework and the project valuation is given and


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considerations regarding the modeling framework and future developments in real

options applied in real estate are discussed. The final conclusion is stated in section

seven.

1.3 Research methodologyAs stated above the purpose of the thesis is to apply real option theory in relation to a

real-world project. The main approach in this thesis is the post positivistic research

methodology approach which is a modification of the positivistic research methodology

approach developed in Cours de philosophie positive (1830-42) by Auguste Comte.

The post positivistic research methodology paradigm was developed by Sir Karl Popper

as a critique of the positivistic paradigm. The post positivistic research methodology

paradigm assumes realism with respect to theoretical entities [Ryan et al.(2002)]. Themain difference between the positivistic paradigm and the post positivistic paradigm is

the critical view of the truth, where the positivistic paradigm only assumes one truth.

This means that the researchers ability to remain objective is questioned, but the

founding belief of an objective reality is still maintained. Thereby the post positivistic

paradigm argues for certain subjectivity in researches, meaning that it is difficult to

maintain totally objectivity during a research [Ryan et al.(2002)]. In relation to the case

study in this thesis the objectivity as a main approach is still maintained, but own

expectations and beliefs might result in possible subjectivity.

The post positivistic research methodology approach induces a deductive ground of

reasoning due to the testing of original theories and beliefs [Guba (1990)]. The

deductive approach is based on a number of expectations formed by common ideas,

such as, principals, theories or laws, which is set up by a deductive ground. Fuglsang &

Bitsch Olsen (2004) argues that the inductive research methodology approach is based

on specific observations and then a general idea is designed. In this thesis a combinationof the two research approaches is adopted because the aim of the thesis is to design and

develop a practical valuation framework to value real estate development projects based

on well founded theories which are applied and verified in practice.

The thesis is combined by an academic and practical orientation. The main focus of the

academic orientation is the theories and the relations between them, whereas the

practical orientation relates to the practical case, with real numbers [Brinberg et al.

(1988)]. The foundation of the thesis is the academic orientation, the practical


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orientation has to be consistent with the theories applied, but certain compromises might

be made in order to satisfy both orientations.

An important issue in the thesis is to ensure validity in the sources used. To ensure this

validity it is intended to incorporate the triangulation principal, implementing that

means that theories, methods and primary data are validated by two additional sources

[Brinberg et al.(1988)].

The modeling framework in the thesis is based on Excel, VBA and Crystal Ball.

1.4 Delimitation

The main focus of the thesis is real options and real option theory, basic classical

finance theory is assumed to be known and is not described in the thesis in detail. The

thesis will only be focusing on financial aspects in the case project, no technical or other

construction specific aspects are considered. Because of limited data available regarding

the project, several assumptions have been made and these are discussed in the relevant

sections. The only project specific information available is published by the project

developer and thus assumed not to be completely unbiased.

2. Real option theory

2.1 Traditional valuation methodology

According to Parthasarathy & Madhumathi (2010) is traditional valuation methodology

insufficient and does not get the value of the flexibility of the assets or investment

opportunities. Applying traditional capital budgeting methods the investment decision

becomes an all-or-nothing strategy and do not account for the flexibility that exists in a

project. There are several problems when using traditional valuation methods for

valuing strategic flexibilities, some of these problems are undervaluing assets with littleor no cash flow, forecast errors in estimating future cash flows and insufficient tests for

plausibility of the final results. When using deterministic models like DCF in a

stochastic world, one may underestimate the value of the asset or project. By using

deterministic models one assumes that that all future outcomes are static and there will

be no fluctuations in the conditions affecting the project. If this assumption holds the

flexibility will have no value, but in the real world this is an unrealistic assumption

[Mun (2006), p. 65-66]. Already in the eighties researchers like Hayes & Abernathy

(1980) and Hodder & Riggs (1985) suggests that the uses of deterministic models are


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Financialoptions Realoptions

Shortmaturity,ussallyinmonths Longermaturity,usuallyinyears

Underlyingvariable drivingitsvalueis

equitypriceorpricesofafinancialasset

Underlyingvariablesarefee cashflows,

wichinturnaredrivenbycompetition, Cannotcontroloptionvalueby

manupulatingstockprices

Canincreasestrategicoptionvalueby

managementdecisionsandflexibility

Valuesareusuallysmall Majormillionandbilliondollardecisions

Competitiveormarketeffectsare

irrelevanttoistvalueandpricing

Competitionandmarketdrivethevalueof

astrategicoption

Havebeenaroundandtradedformore

thanthreedecades

Arecentdevelopmentincorporate finance

withinthelastdecade

Usuallyslovedusingclosedformpartial

differentialequationsan

Usuallysolvedusingclosedformequtions

andbinomiallatticewithsimulationofthe

Marketableandtradedsecuritywith

comparablesandpricinginfo

Nottradedandpropreitaryinnaturewith

nomarketcomparables

Managementassumptionsandactions

havenobearingonvaluation

Managementassumptionsandactions

drivethevalueofarealoption

Source: Mun(2006),p.110

Optiontodefer Optiontoswitch

Optiontoabandon Optiontophase

Optiontocontrack Rainbowoption

Optiontoe xpand Compoundoption

Optiontogrowth Compoundrainbow

Source: InspiredbyMun(2006),p.93

Simpleoptions Advancedoptions

Table 2.1Financial options versus Real options

One of the main differences listed in the table is that in real options usually there is no

traded underlying and no market comparables. Since there is no traded underlying and

no market comparables it can be difficult to get the correct value of the underlying asset

and thereby makes is difficult to estimate the price of the option.

Regarding the option terminology, the two types of options has some similarities, the

following terms are known from financial options theory: call, put, in-the-money, out-

of-the-money, European options and American options. These terms are use in real

option theory as well, so no further introduction of these terms are necessary. But in real

option theory the options are also classified in a different way. They are typically

classified by the type of flexibility that they offer [Copeland & Antikarov, (2003), p 12].

Table 1.2 Types of real options


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Table 2.2 divides the most common types of real options in simple and advanced

options, for a better understanding of the different types of options a detailed

description of the different types of real options is given below.

Option to defer investment:

A deferral option is an American call option found in most projects where one has the

right to delay the start of a project. Its exercise price is the money invested in getting the

project started [Copeland & Antikarov, (2003), p 12]. The option to defer is particularly

valuable in such resource extraction industries, as well as in farming, paper products,

and real estate development, due to high uncertainties and long investment horizons.

[Trigeorgis, L. (2005), p. 3]

Option to abandon:

An option to abandon can be valued as an American put option on current project value

with exercise price the salvage or best alternative use value. Valuable abandonment

options are generally found in capital-intensive industries, such as in airlines and

railroads, in financial services, and in new-product introductions in uncertain markets.

[Trigeorgis, L. (2005), p. 5]

Option to contract (scale back):

An option to contract can also be valued as an American put option on a project by

selling a fraction of it for a fixed price if market conditions turn out to be weaker than

originally expected. The option to contract may be particularly valuable in the case of

new product introductions in uncertain markets. As an example the option to contract

may also be important in choosing among technologies or plants with a different

construction-tomaintenance cost mix, where it may be preferable to build a plant with

lower initial construction costs and higher maintenance expenditures in order to acquire

the flexibility to contract operations by cutting down on maintenance if market

conditions turn out unfavorable. [Trigeorgis, L. (2005), p. 5]

Option to expand (scale up):

The option to expand can be valued as an American call option on a project by paying

more to scale up the operations. As example if mineral prices, reserves, or other marketconditions turn out more favorable than expected, management can accelerate the rate


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or expand the scale of production. This option, which will be exercised only if future

market developments turn out favorable at a future date, but not otherwise, can

oftentimes make a seemingly unprofitable (based on passive NPV) base-case

investment worth undertaking. [Trigeorgis, L. (2005), p. 5]

Option to growth (extend)

The option to growth is also an American call option and is familiar with the option to

expand. Growth options are found in all infrastructure-based or strategic industries,

especially in high tech, R&D, and industries with multiple product generations or

applications (e.g., semiconductors, computers, pharmaceuticals), in multinational

operations, and in strategic acquisitions [Trigeorgis, L. (2005), p. 7]

Option to switch:

Switching options are portfolios of American call and put options that allow to switch at

fixed costs between two modes of operations. This would provide valuable built-in

flexibility to switch from the current input to the cheapest future input, or from the

current output to the most profitable future product mix, as the relative prices of the

inputs or outputs fluctuate over time. As example a multinational company may locate

production facilities in various countries in order to acquire the flexibility to shift

production to the lowest-cost producing facilities as the relative costs, other local

market conditions, or exchange rates change over time. In this way the company can

develop more uses for its assets relative to its competitors and it may be at a significant

comparative advantage for them [Trigeorgis, L. (2005), p. 6].

Option to phase (compound option):

A compound option is an option on an option. As example is a phased or staged

investments were the required investment is not incurred as a single up-front outlay.

The actual phasing of capital investment as a series of outlays over time creates valuable

options to continue with the project or abandon it at any given stage. Each phase can be

viewed as an option on the value of subsequent phases by incurring the next cost outlay

required to proceed to the next phase and. This option is valuable in all R&D-intensive

industries, especially biotechnology and pharmaceuticals; in highly uncertain, long-

development capital-intensive industries, such as energy-generating plants or large-scale


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construction; acquisition or market entry strategies; high-tech startups; and venture

capital [Trigeorgis, L. (2005), p. 6].

The above described classifications of real options are the most common ones, another

classification used for options that are driven by multiple sources of uncertainty or

flexibility is called rainbow options. In most real-life projects, the required investment

is not incurred as a single up-front investment, but a combination of compound options

and rainbow options called compound rainbow options [Copeland & Antikarov, (2003),

p 13].

With a distinction between financial options and real options, and an introduction of the

different types of real options it is relevant to introduce and discuss the valuation

methods and approaches applied in real option theory.

2.3 Real Option Methods and Approaches

In the real option literature different methods and approaches are applied, the most

common methods and approaches are presented in this section.

2.3.1 Methods

Kodukula et al.(2006) argues that three different methods are dominant when using realoptions, the methods are:Partial Differential Equation, Simulation, andLattice. A more

detailed presentation of the methods will be given.

2.3.1.1 Partial Differential Equation

The Partial Differential Equation method can be divided into three sub methods: closed

form solutions, analytical approximations, and numerical methods. In general the

Partial Differential Equation is solved with specifying boundary conditions, these

necessary boundary conditions and constraints, can lead to highly complicated partialdifferential equations when a project has more complex features depending on the type

of option [Land & Pinches (1998), p. 8]. The closed form solution is properly the most

applied Partial Differential Equation method. Generally closed-form solution method

assumes that the value of the underlying asset follows a lognormal distribution or that

the returns are normally distributed [Land & Pinches (1998), p. 7]. A closed form

solution, meaning that numerical values can be found given a set of input assumptions.

Closed form solutions are easy, exact and quick to implement due to the simple input

variables and with some basic programming the implementation is done. On the other


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hand it is difficult to explain these models because they apply highly technical

stochastic calculus mathematics. Closed form solutions are exact for European options

but are only approximations for American options, so if the option type is more

advanced like Compound or an exotic option the standard closed form solution have

limitations [Mun (2006), p. 124]. The most applied closed-form solution method for

valuation of options is Black-Scholes (1973). The paper from Black & Scholes (1973)

was the first to develop a closed form solution that calculates an exact value of an

option. The following model is the standard Black-Scholes model:

e d (2.1)

is the cumulative standard normal distribution function

The model takes five input variables:

Vis the value of the underlying asset or stock price (+)

K is the strike price or the cost of executing the option (-)

rf is the nominal risk-free rate (+)

is the annualized volatility of the underlying asset (+)

T is the time to expiration or the economic life of the strategic option (+)

To understand the model and be able to use it, it is necessary to understand the

assumptions under which the model was constructed. These assumptions are often

violated, but the model is useful as a gross approximation and as a benchmark. The

main assumption is that the price structure of the underlying asset follows a stochastic

process, typically a Geometric Brownian Motion (GBM) with static drift and volatility

parameters and that this motion follows a Markov-Weiner stochastic process [Mun

(2006), p. 213-214].

(2.2)

,


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The other assumptions are standard assumptions known from classical finance, efficient

market with no riskless arbitrage, no transactions costs, no taxes, no dividend, the

exercise price is known and constant and the option can only be exercised at expiration,

for the exact value of a European option [Mun (2006), p. 213-214]. In practice these

strict assumptions are difficult to satisfy as mentioned before. Some of the typical

implications are time for execution, sources of risk, only one risky underlying, efficient

markets, constant variance and exercise price. To circumvent some of these

implications, analytical approximations of the standard closed form solution methods is

necessary. This is the second sub method solving options using Partial Differential

Equation, to circumvent the assumption about time of execution the Black-Scholes

model is approximated to value American options, the two preferred approximations

models are the Bjerksund-Stenland model and Barone-Adesi-Whaley model [Mun

(2006)]. These models accounts for dividends payments as well, and thereby eliminates

two of the implication problems. If the only assumption problem is dividend payments a

Generalized Black-Scholes model can be applied.

The final sub method is a numerical method the most applied is Finite-difference

schemes method developed by Schwartz (1977). The method uses numerical

approximation to value options, the method implies a conversion of the appropriate

continuous-time differential equation into a set of discrete-time difference equations and

then solving these using a backward iterative process. The method can be used to value

both European and American options, the model is more mechanical than intuitive

[Land & Pinches (1998), p. 8]. The two most used finite-difference methods are the

implicit and the explicit method. The implicit method is robust and converges to the

solution of the differential equation, but it is complicated to implement due the required

simultaneous equations and the estimated variance is biased upward. The explicit

method is a simplified method of the implicit method, but it do not always converge to

the differential equation and negative probabilities bias the variance downward [Land &

Pinches (1998), p. 9]. Both methods are more intuitive than the continuous-time

method, but it requires great knowledge of mathematics and they are difficult to

implement and solve [Land & Pinches (1998), p. 9]. As discussed above, the Partial

Differential Equation method requires sophisticated mathematical knowledge even if

the sub methods are relatively easy to use, it is still necessary to understand theassumptions behind the methods in order to set the resulting valuation in appropriate


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perspective. Land & Pinches (1998) finds that closed form solutions are not intuitive,

the least well understood by practitioners and may be difficult to implement.

2.3.1.2 Simulation

Another method to solve option valuation is by using Monte Carlo simulation. Boyle(1977) argues that this method is valid and a solid alternative to the other methods

discussed. Especially within risk management and exercising American option the

method has advantages compared with other methods. Longstaff and Schwartz (2001)

argues that the method also has advantages because it can handle complex options and

keeps track of maximum, minimum and mean, which is necessary in path dependent

options. Mun (2006) argues that simulation is the preferred method for handling

complexity in option models

The theory behind the simulation method is based on financial options and assumes that

the value of a derivative is equal to the expected discounted value of the payoffs, in a

risk neutral world. In real option valuation the method is the same, and the value of a

real option is given by V which is the payoff of the real option, and depends on an

underlying driver, for instance the price(P), Q is the risk neutral probability and is asample form the probability, . The following equations are taken from Longstaff and

Schwartz (2001). The value of the real option is estimated by following function:

V(P1(), P2(), ..., PN()) = V(). In order to estimate the present value of the realoption, the expected value of all samples is discounted back by the risk free rate:

, (2.3)with as the discount factor, in this case the risk free rate. The integral is theaverages of the generated samples from the probability space and V given as:

(2.4)Assuming that the underlying asset follows a Geometric Brownian Motion, the path of

the underlying asset is estimated by:

(2.5)


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Where k = 1, 2, , M, ~0, 1 and is a random draw from a standard normaldistribution. The path is estimated with the time interval [0,T] and is divided into M

units with a length of . The stochastic process is approximated over the interval bya single variable with mean 0 and variance 2.

According to the above equations a payoff function for the real option is V(P,t), thereby

the present value of the real option is estimated by:

, (2.6)In brief this means that the mean of the simulated random samples of a defined

stochastic process discounted back with the risk free rate, is the present value of the real

option. As described above this method has several advantages according to risk

management and complex option structures, but one of the disadvantages is that the

method is less understandable as some of the above equations indicate, thereby it is

difficult to interpret for practical users without the theoretical knowledge.

2.3.1.3 Lattice

The lattice method is like finite-difference schemes building on a numerical solution

technique. Lattice method can handle multiple options, complex option payoffs,

downstream decisions and they give a more intuitive representation of the investments.

The practical implementation and understanding of the lattice method is easier than

Partial Differential Equations [Land & Pinches (1998), p. 10]. A lattice model is a

discrete-time approximation of a continuous-time stochastic process and has like the

other methods some limitations. One of the main limitations is that the lattice tree

quickly becomes very large and unmanageable according to the time period.

In real option analysis the most applied lattice model is the Binomial lattice and it is

also the most applied model within all the available methods. The binomial lattice is

developed by Cox et al. (1979), it is intuitive and requires less mathematical

understanding to develop and use. The arguments against this method are the same as

the general argument for the method, they quickly become large and computationally

demanding because they require several lattice steps to obtain good approximations.

Based on these arguments and according to the arguments in Land & Pinches (1998)

and Mun (2006) the binomial lattice is the preferred method for practical use.


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When using binomial lattice in real option valuation, no matter what type of real option

one is solving, the solution can be obtained in one of two ways: risk-neutral

probabilities or market-replicating portfolios [Mun (2006), p.127].

The market-replicating portfolio approach is using the law of one price and enough

traded assets in the market to replicate all possible payoff profiles, as its main

assumption. The use of replicating portfolios works when using financial securities in a

highly liquid market with no trading restrictions, but in real option context with physical

assets and firm specific projects these assumptions are difficult not to violate, the

mathematics behind the market replicating portfolio is also difficult to apply.

The risk-neutral probabilities approach is using the risk-free rate instead of the risk-

adjusted rate when discounting the cash flows the risk is then accounted for in the risk

adjusted probabilities. The results obtained using this approach is identical with the

results from the market replicating portfolio approach. Compared with the market

replicating portfolio approach the risk-neutral probabilities approach is easier to apply

in practice. No complex mathematics skills are required meaning the model is relatively

easy to explain, this is also one of the reasons why it is highly accepted and the most

applied in practice [Mun (2006), p.128]. Further descriptions and analysis is based on

the risk-neutral probabilities approach.

Constructing a binomial lattice, there is a minimum requirement of at least two lattices.

The first lattice is always the lattice of the underlying asset, while the second is the

option valuation lattice [Mun (2006), p. 128]. Making the binomial models more user-

friendly another lattice can be added, which contains the optimal decision in every given

node.

A binomial lattice is developed by nodes, these nodes indicate the possible outcome in a

given state at a given point in time. In the lattice of the underlying asset every node has

two possible outcomes, up-move (u) or down-move (d) as illustrated below.


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V0 u

V0 u

V0 V0 ud

V0d V0d2

Source:

Mun(2006),p.126

Figure 2.1 Binomial lattice of underlying asset

Figure 2.1 illustrates how a binomial lattice of the underlying asset assumed to be a

multiplicative model. The lattice start in node V0at time t = 0 with the present value of

the underlying asset, as illustrated the value of the underlying has two possible

outcomes an up-movement outcome and a down-movement outcome. This is the case

for every node in the lattice. The illustrated lattice in figure 2.1 is a recombining lattice

meaning that the up-down movement in t = 2 is equal the underlying value in t = 0. The

rationale behind this is the reciprocal relationship between the up- and down-

movements. To calculate the up- and down-movements and construct a lattice model

some input variables are required. In a basic structure of a binomial lattice model there

are six variables: (V) present value of the underlying asset, (K) present value of the

implementation cost of the option, () volatility of the return of the underlying in

percent, (T) Time to expiration in years, (rf) Risk-free rate and (b) dividend outflow in

percent [Mun (2006), p. 129].

When the above input variables are defined the up (u) and down (d) factors can be

calculated and the lattice of the underlying asset can be developed. To calculate the up

and down factors the following equations are applied.

(2.7) is defined as the length of a time-step or sub period. By multiplying the volatilitywith the expression the volatility is divided into time-steps. Calculating the optionvalue in the option lattice, backward induction is used and moving from the right to the

left. The risk-neutral probability is calculated by following equation:

(2.8)


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The notation Ci,t is the value of the option at period t, with i representing the total

number of down outcomes out of tperiods. To estimate the option value in t = 0 one

starts at the terminal period t = T. The option values at the terminal period are equal the

maximum of the immediate exercise or an abandonment value calculated as:

, , ; 0 (2.9)Now the option values for the periods before expiration can be calculated, they should

equal the maximum of the exercise at each period or defer until next period (at least) in

the case of a simple defer option. To calculate the value of deferring the following

equation is applied:, , 1 , (2.10)The value of deferring are compared with immediate exercise value and is expressed as

follows: , , ; , 1 , (2.11)Applying this calculation in every time step, one finds the present value of the option

(C0) using backward induction.

The binomial lattice has been extended by Boyle (1988), he develops a bivariate

binomial lattice which can handle two variables. Copeland and Antikarov (2003) has a

model which is similar to Boyle (1988) named quadranomial model handling two

correlated variables.

The three main valuation methods are now discussed and in order to get correct

estimation results different approaches for valuation are developed. The main

approaches are discussed and analyzed in the next section.

2.3.2 Approaches

Several approaches to analyze real options exists, they can lead to the different results

and are different in their assumptions regarding data, uncertainties and capital markets.

In Borison (2005) five different approaches are described: The Classical, The


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Subjective, The Market Asset Disclaimer (MAD), The Revised Classical and The

Integrated.

2.3.2.1 Classical approach

The Classical approach builds on the no arbitrage assumption from the classic optionpricing finance theory. The approach assumes that the traded replicating portfolio

behaves in standard ways, meaning that price movements of the assets are described by

a Geometric Brownian Motion (GBM). This means that standard Black-Scholes model

can be used, but in a real option perspective the assumption about replicating portfolios

and no arbitrage are often violated and the approach does not account for risk if

uncorrelated with market. The approach cannot handle complex option types like

compound options and Bermuda options, which is why this approach is not preferred in

real option analysis.

2.3.2.2 Subjective approach

The Subjective approach builds on the same assumptions as the Classical, but instead of

using objective data like the Classical, this approach uses subjective data to create

replicating portfolios. Thereby one assumes that even if subjective date is applied the

economical assumptions applied in the classical approach still holds. It is difficult to test

subjective data and thereby confirm that the assumptions are satisfied. This makes the

approach less preferable, because private risk is treaded as market risk, which can lead

to incorrect results.

2.3.2.3 Market Asset Disclaimer approach

Copeland & Antikarov (2003) suggests that the best unbiased estimator of a project is

the project itself. This assumption is called Market Asset Disclaimer (MAD) meaning

that one estimates the market value of the project and treat it as a traded asset and

thereby assumes subjective data. The MAD approach has two problems. Using

subjective data, possible market equivalents is ignored and the model estimates

inaccurate values. The second problem is the GBM assumption, when using subjective

data there is no guarantee that the underlying project follows a GBM. This can result in

nonrandom process and then violates the assumptions. One way to account for the

different risk exposures is by applying an extended MAD approach. Schneider et al.

(2008) has developed an extension of the traditional MAD approach which handles both

market risk and private risk. But this approach becomes complex because of the lattice

structure since it handles both private and market risk.


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Approach Method Data

Simulation

LatticePartialDifferentialEquation

Lattice

Source:

BasedonBorison(2005)

Objective&Subjective

Objective&Subjective

Subjective

Subjective

Objective

Lattice

PartialDifferentialEquation

PartialDifferentialEquation

TheRevisedClassical

TheMarketAssetDisclaimer

Subjective

Classical

TheIntegrated

The three approaches described above all have significant problems with inaccurate and

inconsistent assumptions, which mean that they should be used with care in practice

[Borison (2005) p. 30]. The two remaining approaches are modifications of the three

above, they are developed in order to account for both market- and private risk.

2.3.2.4 Revised Classical approach

The Revised Classical approach recognizes two types of projects, projects with market

risk and projects with private risk. The approach assumes use of Black-Scholes when

the project is dominated by market risk and lattice when risk is subjective or private.

This means that if the majority of risk exposure is market risk, the Classical approach is

applied and if it is private risk, the subjective of MAD is applied. Because the approach

tries to handle both types of risk, it is very difficult to estimate a valid discount rate.

2.3.2.4 Integrated approach

The Integrated approach is like the Revised Classical approach not assuming entirely to

one risk factor. But the approach is difficult to explain and requires complex work for

using it in practice [Borison (2005) p. 29]. The approach suffers from some of the same

disadvantages as the Revised Classical approach.

In relation to the discussion of the different approaches above and the discussion of the

different methods in the previous sub section, an overview of the different methods and

approaches are given in table 2.3.

Table 2.3 Real option Approaches and Methods

Table 2.3 clarifies how the approaches and methods differ from each other. With

reference to this section the different methods and approaches will be discussed in


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relation to real estate and especially in relation to the case project and modifications of

the different methods are introduced in the respective sections.

With the theory, methods and approaches discussed in the previous sections, the first

two research questions have been answered. The different types of real options have

been introduced and the most common methods and approaches have been discussed.

As stated previously, real option analysis have several advantages compared with

traditional valuation methods. Is real option analysis preferred in practice or do the

violated assumptions, discussed above, restrict the application in practice.

2.4 Practical implementation and limitations of real options

Copeland & Antikarov (2003) predicts that real options would be the most applied

method in the capital budgeting process within the next decade. These and other authors

such as Trigeorgis (1993) have pointed out that DCF methods may not consider the

value of flexibility and therefore valuation using real options is needed [Block (2007),

p. 1]. A survey of 451 senior executives covering 30 industries by Teach (2003)

reported that in year 2000 only 9% were using real options in capital budgeting. Ryan &

Ryan (2002) made a survey of 208 CFOs and finds that only 11,4 % were using real

options as capital budgeting method [Block (2007), p. 2]. Is the use of real options as

valuation method increased during the last decade. Block (2007) finds in his survey of

279 respondents that 14.3% of the participants indicated the use of real options. Out of

the 40 users, 18 indicated major utilization, 13 indicated they used real options as a

supplemental tool, and 9 reported using real options to shadow the results of more

commonly used methods. This indicates a minor increase in the use of real options in

the capital budgeting process from previous studies, but it is still not the dominated

method as Copeland & Antikarov (2003) suggested. The survey by Block (2007) also

analyzed reasons why companies not were using real options, this is illustrated in figure

2.2.


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43%

26%

19%

12%

Lackoftopmanagement

support

Discountedcashflowisa

provenmethod

Requirestoomuch

sophistication

Encouragestoouchrisk

taking

Figure 2.2Reasons for not using real options

Source: Block (2007)

Figure 2.2 illustrates that the number one reason with 42,7 % is lack of top management

support. The second reason with 25,6 % is that discounted cash flow methods is a

proven method and therefore a preferred method. The third reason with 19,5 % for not

using the real options is that the real options requires a high degree of sophistication.

With 12,2 % the fourth reason for the nonuse of real options is that they tend to

encourage excessive risk-taking [Block (2007), p. 8-9]. The findings in Block (2007) is

consistent with the findings in Lander & Pinches (1998), they give three main reasons

why real option models not are widely used in practice. The first reason is that the types

of models used are not well known or understood by managers and practitioners and

they do not have the required mathematical skills to use the models. The second reason

is that many of the required modelling assumptions are often and consistently violated

in practical real option application. The last reason they give is that the necessary

assumptions required for mathematical tractability limit the scope of applicability [Land

& Pinches (1998), p. 7].

The surveys state that there still is a lack in the use of real options. Comparing theanswers found in Land & Pinches (1998) with the answers in Block (2007), it can be

concluded that the main reasons for not using real option models in practice are the

same almost a decade later. Block (2007) finds that the use have increased over time

compared with previous studies, but the majority still prefers other and more proven

methods. The increasing trend using real options is assumed to continue in the future as

suggested by Copeland & Antikarov (2003). The complex mathematical requirements

and violated assumptions is mentioned as one of the reason for not using real options,


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according to section 2.3 this was also one of the main drawbacks in real option theory.

But other limitations and critique points regarding real option theory is present as well.

One of the critiques is that in reality it is difficult exercise the option at the right time, to

exercise an option at optimal time, the manager has to follow the expected value of the

project in order to know when it is optimal to exercise. Another critique is the

identification of the real options, this can be difficult for practitioners if they do not

know the theory behind [Triantis (2005)].

The critique and the answers from the surveys above, states that real option theory not is

widely used in practice and the hypothesis in the research question cannot be rejected.

With the different types of options introduced and the valuation theory behind them

discussed, it is relevant to introduce valuation of real estate.

3. Real estate

Investment theory traditionally defines real estate as a triangle of time, money and

space, meaning that in a defined time of space an estimated cash flow is generated for a

specific time period. This gives a relatively deterministic understanding of real estate

where the characteristic features are inflexibility and immobility [Dominik (2000), p. 3].

By this traditional understanding, that real estate is immobile and inflexible, traditional

valuation methods should be appropriate according to Mun (2006), but is it right to

assume that real estate is inflexible and without variability? Gibson (2001) argues that

there are many different aspects in real estate flexibility and the flexibility is important

for the assessment of the assets. To value the flexibility in real estate development the

deterministic methods cannot be used, instead it can be seen from a real option view to

stress the aspects of flexibility and variability. The real option approach focuses on

entrepreneurial flexibility rather than on the traditional characteristic features. Thereby

weaknesses of the deterministic methods can be supported by adding an option value

[Dominik (2000), p. 3]. For better understanding of real estate in a real option

perspective a literature review regarding real options in real estate is given.

3.1 Real options in real estate

Real options in real estate are some of the earliest modeled real options. Within real

estate there are primary seven practical applications today: development, planning,


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investment timing, leasing, operation, funding and industry strategy [Patel et al. (2005)

p. 1]. The literature review will only focus on literature regarding development,

planning, and investment timing. The literature on leasing, operation, funding and

industry strategy is not considered in the thesis. The literature gives different aspects on

the flexibility and thereby complex real option methods for the different types of real

options. The main contributor and pioneer of real option theory in real estate

development was Samuelson (1965), he developed an analytical solution for perpetual

American option to convert land into buildings. Others who have contributed to real

option models applied in real estate are Titman (1985), Williams (1991) and Quigg

(1993). Titman (1985) uses real option analysis in real estate development and argues

undeveloped land can be seen as a defer option, he constructs a binomial model to

explain why the land remain undeveloped. He argues that the option to wait can be

interpreted as an American call option without dividends. The conclusion is the options

to wait contribute significantly to the value of the underlying. The uncertainty about

construction costs and risk free interest affects the option value positively. The findings

in Titman (1985) are confirmed by Williams (1991) and the framework from Titman

(1985) is expanded to analyzing the effects of an option to abandon on the value of

project development. Williams (1991) interprets the abandon option as an American put

option without dividends, he introduces maintenance costs for undeveloped property

and finds that on average the undeveloped properties are more costly to maintain and

are abandoned and developed sooner. Using this framework Williams (1991) finds the

optimal timing for development and abandon. Capozza & Sick (1994) introduces the

conversion option and show that one can convert agricultural land to urban land,

suitable for real estate development and find that the optimal conversion rule depends

on the distance to urban areas. Quigg (1995) presented a method of valuing these

options as perpetual American options. She based her model on the same theoretical

calculus and partial differential equations used in the Black-Scholes option-pricing

model.

The literature review rejects the hypothesis stated by Dominik (2000) about real estate

being inflexible and without variability as mentioned before. In real estate development

flexibility and variability can add significant value to a development project. The

theories from the articles mentioned above all suffer from lack of empirical evidenceand this is also one of the main critiques about real option theory. Some of the reasons


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are that it is difficult to get useful data because every development project is unique and

has different uncertainties and thereby not comparable.

3.2 Empirical evidence

The first empirical research using real option pricing models was Quigg (1993), she was

using data on 2700 land transactions in Seattle. She finds empirical evidence for a

model that incorporates the option wait to develop land. She tests a model with close

resembles to the model from Williams (1991) with some of the implications from the

model in Titman (1985). She finds that the market prices reflect a premium for optimal

development, meaning that there is a premium for the option to defer the investment

with a mean of 6% and an estimated implied volatility ranging from 18%-28%. Hung

Chiang (2006) finds similar evidence with an average option premium of 7,75% basedon Hong Kong data. Sing & Patel (2001) examine the U.K real estate market based on

transaction data, they find that the premium for defer development is between 6% -13%

for office, industry and retail. The empirical findings in Sing & Patel (2000) are

consistent with previous findings in real option theory and find that uncertainty

increases the option value to wait to invest, which is consistent with the evidence from

Quigg (1993) based on U.S data. The newest article regarding defer option premium

and option value is Grovenstein (2011), he finds an average option premium for defer

development of 6,6% based on data from the Chicago area. Yamazaki (2001) is using

data from central Tokyo, he finds that total uncertainty of underlying built asset return

has a substantial effect on increasing land prices, which implies that an increase in

uncertainty leads to an increase in land prices. Bulan et al. (2009) study 1.214

condominium developments in Vancouver from 1979-1998 and finds that increase in

both idiosyncratic and systematic risk lead developers to wait with new real estate

investments. They state that, empirically an increase of one standard deviation in the

return volatility reduces the probability of investment by 13%. These findings are

consistent with the findings in Cunningham (2006). He uses property transactions in

Seattle to test two predictions using previous real option theory in relation with real

estate development. The theory predicts: greater price uncertainty should delay the

timing of development and raise land prices. The article finds evidence that support the

predictions, an increase in uncertainty by one standard deviation reduces the probability

of development by 11% and raises vacant land prices by 1,6%. Fu & Jenne (2009)


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tested real option implications in office construction in Singapore and Hong Kong their

findings are consistent with previous findings.

The empirical evidence confirms that the theories on real options in real estate are

present and the findings are consistent even when compared with data from different

countries, which is essential since Hiang (2005) finds no significant evidence of cross-

volatility among Asian and European markets. But the framework has some

imperfections and researchers have criticized it. In Hui & Fung (2009), they criticize the

valuation framework and the empirical tests especially Quigg (1993) and Williams

(1991). One of the critique points is that the work of Quigg (1993) and Williams (1991)

are technically imperfect. Hui & Fung (2009) find five main issues in Quigg and

Williams models. Three of them directly affect the correctness and strength of Quiggs

results and the last two affects the understanding of Quiggs and Williams models

because of suboptimal densities and the two models are not equivalent. Even if the

models have been criticized and there are some imperfections it is still the best models

available for empirical testing. The empirical evidence proves that real option theory

can be applied in real estate development. Thereby it is highly relevant to discuss how

real options are applied in practical case studies.

3.3 Real options applied

As stated in section 2 by Parthasarathy & Madhumathi (2010), Hayes & Abernathy

(1980) and Hodder & Riggs (1985), the traditional the capital budgeting valuation

method is insufficient and valuation using real options should be preferred if possible.

Researchers have used case studies to examine the application of real options in valuing

real estate projects. Patel & Paxson(2001) applied real options in a case of an office

building and estimated an option to delay to be 270/sf compared with deterministic

DCF of 250/sf. Rocha et al.(2007) apply real option analysis through a case study for

housing investment. They identify the three most relevant options in the project:

Information option, Waiting (defer) option and Abandonment option. By these three

options they make a sequential strategy for the project and divide the project into

phases. They develop a model and identifies the optimal strategy and timing for

simultaneous or sequential investments and improved risk management. By applying

real option analysis the value of the development project increased by 10% and reduced

the risk exposure by more the half compared with traditional DCF valuation method.


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Guthrie (2009) demonstrates the practical application of real option analysis of a

commercial real estate project. He values the flexibility in the construction phases using

a binomial framework. He demonstrates that it is possible to make a relatively

straightforward model to value the flexibility in the real estate project and finds that the

DCF method undervalues the project and leads to too much development. Parthasarathy

& Madhumathi (2010) examine the application of real options in valuing a commercial

real estate mall. They compare the traditional DCF method with three different option

models: Black-scholes, Binomial and Samuelson-McKean. They find that the results

from the Black-Scholes model and Binomial model are consistent with a call option

price of Rs2. 120.2 for the Black-scholes and Rs. 119.58. This indicates that the

uncertainty component in the project adds value to the project. The option premium

estimated by the Samuelson-McKean model was 154,08 and thereby higher than the

two other models, the reason for this is that the Samuelson-McKean model is a

perpetual option pricing model and has no time limit which increases the value and adds

additional value to the project.

The reviewed papers have all applied real option methods to value real estate projects,

the conclusion from the papers is that it is possible to add additional value to a project

by indentifying the flexibility and make decisions regarding the uncertainty. By doing

this it is possible to reduce the projects risk exposure and increase the net present value.

4. Models applied in real estate construction & planning

The literature on real options in the real estate planning and construction is rather

limited, most of the literature concerns development of undeveloped land or conversion

of agricultural land to urban land. The most applied models considering planning and

construction is compound options, adopted from R&D real option literature. In section

4.1 the theory behind compound option models is discussed, the section will focus on

closed-form solutions since compound options in a lattice model is relatively straight

forward and the theory is similar to the one described in section 2.3.

4.1 Compound options

Real options in R&D is normally valued using compound options, the R&D expense is

considered a premium for the new product option. These expenses is divided into

2Rs.=IndianRupee


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PhaseIlaunch

Failure Information

gathering

Deferoption

Favourablemarket

Nextphase

Unfavorablemarket

Abandonoption

Abandonoption

Success Information

gathering Nextphase

phases or stages depending on the pipeline and the approval from the authorities, if a

product fails an approval or the market conditions are unfavorable the R&D can be

abandoned or deferred at a given phase. By having this flexibility to only commit to

some of the R&D expense and the opportunity to abandon, adds value to the project. A

real estate development project can be seen in the same way, the construction of the

project can be divided into phases and if the market conditions turn out unfavorable the

next phase can be deferred or the whole project can be abandoned. Figure 4.1 illustrates

how a two phased compound option strategy can look like. The sequential compound

decision strategy includes three real options available to the developer, information

option, defer option and abandonment option. Phase one provides important information

for the next phase etc.

Figure 4.1 Sequential compound strategy

Source:Rochaet al. (2007)

The theory behind compound options was developed by Geske (1979), he developed a

European compound option model to view the option value of R&D and investment and

implementation phases. The model from Geske (1979) is based on a model developed

by Margrabe (1978), which is a closed-form solution on a European option to exchange

one asset for another. Applying the traditional option theory, the Black-Scholes formula

can be seen as a one-fold option. The compound option model Geske (1979) developed

is a two-fold option. The model assumes the same traditional economic assumptions as


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the Black-Scholes formula3 and the Gekse (1979) model is a closed-form solution as

well and assumes the flowing assumptions:

V: present value of underlying asset

ti: maturity date for compound option Ci

Ki: exercise price for compound option Ci

Ci: present value of the compound call option on the option Ci+1

r: risk free interest rate

: dividend yield

2: variance of the returns on the underlying asset

N: cumulative normal distribution function

N2: bivariate cumulative normal distribution function with a1and b1as upper

limits and as the corretation coefficient between the two variables. For

overlapping Brownian increments, the correlation coefficient () is The price for the two-fold compound option then becomes (Eq 4.1):

, ; ; ; ; where ln 2

t t , a b t t ln 2

t t , a b t tV*= the solution of C1 (V,t1) - K1= 0.

Since the development of the closed-form compound option model several refinements

have been made. Carr (1988) developed a model which integrates elements from

Margrabe (1978) exchange option model, and Geske (1979) compound option model.

Another refinement is developed by Lee & Paxson (2001), they developed a model

which is extended to a sequential American exchange option. To value this option a

3investorsareunsatiated,perfect capitalmarkets,unrestrictedshortsaleswithfulluseofproceedsare

allowed,riskfreeinterestrateandvolatilityareknownandconstant,thetradingiscontinuouslyintime,thefirmhasnopayouts


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combination of an American exchange option and a European sequential exchange

option is applied4.

In the American sequential exchange option model, the first compound option can be

exercised at t1and then an underlying American exchange option is exercisable anytime

during t2-t1. The value of such an option is equal to the sum of the Carr (1988) European

compound exchange option and an early exercise premium for the underlying exchange

option [Lee & Paxson (2001)]. The value of the American sequential exchange option is

given by following formula:

,, , , ,, , , (4.2),, , , ,, , ,,, 0 , , ,, , ,

,,0 , , ,, , ,The above formula is a combination of American exchange option and a European

sequential exchange option. The first part of the formula is the European compound

option with K1from Carr (1988) and the two other parts are the early exercise premiums

in an underlying American exchange option with time interval t2-t1, assuming K1 = 0

[Lee & Paxson (2001)].

The American sequential exchange option model was developed to value R&D projects,

but as mentioned earlier, a real estate construction project is similar with R&D projects.

Some of the limitations in the model are the limited number of stages it handles. In

order to handle several stages another models is applied.

4.1.1 n-fold compound option

A model that accounts for several stages is developed by Cassimon et al. (2004) the

model builds on the traditional Geske (1979) model, but as mentioned before the Geske

(1979) model can only handle 2-fold compound options, this is a problem because many

projects contains several phases. So instead of having only a 2-fold compound option an

n-fold compound option can be constructed. The model uses the same notation as the

Geske (1979) model described above, but instead of using the bi-variate cumulative

4SeeAppendixI


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normal distribution function, the n-fold compound option model uses an n-variate

cumulative normal distribution function with aias upper limit, as lower limit and as correlation matrix. All the Ci are a function of V and time t, so the calls have the

same partial differential equation:

(4.3)With boundary conditions: , max 0, , To get the formula for the n-fold compound option Cassimon et al. (2004) derive the

call Cnusing the standard Black & Scholes (1973) model and then define the next call

option Cn-1, which has the call Cnas underlying, using the model developed by Geske

(1979), the process is repeated by adding the respective time steps. The value of the n-

fold compound option is given by:

, , ; , ; (4.4)A proof can be found in Cassimon et al. (2004).

With the n-fold compound option model by Cassimon et al. (2004) it is now possible to

get a closed-form solution for a compound option which has more than 2-folds like the

Geske (1979) model. A major drawback of the model is that it is very difficult to

implement because of the n-variate cumulative normal distribution function, a way to

get around the implementation of the n-variate cumulative normal distribution function

is to merge some of the stages for instance stage 2 and 3, assuming that the decision to

undertake both is made at stage 2. A new exercise price is then needed and is calculated

as:

, (4.5)

With the merged exercise prices it is possible to value multi-staged compound options

using the traditional Geske (1979) model, but the precision of the calculated value is

reduced due to the merged exercise prices and if possible the complete model from

Cassimon et al. (2004) should be preferred. The model has been refined since it was

developed, in Cassimon et al. (2007) the model is modified to account for American

options with multiple dividend payments. And in Cassimon et al. (2011) the model ismodified to incorporate both commercial (market) and technical (private) risk in one


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risk measure. In relation to the different approaches discussed in section 2.3, the one

risk measure gives an advantage since the only approach that can handle both types of

risk at once is the extended MAD approach by Schneider et al.(2008). In the extended

MAD approach the risk is still divided in two risk measures and can thereby only be

solved by lattice method. When having one risk measure the Partial Differential

Equation method can be applied. From a real estate development point of view, the

Cassimon et al. (2011) model can be useful since construction of a real estate project

often is exposed to both market risk and technical risk. Another way to account for

different risk aspects is to change the volatility during the development. One of the main

assumptions of the above described models is constant volatility, Lint & Pennings

(2003) developed a model which accounted for reducing uncertainty over time, but no

further analysis is given regarding this model. The different theories and models

introduced and discussed above are all highly relevant in planning construction of real

estate projects. According to the case study the most relevant models are incorporated

and applied in the valuation.

5. Lighthouse* case

As stated in the literature review valuing flexibility is crucial in construction of real

estate projects. In order to account for the value of flexibility, Trigeorgis (2005)

developed a formula that combines the static NPV with the flexibility value:

(5.1)In this section a real world case is applied, the case is a real estate construction project

at the harbor front in Aarhus. The objective of the case is to make a valuation of the

project, and thereby reject or confirm the theoretical hypothesis about increasing value

by adding flexibility. The flexibility is generated by by dividing the construction of the

project into phases and make sequential decisions. Different valuation methods are

applied and the results are analyzed and compared with previous findings. The methods

are evaluated in relation to the practical utility for decision makers.

The illustration on the front page shows the Lighthouse* project when the project is

completed. The project contains five buildings as illustrated. The separate buildings are

named L,I,G,H and T where building L is the first from the left and building T is the

tower. The construction of the project is divided into three phases, phase I is the


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Total PhaseI(I)&II(L) PhaseIII(HT)

Totalgrossfloorarea 62.000 m2 GrossFloorarea 12.000 m2 Floorearea 38.000 m2

GrossCondomi ni um 47.840 m2 GrossCondominium 11.040 m2 Condominium 25.760 m2

Condominiumnetare a 40.848 units Condonetarea 8.832 m2 Condonetarea 23.184 m2

Condominium 350 units Condominium 90 units Condominium 170 units

Officearea 10.000 m2 Leasableofficearea 10.000 m2

Constructiontime 4 years Constructiontime 1 years Constructiontime 2 years

Source: AarhusKommuneLokalplannr.815 and AktieselskabetHavneinvest

construction of building I and G, phase II is the construction of building L and phase III

is the construction of building H and T. Building G has been sold to another developer

and is not considered in the case. The project area and construction schedule are

described in section 5.1.

5.1 Project description

The construction of the project was started ultimo 2010 by beginning the construction of

phase I (building I), in the case study it is assumed that phase I is not started yet and

thereby it is still possible make decisions regarding phase I. It is also assumed that the

construction of a phase can only be exercised when then previous phase is fully

committed and construction is started.

Table 5.1 Area and construction schedule

Table 5.1 shows in details how the buildings are phased. The area schedule for phase II

is equivalent to phase I, the information in the second column is per phase. The project

is estimated to be completed in four years in the inflexible situation with simultaneous

decisions. One year for phase I and II respectively and two years for phase III.

Before the underlying drivers in the project are estimated, the approach for the case is

defined. As discussed in section 2.3 there are different approaches depending on the

project assumptions, uncertainties and source of data. In the literature review it was

found that several approaches could be applied when defining the valuation process.

Quigg (1993), and Sing & Patel (2001) used the Classical approach with objective

market data and the no arbitrage assumption. While Rocha et al. (2007) based their

study on the Integrated approach, in their study the underlying asset was estimated as a

function of price/m2, sales speed and typical characteristics of housing development

based on the market data. In Parthasarathy & Madhumathi (2010) the Market Asset

Disclaimer approach is applied, they use subjective data and estimate the value of the

underlying asset based on management assumptions.


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In relation to this case the applied approach is difficult to explain. Real estate is not

directly a part of the capital market and it is difficult to believe that the assumptions

from classical finance theory about efficient markets are satisfied. One could argue that

the Revised Classical or Integrated approach should be applied because the assumption

of replicating portfolio is difficult to satisfy and sale prices is anchored in the specific

project and local area sale prices and thereby the risk should be considered as private

risk. In this case a combination of the Revised Classical and Integrated approach is

applied, the two approaches are very similar and according to theory the results obtained

are approximately the same.

A real estate project has two important parameters, the sale price/m2 and the

construction costs. Several papers and econometric models have been developed to

predict real estate prices, some of the most famous models regarding the Danish real

estate market are: ADAM, MONA, SMEC and DREAM 5. They are all very complex

models and are not considered in this case because of the complexity, instead historical

data is applied based on the assumption from DiPasquale & Wheaton (1992) stating that

sale prices/m2 can be applied as a single source of uncertainty. There is no available

data for sale prices/m2of new constructed condominiums at prime locations, such as the

Lighthouse* project. As proxy the data applied for the project consists of quarterly sale

price/m2 for condominiums in the Aarhus 8000 area from 1992 to 2010, gathered from

The Association of Danish Mortgage Banks. Regarding the construction cost, quarterly

data of the construction costs index and CPI are collected, the figure below illustrates

the historical trends over the given time period.

5ADAM=AnnualDanishAggregateModel(StatisticsDenmark),MONA=ThecentralbankofDenmark,

SMEC=SimulationModeloftheEconomicCouncil(EconomicCouncil)andDREAM=DanishRationalEconomicAgentsModel(independent)


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0

50

100

150

200

250

300

Q11992

Q11993

Q11994

Q11995

Q11996

Q11997

Q11998

Q11999

Q12000

Q12001

Q12002

Q12003

Q12004

Q12005

Q12006

Q12007

Q12008

Q12009

Q12010

Q12011

Index

Time

CPI

Constructioncostindex

PropertypricesAarhus8000

Expectedgrowthrate

Figure 5.1 Sale-price/m2

Source: The Association of Danish Mortgage Banks and Statistics Denmark

As stated above, figure 5.1 illustrates the movements in the historical sale price/m2,

construction cost index and CPI. The movements in the sale price/m2indicate no stable

pattern. The average growth per year over the period was 6,56% for the sale price/m2.6

The average sale price/m2in Q4-2010 for condominiums in the Aarhus 8000 area was

25.563 kr./m2based on 154 trades7. It is assumed that growth in sale price/m2between

the years 2003-2008 is abnormal and the future expected growth rate is estimated on

data in the time interval from 1992-2003 and the future expected growth is thereby

estimated to be 4,02% and is assumed to be constant during the case projection. The

trend lines for the construction cost index and CPI indicates relatively constant

m

master thesis u. appendix

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