investing in the competitive agricultural market

7/30/2019 Investing in the Competitive Agricultural Market

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Masters Thesis

Investing in the Competitive

Agricultural Market

Using Real Options Theory and Stochastic Simulation

by

Christiaan Paul Roos

Bachelor of Science in Econometrie

0440892

Primary Supervisor Secondary Supervisor

Dr. R. Ramer Dr. J. Tuinstra

Submitted to the Board of Examiners

In partial fulfilment of the requirements

For the Degree of Master of Science in Econometrics

At the Faculty of Economics and Business

University of Amsterdam

January 14, 2009


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Abstract

Investing in the Competitive Agricultural Market

This study concentrates on the valuation of a decisions process in an agricultural

competitive market where time-to-build lags delay these decisions. Classical

Real Options theory is appreciated above the Net Present Value approach, but

still extensions of the Real Options theory need to be made. Especially deriving

an option to switch model under competition is inaccessible.

This study presents a stochastic competitive model where by simula-

tion farmers can switch between two market producing grain and maize. More

volatility and mean-reversion makes the farmers more willing to switch between

high-priced and low-priced markets, but if marginal costs, switch costs or wages

in one market increase, the gain in the option value to switch decreases. If

the competition is severe and more farmers switch each year, the single farmer

has to follow the competition and also switch to the high-priced market. If the

farmer does not switch, he will lose much more in the market where competition

switch, than in the market where switching is not allowed. It seems that due to

competition markets are even more volatile and reactions on price differences

are justified.

The effect of different types of underlying stochastic processes, such as

mean-reverting or mixed jump diffusion processes needs more clarification, but

can easily be done in simulation. The empirical evidence of this switching pat-

tern in a competitive market environment is yet to be found. The recent switches

of farmers in the world food market to produce crops with high concentration

of sugar, used for biofuel seems to be clarified by the simulation.

ii


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Contents

Page

Abstract ii

List of Figures v

List of Tables vi

Acknowledgements vii

1 Introduction: Preliminary Overview of Real Options in Agricul-

tural Market 1

2 Valuing Investment Decisions Using Real Options Theory 3

2.1 Real Options Theory Versus the Net Present Value Ap-

proach 3

2.2 Model of the Option to Switch 5

2.3 Model for Time-to-Build Lag 11

2.4 Model for Exogenous Competition 14

3 Simulation Results of Classical Real Options Models 18

3.1 Influence of Option to Switch on Decisions Process 19

3.2 Influence of Time Delay on Decisions Process 25

3.3 Influence of Competition on Decisions Process 30

4 Simulations of Stochastic Model 34

4.1 A Stochastic Model for Competitive Agricultural Market 34

4.2 Simulation Results of Stochastic Model 38

4.3 Discussion of Assumptions and Results 47

4.4 Additional Model for Time Delay and Competition 50

iii


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Contents iv

5 Application of Real Options Theory in Empirical Work 53

5.1 Discussion of Real Options Models in General 53

5.2 Agricultural Case Study 54

5.3 Real Estate Case Study 56

6 Conclusions of Valuing Decisions 59

6.1 Decisions Process Under Competition 60

6.2 Further Research 61

Appendices

A Definitions and Derivations of Mathematical Models xi

B Additional Figures xiii

C Software Programs xv

References xx

Used symbols xxii

Index xxv


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List of Figures

Figure Page

2.1 Option to switch between grain and maize market 7

2.2 Option to switch with time-to-build lag 12

2.3 Example of the influence of competition on the price process 15

3.1 Firm values in maize and grain market, split up for different costs 21

3.2 Influence of switch cost to grain market on price thresholds 22

3.3 Influence of mean-reverting speed parameter on price thresholds 23

3.4 Influence of risk-free rate on price thresholds 24

3.5 Influence of volatility on price thresholds 25

3.6 Firm values in grain and maize market with and without time delay 26

3.7 Influence of time-to-build parameter on price thresholds 273.8 Influence of volatility on price thresholds under time delay 28

3.9 Influence of risk-free rate on price thresholds under time delay 29

3.10 Firm values in maize market in competitive industry 31

3.11 Influence of arrival rate of competition on firm value 32

3.12 Influence of volatility on firm value in competition 33

4.1 Example of one simulation run in a stochastic model 39

4.2 Influence of mean-revering speed parameter on the gain in option value 43

4.3 Influence of volatility on the gain in option value 43

4.4 Influence of intensity of switching on the gain in option value 44

4.5 Influence of switch cost on the gain in option value 46

4.6 Influence of wages on the gain in option value 46

B.1 Influence of switch cost to maize market on price thresholds xiii

B.2 Influence of variable cost on price thresholds xiii

B.3 Influence of constant cost on difference in option value for switch

market in stochastic model xiv

v


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List of Tables

Table Page

2.1 Comparison between a financial option and a real option to invest 4

3.1 Basic parameter values for option to switch 19

3.2 Basic parameter value under time delay 26

3.3 Basic parameter value under competition 30

4.1 Basic parameters of simulation in a stochastic model 40

vi


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Acknowledgements

From the beginning of my study Econometrics I am fascinated by world topi-

calities. During the time I wrote this Masters thesis prices of commodities rose

severely. Without the encouragement of especially two people, I would not be

able to capture the effects of this trend in a prudent way.

First, I am most grateful for the support of my supervisor Dr. Roald

Ramer. Thank you for your helpful comments, questions, and suggestions during

this project. I really enjoyed spending time with you arguing and discussing how

mathematical models could influence the way we think about financial markets.

Above all you sharpened my critical thinking.

Second, I dedicate this work to my beloved friend Gabriella, the one

I truly love. Thank you for being patient and caring while I worked on this

project and should be with you. You really inspire me to concentrate on the

important things of life.

vii


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1 Introduction: Preliminary Overviewof Real Options in Agricultural Market

From 1990 till the spring of 2008, consumer prices of commodities increased

more than fifty percent1 and some commodity prices doubled. This raised se-

rious concerns about the food situation of poor people in developing countries.

Changes in supply and demand are the main reasons for this large increase in

volatility.2

On the supply side one important factor behind this rising of food prices

is the increasing price of oil.3 The high oil price has a direct impact on the

costs of necessary inputs like fertilizers, pesticides, and transportation costs. In

addition U.S. farmers have shifted their cultivation toward biofuel feedstock toproduce ethanol.

Also the growing world population is demanding more and different

kinds of food. The rapid economic growth in many developing countries has

given consumers more purchasing power. This shifted their demand from tra-

ditional staples toward higher-value foods like meat and milk. More over the

number of traders at commodity markets has enlarged and the quantity being

traded at commodity markets are making the commodities more volatile.

Due to this increased volatility more farmers familiarise themselves as

an investor and are aware that they can switch from harvesting different types

of crops. The traditional methods of valuing such projects, e.g. the Net Present

Value (NPV) which discount future cash flows, are incomplete; these methods

dont take into account uncertainty of the future cash flows and are not able to

1These consumer prices are derived from the food price index calculated by the Food andAgriculture Organization of the United Nations.

2Pinstrup-Andersen, Pandya-Lorch, and Rosegrant (1999) discuss a more fundamentalprediction of the world food situation until 2020.

3Braun (2008) suggests an intertwined effect of the oil price and agricultural prices.

1


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1 Introduction: Preliminary Overview of Real Options in Agricultural Market 2

adapt for changes in future prices. Conversely, we can extent the NPV method

with Real Options theory to give a more reliable valuation of cash flows, by

coping with uncertainty, flexibility and irreversibility.

Real Options theory has been used in a variety of industries such as

the Manufacturing Industry, Research and Development, and to Oil & Energy

Industry4. This study though focuses on decision making in the agricultural

market. Especially I concentrate on valuing investment decisions in exogenous

competition5 where a time-to-build lag delays the effects of these investments.

The organization of the study is as follows. Chapter 2 gives a theoretical

overview of the literature by presenting some important contributions for this

study. Mathematical techniques are used to present closed-form solutions for

the valuation of the option to switch and the time-to-build model. Furthermore

I propose a model which deals with a competitive market environment.

Chapter 3 explains the simulation results of the models for the switch

option, the construction lag and the competition approach. Chapter 4 presents

the results a stochastic model where by simulation the behaviour of farmers in

markets for different crops is analysed.

Chapter 5 discusses the difficulties in applying Real Option theory inempirical work. Finally, Chapter 6 concludes with an overview of the models,

results, and suggestions for further research. The table of Used symbols after

the appendix beginning on page xxiv and the Index on page xxv can be used

for easy referencing.

4See e.g. the classical example for the valuation of oil fields Paddock, Siegel, and Smith(1988) or a more recent contribution of Armstrong, Galli, Bailey, and Cout (2004) whoincorporate technical uncertainties in oil projects.

5In exogenous competition the farmer needs to decide whether he switches or waits, while

if he waits competitors enter the market and take a share of the gross market value.


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2 Valuing Investment DecisionsUsing Real Options Theory

In this chapter I present some contributions in Real Options theory. In the next

section the implications and advantages of Real Options theory over the NPV

approach are discussed. Section 2.2 presents the model that values the option

to switch. Also closed-form solutions of the construction time are derived in

Section 2.3 and I present an approach to model competition in Section 2.4.

2.1 Real Options Theory Versus the Net Present Value Approach

While an call (put) option gives the holder the right to buy (sell) the underlying

asset by a certain date for a certain price,1 a real option gives the holder the

right, but not the obligation, to undertake capital investment opportunities in

real assets such as land, buildings, power plants and equipment.2 Vollert (2003)

presents a resemblance between a regular financial option and a real option to

invest, see Table 2.1. However a comparison between financial and real option is

not straightforward because the absence of transaction costs in financial options

is not valid for real options. Moreover, real options are strictly related to the

particular firm that the option obtains, so real options are not tradeable.

On the other hand valuation of real options by means of traditional cap-

ital investment appraisal techniques is almost impossible and often wrong.3 Tra-

ditional investment techniques like the NPV and Discounted Cash Flow (DCF)

1See Hull (2006) for a fundamental learning in derivatives.2See also A. K. Dixit and Pindyck (1994) and Vollert (2003, chapter 2) and references

therein for a basic intuition of Real Options theory.3See Lander and Pinches (1998); Trigeorgis (2000); Schwartz and Trigeorgis (2001) for an

overview of real options, challenges to implementing these options, and types of real options.

3


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2 Valuing Investment Decisions Using Real Options Theory Contrary to NPV 4

Financial option Real Optionon a stock to Invest

Underlying asset Current stock value Static NPV of future cash flowsExercise price Fixed stock price Investment costsTime of expiration Exercise date Time until opportunity of

investing disappearsRisk Stock value uncertainty Pro ject value uncertaintyInterest rate Risk less interest rate Risk less interest rateDividend payments Payments to stock holder Payments lost through waiting

Table 2.1 Comparison between financial option and real option to invest(Vollert, 2003, p. 16)

method focus on an expected scenario and a linear operating strategy. In the

real world however decision making practitioners have to deal with uncertainty

and a competitive market environment. One could therefore extend the NPV

approach with the value of the option from active management (Trigeorgis,

2000):4

expanded NPV = static NPV of future cash flows

+ value of all relevant managerial real options

Real Options theory takes account for flexibility, irreversibility and uncertainty

which determines the intensity of the investment decision and the value of all

relevant managerial real options.

Just like financial options, flexibility limits the downside losses but ex-

ploit its upside potential. Flexibility gives the investor the opportunity to wait

whether market conditions turn more favourable to invest or turn out negatively

so no investment will take place. Only in the case of perfect information the

investor will be able to discount future cash flows to the present correctly.

With uncertainty it is impossible to determine the value of future cash

flows if the underlying probability distribution is unknown. However, even if

the distribution is known, it is difficult to take investment decisions because

the actual state of the investment can largely deviate from the prediction. If

the decision is irreversible the NPV could be negative and management could

decide not to take the investment, where with active management decisions can

be taken if market conditions get more favourable.

4For an intuitive example of an NPV approach compared to a Real Options approach, seeDias (2004). Even more (discrete world) examples are in Trigeorgis (1996).


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2 Valuing Investment Decisions Using Real Options Theory Model of the Option to Switch 5

Irreversibility of an investment means that the decision once made and

implemented, the investment cannot be changed without additional costs. While

the NPV approach is not able to eliminate irreversibility, Real Options theory

can cope with various paths of different investment decisions.

2.2 Model of Option to Switch Between Markets

Due to large changes in supply and demand for commodities, a farmer could add

value to his firm if he harvests the right crop: the crop that gives him the most

profit. There could be option value in this investment decision, if he switches

between two markets. Say that the maize market and the grain market are

both within reach for the farmerconsidering switching costswhere he yields

higher cash flows in the grain market. These two commodities can be used for

biofuel because of the high concentration of sugar, where yeast fermentation is

used to produce ethyl alcohol (ethanol).

I follow here the general entry-exit model of A. Dixit (1989).5 Assume

that the price of grain is given by P(g)t and the price of maize by P(m)t . The

differential is the difference between one unit grain and one unit maize: Pt =P(g)t P(m)t where additional cost is such that the price of grain per ton can becompared with the price of maize per ton. One could see as the representation

that the producing maize can be more expensive6. I cannot compare just the

price of maize and that of grain, so after scaling these prices are comparable.

Another way around could be that I model the revenue of one ton grain and

maize.

I assume for now that the decisions of the farmer cannot influence the

price and that there is no competition where competitors can enter or leave the

marketnevertheless the assumption of competition will be relaxed later on.

There is no construction time or grow time. Because of these assumptions I

assume that the price is exogenous for a single farmer.

The shocks in the differential of the price of maize and price of grain

5While Vollert (2003, chapter 3) presents Real Options theory using impulse control andoptimal stopping theory in a very general way, I only use some concept of Stochastic Calculusto come to the solution of the three models. Interested readers can consult Mikosch (1999) orEtheridge (2002) for a introduction course in Stochastic Calculus.

6If grain would be more expensive, could be negative.


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follows a geometric Brownian motion where these fluctuations are spanned7 by

other assets in the economy:

dPt = Pt dt + Pt dBt, P0 = p (2.1)

where is the constant mean-reverting speed parameter and a constant mea-

sure of volatility; dt is the time increment and dBt is the increment of a standard

Brownian motion under the equivalent martingale measure P. This is an ab-

straction of reality; it is more realistic that commodity prices are mean-reverting

so that a mean-reverting process is more applicable. Though the advantage of

the Brownian motion is that it is easy to use and the solution is used further onin more advanced models. Later on we will see how a mean-reverting process

can be used.

Besides that producing maize could be more costly than producing

grain, there are switching costs only to be paid if the farmer switches from

producing one crop to the other. The switching cost function for the states

producing maize (Zt = 0) and producing grain (Zt = 1) is given by the matrix

H(Zt, 1

Zt) = 0 GM 0

(2.2)

where fixed costs M > 0 applies whenever the farmer switches from the grain

market to the maize market and G > 0 is a equivalent fixed cost when the

farmer switches from the maize to grain market. I rule out the possibility of a

money machine by assuming M + G > 0.

Say there are two barriers: P(h) and P(l) which are the high and low

price thresholds respectively. I assume that the farmer begins in a grain market,

so between P(l)

P0

P(h) the farmer harvests grain; subscript 0 indicates

the price of the differential at time zero. The farmer only switches if the price

differential is higher than P(h) (not while he is already harvesting grain; he

switches only to grain, if he was producing maize) or lower than P(l) and no

switches occur if P(l) P0 P(h), i.e. the farmer holds an option to switch.So there is path dependency: the set of decisions of the farmer for a given

7It is worth noting that even if the risk ofPt is not directly traded in the market, it sufficesto be able to trade some other asset whose risk tracks or spans the uncertainty of Pt by usingthe spanning condition (A. K. Dixit & Pindyck, 1994, p. 117).


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t1t2

P(L) Maize

Grain

Grain

P(H)

Figure 2.1 The option to switch between producing grain and maize underfluctuating market prices

circumstance is limited by the decisions the farmer made in the past even when

these circumstances are no longer relevant. P(l) and P(h) are independent of t

and assumed stationary.

An example of a possible development of the price differential and theswitches of the farmer are illustrated in Figure 2.1. The farmer begins in the

grain market, but as the price of maize rises, the differential lowers and the

farmer switches to the maize market at time t1. Later, when the price of dif-

ferential rises he switches back at time t2. Note, that the farmer only focus on

todays price; not on prices one period ahead. Moreover there are spill effects, so

called hysteresis. If a farmer is actively producing maize while P(l) Pt P(h)and the price rises sharply so that Pt > P(h) the farmer will switch to the grain

market. After investing, the price differential could fall and return to the origi-

nal value where P(l) Pt P(h). But it is unlikely that the farmer will take hisinvestment back. He continues to produce in the grain market where additional

cash flows are yielded. So the underlying cause is back to the original level, but

the effect is not; this effect is called hysteresis. It is a very common process in

the economy and I will come back at this effect, especially in the discussion of

the results in Chapter 3 and 4.

Although a farmer can only harvest his crops once or twice per year and

gain a short cash flow, I assume that he generates continuous cash flows. One


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could see this as interest he gets on his saved money at a bank account. Future

cash flows are discounted at a constant risk-free rate .

I can introduce V(g)(Pt) as the value of the firm in the grain market at

price level Pt and V(m)(Pt) as the value of the firm in the maize market at price

level Pt. The form of the additional cash flows in the grain market is Pt c.This is the revenue difference corrected by the additional costs per unit.

A farmer wants to maximise his investment opportunity, so the value

of the firm in the grain and maize market are found by Its lemma (see ap-

pendix A):

1

2 2

P2t

d2V(m)(Pt)

dP2t + Pt

dV(m)(Pt))

dPt V(m)

(Pt) = 0 (2.3)

1

22P2t

d2V(g)(Pt)

dP2t+ Pt

dV(g)(Pt)

dPt V(g)(Pt) + Pt c = 0 (2.4)

To obtain closed form solutions for the value of the firm in the grain and maize

market, I need conditions at the moment that a farmer switches from one crop

to the other. At threshold P(h)i.e. if the farmer switches from the maize to

grain marketthe value of the firm in the maize market must be the same as

the value of the firm in the grain market less the switching costs: V(m)(P(h)) =

V(g)(P(h)) G; this is the value matching condition. At this point the twofunctions must meet tangentially at P(h): V(m)

(P(h)) = V(g)

(P(h)); this is the

smooth pasting condition. There are equivalent conditions for the switch point

at P(l) where the farmer switches from the grain to the maize market. This

gives the following conditions:

V(m)(P(h)) = V(g)(P(h)) G, V(m)(P(l)) M = V(g)(P(l))V(g)

(P(h)) = V(m)

(P(h)), V(g)

(P(l)) = V(m)

(P(l)) (2.5)

The general solution of Eq. (2.3) and (2.4) is (derived in appendix A):

V(m)(Pt) = AmPmt , (2.6)

and

V(g)(Pt) = AgPgt +

Pt

c

. (2.7)


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where g and m are:

g,m =12

2

2

12

2

+ 22

, (2.8)

where I assume that m > 0 and g < 0 and where Ag and Am are constants

determining the value of the firm. Now the constants Ag, Am, P(h) and P(l)

can be determined by writing out the matching and smooth pasting conditions

of Eq. (2.5) giving us four equations and four unknown parameters:

Ag(P(h))

g+

P(h)

c

= Am(P(h))

m+ G, (2.9)

Ag(P(l))

g+ P

(l)

c

= Am(P(l))

m M, (2.10)

Agg(P(h))

g1+

1

= Amm(P(h))

m1, (2.11)

Agg(P(l))

g1+

1

= Amm(P(l))

m1. (2.12)

Numerical solution methods are needed to solve this system of non-linear equa-

tions.8

Ornstein-Uhlenbeck process A geometric Brownian motion can move veryfar away from its initial starting point. Contrary for commodities it can be

assumed that the long-run marginal costs are constant so that the use of a

mean-revering process is more applicable.9

Now assume the mean-reverting process is an Ornstein-Uhlenbeck pro-

cess (O-U process) where in Eq. (2.1) has the term Pt itself in it. The dynamics

of the differential is then given by:

dPt = (m Pt)Pt dt + dBt, (2.13)

with m the long-run mean (see it as the long-run marginal production costs),

a constant mean-reverting speed parameter and a constant measure of

volatility, although in some cases the mean-reverting parameter can be very

low. dBt is again the increment of a standard Brownian motion under the

8A. K. Dixit and Pindyck give two ways of determining the value ofV(Pt): with dynamicprogramming and with replicating portfolio theory.

9See e.g. the fundamental article about the stochastic behaviour of commodity pricesSchwartz (1997) or Srensen (2002) for more fundamental insight on mean-reversion.


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equivalent martingale measure P.

Sdal, Koekebakker, and Aadland (2008) models using Real Options

theory the differential spread for valuing shipping carriers in two markets as an

O-U process. The solution to Eq. (2.13) is:

Pt = etP0 + m(1 et) +

t0

e(ts) dBs.

One can derive (see e.g. the appendix of Sdal et al. (2008)) that the

closed form solution for switching under an O-U process is:

V(m)(Pt) = AmK 2

,1

2

,

2(m

Pt)

2+ Bm(m Pt)K

1

2

1 +

,

3

2,

2(m Pt)2

(2.14)

V(g)(Pt) = AgK

2,

1

2,

2(m Pt)2

+ Bg(m Pt)K

1

2

1 +

,

3

2,

2(m Pt)2

+m c

+

Pt m +

,

where Am, Bm, Ag, and Bg are constants and K is the Kummer function or con-

fluent hypergeometric function (see appendix A for the definition and necessary

conditions) and all other parameters are defined as before. Now with Eq. (2.5)

I have matching value and smooth pasting conditions and with Eq. (A.2) in the

appendixi.e. how the value of Bm and Bg depends on a constant times Am

and Ag respectivelyI have four equation and four unknown constants. So the

constants P(h) and P(l) can be determined.

I have now given the solutions for two different underlying It processes.

I could extent this analysis further and for example assume that the farmer wants

to stop farming totally if the price of maize and grain are very low; this process

of abandoning his investment for a while is called mothballing. I will not discuss

this process of mothballing further more, because it is not substantially more

difficult: the solution of this problem can be found with eight equations and

eight unknown constants (A. K. Dixit & Pindyck, 1994, chapter 7).


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2 Valuing Investment Decisions Using Real Options Theory Model for Time-to-Build Lag 11

2.3 Modelling Decisions under Time Delay

Intuition Vollert (2003, section 5.2) assumes there is only a time lag between

the date of the exercise decision and the time when the decision finally takes

place. This is at some point applicable in the farmer story; this time lag can be

seen as the time between sowing and harvesting a crop.

In spite of this a farmer is even more restricted to time: for several

months in the year it is not possible for him to plant new crops because for maize

and grain these harvest periods differ. In the classical Real Options models it is

assumed that the decision maker can act continuously at any moment; however

this assumption is in our model not applicable. It is important to see that the

decisions are discrete, while the price process can be continuous in time. Further

research is needed to know the scale of the impact of these discrete decision

moments. Such a analysis could be difficult if decisions moments stretch over

several years and one also keeps track of the competition. At this moment I will

relax the assumption that a farmer can only plant new crops at specific moments

in time. This could be modelled in discrete time easier than in continuous case

here. I assume that time lags represent the waiting time for a farmer between

sowing his new crop and harvesting it.Following Bar-Ilan and Strange (1996) we can see that there are four

time-to-build lags in the agricultural market for a farmer. The first stage is

producing in the maize market, the second is producing in the grain market,

the third is in the time-to-build phase, where a farmer switches from grain to

maize (at the low price threshold) and the fourth phase is in the time-to-build

phase, switching from maize to grain (at the high price threshold).

One can see in Figure 2.2 how the situation for a farmer changes under

a time of construction, where the price differential is just an example of how

it can evolve. The difference with Figure 2.1 is that we can see a construction

phase, indicated by the term Construction Time. In this phase the firm is under

construction: between t1 and t1+h switching from the grain to the maize market

and between t2 and t2 + h switching from the maize to the grain market.

In both situations the farmer really switches, because at t1+ h the price

difference is lower than the price difference once the farmer initiated the switch

(at time t1). At t2 + h the price difference is higher than the price difference

once the farmer initiated the switch (at time t2). It could be the case however


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P(h)

Construction Time

Construction Time

t1+ht1

Maize

Grain

P(l)

PsP(h)

t2 t2+h

Grain

Figure 2.2 The option to switching between producing grain and maize mar-ket under fluctuating market price and under construction time

that after the construction phase, the price difference is back at the level of the

initial start of the switch: then the farmer does not want to switch.

Time Delay More Formally Because I want to build on the preliminary

analysis of the general switch options. I focus on the construction phase and

assume all other parameters are equivalent as in the option to switch. I derive

now the farm value being in the construction phase V(c)(s, Pt). 0 s h isthe remaining time to complete construction and h the time of the investment

lags. Actually, in terms of four phases, this is the construction phase switching

from maize to grain at P(h). In further analysis I will mention this construction

phase as V(c)(s, Pt)mg, but for readability I use here V(c)(s, Pt).

At the end of the construction phase the farm stays in the grain mar-

ket or directly switches back to the maize market at P(l), so V(g)(P(l)) =

V(m)(P(l)) M, thus V(c) can be found by:

V(c)(s, Pt) = E

es max{V(m)(Ps) M, V(g)(Ps)}

. (2.15)

Conditioning on Ps < P(l) (the farmer will switch to the maize market) and

Ps > P(l) (the farmer will stay in the grain market) and writing out the values


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of V(m) and V(g) of Eq. (2.6) and (2.7) gives:

= E

esV(g)(Ps) | Ps > P(l)+ E

es(V(m)(Ps) M) | Ps P(l)

=

P(l)es

AgP

gs +

Ps

c

log(Ps) dPs

+

P(l)0

es

AmPms M

log(Ps) dPs,

where log(Ps) is the density function of the output price Ps. Bar-Ilan and

Strange (1996) derive in their appendix the solution to Eq. (2.15), however they

use a slightly different interpretation of V(m) and V(g):

V(c)(s, Pt)mg = Ag

1 (u(l) g)

Pgt

1 (u(l))

ces

+

1 (u(l) ) Pte()s

+ Am((u

(l) m))Pmt (u(l))Mes, (2.16)

where () is the cumulative standard normal distribution and u is defined as

u(l) = log(P(l)

Pt ) ( 12

2

)s

s.

One could derive an equivalent expression for the value of the farmer during

construction phase switching from the grain to maize market. The function u(l)

changes while following the same derivation as above:

u(h) =log(P

(h)

Pt) ( 122)s

s.

The function V(c)

(s, Pt)gm

is:

V(c)(s, Pt)gm = Ag

1 (u(h) g)

Pgt +

1 (u(h))

ces

+

1 (u(h) ) Pte()s

+ Am((u

(h) m))Pmt (1 (u(h)))Ges,

The conditions for the value of the farm being in the grain or maize market are

equivalent to the matching value and smooth pasting conditions of the switching


21/85

2 Valuing Investment Decisions Using Real Options Theory Model for Exogenous Competition 14

option in Eq. (2.5):

V(m)(P(h)) = V(c)(h, P(h))mg Geh,V(m)

(P(h)) = V(c)

(h, P(h))mg,

V(g)(P(l)) = V(c)(h, P(l))gm M eh,V(g)

(P(l)) = V(c)

(h, P(l))gm, (2.17)

where V(c)

(h, P(h))mg is the partial derivative to Pt in P(h): V

(c)(h,Pt)Pt

P(h)

.

With these boundary conditions I have a system of equations that can be

solved numerically to determine Am, Ag, P(h) and P(l). m and g are calculated

via Eq. (2.8). One could find the whole system of non-linear equations written

out in Bar-Ilan and Strange (1996, p. 641), although they build an entry-exit

decision model and use only one time-to-build phase.

2.4 Modelling Decisions Under Exogenous Competition

Classical Real Options models take into account that only one firm has the

right to make one decision, not influenced by the decisions of other competi-tors. In terms of market structure one could say, the farmer acts as a single

firm. Therefore the price movement is assumed to be exogenous for it cannot

be influenced by the farmer. But this assumption is far from realistic in the

agricultural environment.

Intuition A very common process in the agricultural market is for example

that farmers have actually impact on the price. If farmers face a high price for

their crop, more farmers invest in the market. But once the crops are harvested

the supply of this type of crop increasesthe supply curve shifts to the right

increasing the quantity produced and decreasing the price. When the prices are

low, farmers will leave the market and prices will rise again.

This pattern can be seen in Figure 2.3, where an example of the price

process, the mean of the price, and the two floors are indicated. The figure

indicates that once the price reaches the upper floor, price is high and com-

petitors will enter the market. Price will change thereafter and reach the lower

floor, were competitors will leave the market. Later on the price rises again,


22/85


upper floor

lower floor

mean

Figure 2.3 Example of the influence of competition on the price process

competition enters, and so on and so forth. This effect is especially strong in

farming industries where a time delay is present; it is sometimes called the pork

cycle10.

Modelling a pork cycle with Real Options theory is very difficult. It

is tried by Grenadier (2002) with a Game Theoretic approach. He finds that

it is very difficult for the classical Real Options approach to explain boom-

and-bust markets, where periodic bouts of overbuilding result in waves of high

vacancy and foreclosure rates. Contrary to competition with many competitors,

in oligopolistic markets actions taken by the firm may likely result in strategic

answers by its competitors. The firm has to take account for possible reactions

of the competitors in the industry.11

Competition More Formally Following Vollert (2003, section 5.1.1) a far-

mer can make an estimate of the intensity and impact of the competition even

without having too much information. At a very high price for example, he can

predict that a competitive farmer will join him at the market. He knows that

10Formally it is the Cobweb model; see e.g. Rosen, Murphy, and Scheinkman (1994).11The theory about competition in Real Options theory is scarce, however contribution

are made by Smit and Ankum (1993); Vollert (2003) about a Game Theoretic approach ofoligopolistic markets and Kulatilaka and Perotti (1998); Lambrecht and Perraudin (2003)about imperfect markets and information.


23/85


competitors will exercise their option to invest, he only doesnt know when.

Implementing switching is not possible any more with this approach. So I

investigate the competition impact for now only in one market, say the maize

market.

Assume further that Qt represents the gross market value of the maize

market. New competitors can enter or exit the market. If a new competitor

enters the market, a farm possesses a shared option to invest in a new market.

We can see the entries of new competitors as Poisson events12: with rate

competitors are entering the market, this is exogenous uncertain competition.

If a new competitor enters the market the gross market value drops for the

firm with (1 )Qt. The next competitor which enters causes another dropto (1 )2Qt. One can see therefore the gross market value as a mixed-jumpdiffusion process13 under the probability measure P:14

dQt = Qt dt + Qt dBt Qt dt, Q0 = q, (2.18)

where [0, 1] and dt is defined as:

dt =

0 with probability 1 dt

1 with probability dt.

This is a mixed-jump diffusion process with constant and and Bt has incre-

ments of a standard Wiener process. Moreover dt and dBt are independent.

I assume that the Poisson risk is private to the firm, so that the Poisson

risk can be diversified away by investing in direct competitors. Uncertainty

about the future gross market value is represented by volatility in Eq. (2.18),

still demanding a risk premium. Note that if every farmer takes decisions on

this process then every farmer makes the same mistakes, resulting in systematic

errors.

Assuming that V(m) is the values of the farm in the maize market we

12It is common in literature to use Poisson processes for suddenly occurring events, seefor example Weeds (2000) who uses a Poisson process for the moment that an invention hasmade.

13See e.g. Trigeorgis (1991) for an extensive analysis of a jump diffusion process in RealOptions theory or Chang and Chen (2006) who examines the influence of mixed-jump diffusionprocesses contrary to more standard price processes.

14The equivalent martingale measure is derived by Vollert (2003, pp. 131133).


24/85


have equivalent to the former equations:

12

2Q2t d

2

V(m)

(Qt)dQ2t

+ ( + )Qt dV(m)

(Qt))dQt

( + )V(m)(Qt)+ V(m)((1 )Qt) = 0

Equivalent to Eq. (2.6) and (2.7) we get

V(m)(Qt) = AmQmt

where m is determined by the roots of the characteristic equation:15

12

2m(m 1) + m ( + ) + (1 g)m = 0,

where I assume as before that m > 0.

Closed-form solutions for m are not obtainable for = 0, 1: it has tobe solved numerically. The price thresholds P(h) and P(l) are not obtainable

any more, so deriving an equivalent system of non-linear equations as before is

inaccessible. In addition the way I implemented the additional market share in

the grain market cannot be fully understood. Even if I would able to state a non-

linear system of equations here, the interpretation of a high and low threshold

is difficult. I suggest another approach to model competition in Chapter 4.

15Compare Chang and Chen (2006, equation 23) who give an equivalent result for a mean-

reverting mixed-jump diffusion process.


25/85

3 Simulation Results of ClassicalReal Options Models

Like many recent contributions in Real Options theory I use simulation tech-

niques for our agricultural market models. An empirical analysis of the agricul-

tural market is troublesome for two reasons.

First, finding good data is challenging and even if good data is available,

interpreting data cannot be done well in the time frame of this study. Because

the models in this study can cover several years, a set of panel data where several

years and several firms are measured is desirable. Creating such a panel data

set is costly in time and in money.

Second, even if good data would be available, the subject of agricultureis not commonly used in Real Options theory for it is hard to value the real

option value of the land. To intuitively understand how data can be translated

to value options decisions, it is worthy to look at an example from the filming

industry.

Say a company has bought the rights to make a film of a best selling

book. The film industry can decide to exercise this option to make the film,

or can decide to wait on more favourable market conditions. In a data set one

could derive the buying date of the rights (i.e. the buying date of the option)

and derive the exercise date. Once the decision to start the production of the

film has been made, the film gets in production, generates a cash flow and the

option value is calculated. In this example there is a strict separation between

buying and exercising the option.

Conversely in the agricultural market, this separation is not so clear:

when does the farmer really buy the option to switch? Does he holds the option

already or is he able to buy it? Moreover, real options are not tradeable and

cannot be separated from the farm. If the right data is not available, it can

18


26/85

3 Simulation Results of Classical Real Options Mo dels Option to Switch 19

Parameter Basic Value IntervalMean reverting = 0 annually (0, 0.16)

Volatility =

0.0278 annually (0.1, 1)Discount rate = 0.08 annually (0, 0.2)Operation cost in grain market c = 13.44 Euro per unit produced (4, 18)Switch cost to grain market G = 37.59 Euro per switch (0, 80)Switch cost to maize market M = 25.59 Euro per switch (0, 45)

Table 3.1 Basic parameter value for option to switch, based on dataof Tauer

be still worthwhile to investigate the model and the underlying processes. In

addition, we want to know how the model reacts on changing parameters: to

be precise, I make a model where simulation is used to present features and

characteristics of the model and underlying process.

In the following three sections I present simulation results of the option

to switch, the influence of time delay, and the influence of competition on the

high and low thresholds. The further outline of this study is as follows. Chap-

ter 4 gives an outline for a stochastic model for the valuation of a farmers firm

and presents preliminary results of this simulation. Chapter 5 gives a discussion

about the application of the Real Options framework in empirical work. Finally,

Chapter 6 summarises the important conclusions of this study and presents sug-

gestion for further research.

3.1 Influence of Option to Switch on Decisions Process

Although for simulation one could use all kind of parameter values, I use for this

section the data of Tauer (2006), who investigates the decision process in Dairy

Farming.1 However, I have slightly different interpretations of the parameter

value. The switch cost function in Eq. (2.2) has values of switch cost to the

maize market M and switch cost to the grain market G and are not entry or

abandon costs respectively like in Tauers article. Further more, I use a larger

interval for his parameter values, to obtain more values, so graphically results

are better interpretable. The basic parameter values and the intervals are given

in Table 3.1. I solve the system of non-linear equations from (2.9)(2.12) with

1Another example of an empirical work in the agricultural market is that of Wossink andKuminoff (2007). They examine investing in organic farming with the Real Options theory.


27/85


the Matlab program in appendix C.

Value of the Firm For different variable cost c, and switch costs G, and M

the firm values are displayed in Figure 3.1. Here for all values of the costs, the

firm value in maize market (the dotted line) and in the grain market (straight

line) is given. The left dashed line is the appropriate low threshold P(l) and the

right dashed line the high threshold P(h). Costs are increasing from left-to-right

and above-to-below.

The slope and pattern of the firm value in the grain market V(g) and

maize market V(g) is different: this is because of the definition of these two

functions, see Equations (2.6) and (2.7). Because Am and m are positive, V(m)

is a positive valued function, increasing in Pt. However, for a certain interval of

Pt the function V(g) is larger. And due to the negative values of Ag and g, this

function is also increasing in Pt, but only positive if Pt is larger than fifteen.

It seems that V(g) diverge to one point: this is because of the term Pt cand this will be larger than AgP

gt . The convergence value is larger for lower c,

which makes sense because the differential Pt c is lower and and arethe same for = 0.

One can see that the increasing costs widens the spread between P(h)

and P(l). At Pt = P(l) the function V(m) exceeds V(g) exactly by the amount

of M; it is optimal to exercise the switch option to the maize market, giving

up V(g) + M and receiving V(m). The switch at Pt = P(h) is equivalent: it is

optimal to switch to the grain market while V(g) = V(m) + G.

For a small area of Pt (depending on c, G and M) it is worthwhile to

be in the maize market, and for rising Pt one should switch to the grain market.

That eventually V(m) is higher is because of the form of the function and cannot

be seen from the fact that one should switch to the maize market, because one

could only switch if V(m)(P(l)) M = V(g)(P(l)).

Influence of Parameters on Low and High Threshold Figure 3.2 dis-

plays the thresholds at rising switch cost to the grain market G. The figure

shows that the larger the switching cost to the grain market, the higher the

high threshold and the option value is. The switch trigger to the maize market

is untouched, but the spread between P(h) and P(l) is greater, so the reluctance

to abandon is higher. The influence on P(h) is much larger than on P(l), mostly

because the low threshold is unaffected by the increasing switch costs. The rea-


28/85

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

0

20

40

0

100

200

300

400

500

600

P

FirmValue

Figure

3.

1

Firm

valuesinmaizea

ndgrainmarket,splitupfordiffe

rentcosts

21


29/85


0 10 20 30 40 50 60 70 8016

18

20

22

24

26

28

30

32

34

G

P

P(L)

P(H)

Figure 3.2 Influence of switch cost to grain market on price thresholds

son for the rise of P(h) is because the farmer must pay more to switch to the

active grain market, so price must rise to earn the switch costs back.

The influences of the variable cost and the switch cost to the maize

market are as intuitive as the influences of the switch cost to the grain market,

so figures will not be presented here. In the appendix Figure B.1 shows that

increasing switch cost to the maize market M influence P(h) slightly, but has a

dramatically effect on P(l) as in the case of the effect of G on the thresholds.

In the appendix Figure B.2 shows that increasing production costs c rises the

price thresholds P(l) and P(h), the threshold P(h) slightly more than P(l): the

zone is slightly larger at c = 18. The interpretation is as follows. With higher

production costs in the grain market, the expected cash flows from this market

will be less, and the value of the project will diminish. Therefore, a higher

price level P(h) is required before the farmer is willing to switch to this market.

Equivalently, a higher production cost in the grain market, will give reason for

the farmer to switch to the maize market more quickly, because of the loss of

money in the grain market. The farmer switches more reluctantly to the market

with higher production cost and abandons it sooner.

In addition, there is a large difference in interpretation of G and M.

The switch cost to the grain market must be paid immediately, while the switch

to the maize market affects the farmer only through the prospect that he will


30/85


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1612

14

16

18

20

22

24

26

28

30

P

P(L)

P(H)

Figure 3.3 Influence of mean-reverting speed parameter on price thresholds

have to pay the switching costs at some time in the future2. Because of this

discounting effect, impact of G on the price thresholds is larger than the impact

of M.

The influence of the mean-reverting speed parameter on the thresh-

olds is displayed in Figure 3.3. Only for the interval (0, 0.16) the solutions are

stableholding other parameter values constant. This is probably because of

the underlying Brownian motion in the price differential process: higher values

of give a higher trend in the process. And while other parameters are held

constant, the term Pt in the function V(g) of Eq. (2.7) can get troublesome, if

the denominator goes to zero and the system gets unstable. A higher value of

lowers the two thresholds; this is mainly because of the fraction Pt explained

above. Moreover the higher the mean-reverting parameter, the more predictable

the price differential becomes: there is a higher trend. So the uncertainty about

future cash flows is less, and the farmer is willing to switch at a lower price.

Also, the difference between the discount rate and the mean-reverting parame-

ter is less: the farmer has to switch at a lower price, for the margins between

the markets are lower.

The influence of the risk-free rate on the thresholds is displayed in

2This is true because the definition of the entry and exit decisions by (A. Dixit, 1989) aredifferent.


31/85


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

10

20

30

40

50

60

70

80

P

P(L)

P(H)

Figure 3.4 Influence of risk-free rate on price thresholds

Figure 3.4. The high threshold seems to have its minimum at = 0.13. For

a low value of the fraction Pt rises, but this effect is partially offset by the

rising variable costs of c . The low threshold keeps diminishing for rising risk-

free rate. The effect of the rising will be practically lower the thresholds, for a

risk-free rate of more than 20% is unrealistic. The spread widens for increasing

, so to switch between markets is less important.

Finally, for this section I present the influence of the volatility parameter

on the thresholds in Figure 3.5. A larger value of influence the thresholds

to rise, the high threshold more than the low.3 A moderate change in causes

the spread of the thresholds to increase dramatically. The reason is that with

higher volatility, the price difference fluctuates more and because the farmer

faces switch costs, he dont want to switch frequently. He rather prefers to stay

in the market where he is because he knows that as soon as he switches with

probability one the price will get back at the original level under the assumption

of a Brownian motion with high volatility.

We saw above in Chapter 2 that uncertainty is one of the three key

elements in investment decision making. We can also see here that more un-

certainty delays switching decisions leading to a larger hysteresis effect. While

3Compare the same effect of on regular option values on stocks: increasing the volatilityof an option (increasing risk) will higher the price of the option.


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3 Simulation Results of Classical Real Options Models Influence of Time Delay 25

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.265

10

15

20

25

30

P

P(L)

P(H)

Figure 3.5 Influence of volatility on price thresholds

the focus of this study is more on the impact of competition and time delay,

it is important for further research to investigate the influence of the under-

lying stochastic process, such as a mean-reverting process; interested reader

can consult A. K. Dixit and Pindyck (1994, section 5A) for a starting point.

Programming the Kummer function in Matlab4 could be cumbersome though,

possibly by using large numbers in the Kummer and fsolve function.

3.2 Influence of Time Delay on Decisions Process

In this subsection the influence of a construction phase is discussed. I stated

in Chapter 2 that the farmer has a construction time from the grain to maize

market V(c)gm

and a construction lag switching back from maize to grain

market V(c)mg

. However, it seems impossible to solve the system of non-

linear equations in (2.17). The Matlab program in see appendix C that solves

this system only takes into account one time lag at the switch to the more cash

flow generating grain market.5 For the simulation data I use the same data as

Bar-Ilan and Strange (1996). The base parameter values are in Table 3.2; every

4Downloadable packages are available, see for example the Confluent hypergeometric func-tion by Stepan Yanchenko at http://www.mathworks.com/matlabcentral/fileexchange/12665.

5It is possible to change the function V(c)mg

so the farmer switches from grain to maize.


33/85


Parameter Basic Value IntervalMean-reverting = 0 annually (0, 0.018)

Volatility =

0.02 annually (

0.006,

0.06)Discount rate = 0.025 annually (0.021, 0.3)Operation cost in grain market c = 1 Euro per unit produced (4, 18)Switch cost to grain market G = 1 Euro per switch (0.4, 3.5)Switch cost to maize market M = 0 Euro per switch (0, 3.5)Time or construction lag h = 6 year (0, 11)

Table 3.2 Basic parameter value under time delay

0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

80

P

No Delay

0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

80

P

Construction time of 3 years

0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

80

P


0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

80

P


Maize FirmGrain Firm

Maize FirmGrain Firm

Maize Firm

Grain Firm

Maize Firm

Grain Firm

Figure 3.6 Firm values in grain and maize market with and withouttime delay

time I change the parameter value I keep the other values constant.

Value of the Firm and Time Lag The construction time has a severe

impact on the price thresholds P(h)

and P(l)

. One can see if Figure 3.6 thefirm value in the maize (dashed line) and the firm value in the grain market

(straight line). Also the high and low price thresholds are displayed (vertical

dotted lines). One can see that while the delay period is relatively short that

there is not much impact on the price levels. But as the delay becomes larger

(up to six and nine years) in the lower graphs, the spread between the high

and low price threshold becomes smaller and smaller. However, there is not

much influence on the actual value of the firm in either market, for the price of

producing units in the grain market c, and the switch costs G and M are kept


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0 2 4 6 8 10 120.7

0.8

0.9

1

1.1

1.2

1.3

1.4

h, time lag

P

P(L)

P(H)

Figure 3.7 Influence of time-to-build parameter on price thresholds

constant.

Although a construction time of nine years is unrealistic in the agri-

cultural market one can see the dramatic effect of the time lag on the high

price threshold. In Figure 3.7 the low price threshold is less affected, because

I only use one construction time. However, even the low price threshold is af-

fected, because of the opportunity to switch back. Especially after six years the

price threshold drops enormously. I present one more graph of the influence of

the volatility on the price thresholds before discussing why these effects seems

reasonable.

I present in Figure 3.8 the influence of the volatility on the price thresh-

olds. One can see that the high price threshold (dashed line) is less affected by

the increase of volatility than the low price threshold (straight line). Like in the

option to switch, the low threshold declines with increasing volatility and the

spread between the two threshold rises.6

What we can see from above figures is that the implemented time lags

lead to earlier switching for the farmer than if there is no construction time. Ac-

tually I find this resultin the case of agricultural environmentrather strange:

a farmer would switch harvesting another crop if construction time is longer.

6This result looks similar to the case of Bar-Ilan and Strange (1996), where the exit costis one.


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0 0.01 0.02 0.03 0.04 0.05 0.06

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

P

P(L)

(h = 0)

P(H)

(h = 0)

P(L)

(h = 6)P

(H)(h = 6)

Figure 3.8 Influence of volatility on price thresholds under time delay

Besides more uncertainty about future prices leads to fewer switches at no time

lag, than with a time lag. So in the presence of time lags, the farmer switches

earlier under uncertainty than under certainty. It seems that switching is has-

tened under a time lag.

This is contrary to the standard switch options results in literature

where more uncertainty leads to longer waiting to switch, as we have seen in

the previous section. It is known that uncertainty raises the benefit of waiting

but leaves the opportunity cost of a further delay uninfluenced. This is because

the farmer can wait: he limits his downside potential, but can use unlimited

upside potential.

Under a time lag however, the farmer wants to switch earlier, because

he has to face a construction time, where his farm is worthless. The opportunity

cost of waiting to switch increases, and while the farmer has the option to switch

or stay in the maize market, future profits are truncated from below by the time

lag. Under a time lag, higher uncertainty leads to a lower high threshold and a

lower low threshold. The farmer will switch even at a lower price to the grain

market and wants to stay there, because prises must drop dramatically will the

farmer switch again back to the maize market. And the longer this time lag is,

the sooner the farmer wants to face the reality of in the construction phase and

the sooner he switches to the grain market.


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0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

P

P(L)

P(H)

Figure 3.9 Influence of risk-free rate on price thresholds under time delay

Influence of Risk-free Rate on High and Low Threshold Figure 3.9

displays the influence of the discount rate on the price levels under time delay

is presented. One can see that this effect is much different than the effect the

discount rate has on the price levels without a time lag (see Figure 3.4). Here

a increasing discount rate increases the high and low price level, where without

time lag, it lowers the thresholds. The reason is the does not only appears in

the function for the firm value in the grain market V(g), but also in the construc-

tion time V(c) and even in the value matching and smooth pasting functions

(the switch costs are discounted back to the time they are implemented).

However, the switch costs are smaller with higher time lag because of

the exponent function which discount these costs back at the moment it is

implemented. Also the exponent term in Pte()s

of Eq. (2.16) lowers with

increasing . Moreover the whole term lowers by dividing by a higher . Maybe

because all terms with such a exponential function in this equation lower, the

term Ag

1 (u(l) g)

Pgt rise and therefore the thresholds are higher.

Apparently, the cash flows are not offset by the increasing variable costs. In

the construction phase the farmer does not make a profit so the high threshold

must be higher for him to switch to gain back this expenditure.

The influences of , M, G are almost the same as in the Option to

Switch, increasing lowers the thresholds; increasing G raises the high threshold

very much and the low threshold keeps almost unaffected; increasing M rises


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3 Simulation Results of Classical Real Options Models Influence of Competition 30

Parameter Basic Value IntervalMean reverting = 0 annually Volatility = 0.2 annually (0, 1)Discount rate = 0.04 annually Switch cost to grain market G = 1 Euro per switch Competition erosion = 0.5 annually (0, 1)Arrival rate of competitors = 0.1 annually (0, 1)

Table 3.3 Basic parameter value under competition

the high threshold slightly more than the decrease of the low threshold. The

reasons are given in Section 3.1 and need not to be repeated here. It seems that

the influence of time delay as it is modelled here does not have a large impact

on the decisions of a farmer. Only if the farmer has to switch his whole farm

and therefore faces a large time delay, the time lag has a severe impact.

3.3 Influence of Competition on Decisions Process

Like I stated above solutions for a system of non-linear equations in Section 2.4 is

not feasible but I give some results here about the value of the farm in the maize

market and discuss the influence of competition and volatility on the firm value.Note again that exogenous competition is modelled here; only the decisions of

the single farmer are analysed and not the decisions of the competition.

Value of the Firm in the Maize Market I use basic parameters as in

Table 3.3 and keep these constant as one parameter at the time will be changed

over its interval. The impact of the scale of the competition in the maize market

is visualised in Figure 3.10.

In this figure one can see the trigger value of the Gross Market Q (top

figure) and the value of the maize firm for different market sizes (bottom figure).

The trigger value has been derived from the value matching and smooth pasting

condition for a general timing option (A. K. Dixit & Pindyck, 1994, p. 141):

V(m)(Q) = Q GV(m)

(Q) = 1.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.4

1.5

1.6

1.7

1.8

1.9

2

Q*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

FirmV

alue

Q*()

Q*( = 1)

Figure 3.10 Top: Influence of competition Level on trigger level Q; Bot-tom: Influence of competition level on firm value of maize for different grossmarket values Q

And the solution for these conditions is:

Q =m

m 1G

V(m)(Qt) =G

m 1

QtQ

m. (3.1)

One can see from the figure that if the competition increases and takes

a larger value of the gross market value, the trigger level Q decreases. So

competition in this form lowers the trigger level and makes the farmer willing

to invest earlier. Also the firm value lowers in the same rate as the trigger level.

The effect of diminishing trigger level is equivalent for the firm value: the more

competition there is, the lower the firm value will be. At this point the firm

suffers from the competition, because a large part of the gross market value has

been taken over by competitors.

In essence the value of waiting is reduced by the threat of a loss in

market value due to the entry of competitors. Therefore the opportunity cost of

waiting rises. By early exercising the option the farmer may have early mover

advantage and he can be a single farmer until other competitors enter.

Another parameter which influences the trigger level is : the Poisson

rate of competition arriving in the market, see Figure 3.11. Here one can see


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.2

1.4

1.6

1.8

2

2.2

2.4

Q*

= 0

= 0.1

= 0.5 = 1

Figure 3.11 Influence of arrival rate of competition on firm value

that increasing has a decreasing effect on the trigger level of investing. And

the more competition there is in the market, the lower the trigger level will be.

The reason for this is that with competition the impact of the arrival rate is

higher. More competition and more competitors in the market make the farmer

wants to invest earlier; if he does not invest, the competitors will take over that

part of the gross market value. Equivalently by waiting the farmer erodes his

option value to invest.

It is interesting to see that the effect of = 1 is almost equal to the

effect of = 0.4 with severe competition.7 At = 1 the investment trigger is

less than 1.2, for = 0.4 the trigger is still slightly above 1.2. Therefore the

effect of the competition intensity or erosion is much more intense. So it is

necessary to have correct information about and once doing empirical work,

for the intensity of competition could have a dominant effect on the trigger level

and firm level.

In Figure 3.12 one can see the influence of the volatility parameter

on the trigger level Q. The influence of the volatility makes the trigger level to

rise, though more without competition than with competition. So competition

7The meaning of = 1 is the every year a competitor arrives, and in case of = 0.4 onaverage every four years one competitor arrives.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

Q*

= 0

= 0.1

= 0.5

= 1

Figure 3.12 Influence of volatility on firm value in competition

decreases the impact of higher volatility.

We saw earlier that without competition and absence of time delay, the

high price threshold P(h) rises with increasing . In the presence of a time

lag, the price threshold rises much less (it slightly levels) compared with the

model without time lags. The effect in a competitive industry equilibrium is

comparable with the presence of a time lagfor the trigger level rises a little

bit.

However one can see that uncertainty is less important in the presence

of competition than in the absence of it. The most important reason for this

effect is that under competition the farmer just has to invest if the opportunity

is available where he tries to invest ahead of the competition and has no room

left to think about uncertainty.

I will not present the influences of the speed parameter or the discount

rate here, because their influence does not contribute much to the discussion

about competition. Anyhow, the results of this section about competition are

quite unsatisfactory for two reasons. First, the model itself could not be able to

capture the dynamics of a perfect competitive market. Second, previous results

only give some insight in the firm value in the maize market and I am not able

to cover the dynamics for switching between the competitive maize and grain

market.


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4 Simulation of Investment Decisionswith a Stochastic Model

While classical Real Options theory derives high and low price thresholds for a

single farmer when to invest, in this chapter I give a preliminary outline of a

stochastic decision model under exogenous competition.1 It presents the model

assumptions and the model itself in Section 4.1. The results are presented in

Section 4.2 and finally another model is discussed in Section 4.3.

4.1 A Stochastic Model for Competitive Agricultural Market

It is worthwhile to look at the set-up, assumptions and limitations of a stochastic

model for an agricultural market under exogenous competition. Albeit compe-

tition is a very difficult concept in Real Option theory, some progress is made

with oligopolistic markets using Game Theoretic approaches2 and the effects and

implications of perfect competition in Real Options theory could need further

research. It is therefore important to investigate this matter further.

Assumptions for a Perfect Competitive Industry The agricultural mar-

ket consists of a large number of producers and consumers. And most farmers

produce homogeneous crops. I assume that perfect information is available

for both consumers and producers, and there are low entry and exit barriers.

Farmers try to maximise their profits. With these conditions, it is reasonable

to assume that the competition in this agricultural market is exogenous.

1See e.g. Thornton and Jones (1998) for an agricultural equivalent simulation, where adynamic land-use model is presented and limitations of with such a model are discussed.

2See e.g. Smit and Ankum (1993); Huisman (2001) about a Game Theoretic approach inReal Options theory.

34


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4 Simulations of Stochastic Model Set up and Assumptions 35

Leahy (1993) demonstrates that the equilibrium investment policy of

an individual firm is identical to a myopic strategy in which a firm ignores the

effect that other firms react on the price process. A. K. Dixit and Pindyck

model perfect competition using price floorsgenerated by the exits of other

firmsand price ceilingsgenerated by the entries of other firms. They find

that the option to remain uncommitted is worthless since competition eliminates

all profits.

I investigate the determination of the equilibrium strategy where a

farmer needs to account for the actions of its competitors when determining

its optimal investment strategy. Farmers produce a good that they sell in a

competitive market at the market clearing price, which is determined by sup-

ply and demand. While demand is exogenous, supply resulting from firms is

endogenous. Farmers consequently must invest strategically, accounting for the

investment strategy of other farmers in the industry and the impact of other

farmers.3

While the price is endogenous under exogenous competition, I assume

that shocks are exogenous. For now a geometric Brownian motion is a simple

process to assume, but other processes could be used as well: e.g. the mean-

reverting O-U process or a more economic realistic mixed-jump diffusion process.I use constant returns to scale between capital and labour, however, this could

be changed.

The Construction of a Stochastic Model Following e.g. A. Dixit (1991)

I split the agricultural market for grain and maize in infinitesimal number of

parts, where a farmer can hold one or more small parts. A constant number

of farmers are in the whole market of grain and maize while the supply is only

from these farmers. Every producer in the market is price taker and only the

industry equilibrium is investigated where the chosen capacity to produce a

goodunder rational expectationsis the equilibrium between the marginal

costs of the farmers firm and the demand for that good. The assumption of

rational expectations determines the equilibrium of prices and entry strategies

of the firms.

I assume that a farmer chooses between two goods, grain and maize

assumed to be substitutes. So once a demand shock is positive, the demand for

3

See for a more fundamental explanation also Novy-Marx (2002).


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one good rises, but the demand of the other falls.4 One could choose a unit

of good maize qm such that average costs to produce qm are equal to one, if

the firm produces at minimum average costs. In the same way one could do

this for the other good grain qg. As the demand is exogenously given, the total

production of the farmers does not influence this. The industry production is

given by a Cobb-Douglas function:

qm = LmK

1m where 0 < < 1, and

qg = LgK

1g ,

where total qm and qg are the quantity of maize and grain produced respec-tively and Lm, Km, Lg, Kg are respectively the labour input and capital input

to produce one unit of maize and grain. All labour and capital input is widely

available.

The relation between the price and quantity is given by an inverse de-

mand function assuming that all of the capital is used in production, so reducing

output in a farm means scrapping capital:

P(m)t = D(Xt, qm), for example use:

P(m)t =

1

Xtqm ,

where is the elasticity of demand, P(m)t is the price of one unit of maize and

X the exogenous demand shock5, modelled as a geometric Brownian motion:

dXtXt

= dt + dBt, (4.1)

where Bt is the increment of a standard Brownian motion and and assumed

to be constant. And equivalent for the grain market we find:

P(g)t = D(Xt, qg), for example use:

P(g)t = Xtq

g ,

4In terms of A. K. Dixit and Pindyck (1994, chapter 8) I use here only industry wideshocks and not farm specific shocks.

5A. K. Dixit and Pindyck (1994, chapter 8) distinguish two types of exogenous shocks:one for the specific firm and one for the total market the firm is part of. However, I only takethe whole market demand shock into account.


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where P(g)t is the price of one unit of grain and Xt as above in Eq. (4.1).

Xt is the source of uncertainty in the modelchanging because of the

changing demand of the total demand of maizeand could be model differently

as well: using XXt as in Eq. (4.1) we get a mean-reverting process. For nowwe assume, like I stated above, that the shock is for both goods the equivalent

so a positive shock in demand will change the demand for good qm to the right

and the demand for good qg to the left.

Marginal costs of producing one unit of maize and grain are defined as

a function of the production times the wages w:6

MCm = w

qmKm

1+ cm, (4.2)

MCg =w

qgKg

1

+ cg, (4.3)

where the constant costs cm > cg are used for modelling that the grain market

generates a higher cash flow. Note that these marginal costs are for every farmer

the same. Different marginal costs for farmers specialised in one market can give

this simulation more depth.

The farmer can switch for significantly switching costs M and G, where

M is the switching costs from switching from the grain market to the maize

market and visa versa. Because there are very large amount of firms, investment

equals entry by infinitesimal firms and disinvestment equals exit by infinitesimal

firms. Moreover, I add an additional stochastic element in this switching: I

assume that the number of switching farmers is Poisson distributed with Poisson

rate #, because for some unknown reasons the farmer wants or wants not to

switch to produce a different crop. He is not able to know all his motivations

for switching. One could see this parameter also as the rate that farmers are

ready to switch and want to switch.

The simulation consists several years of farming where farmers can

switch from the maize market to the grain market if the price minus the marginal

costs in one market are larger than the other accounting for switching costs. In

each year that a market is more favourable the number of farmers who switch

is Poisson distributed.

6See e.g. A. Dixit (1991), where equivalent derivations are used to model the impactof price ceilings on the equilibrium in the market. Though, the price process is much moresubtle. Also demand functions can be based on the impact and type of switching.


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4 Simulations of Stochastic Model Simulation Results of Stochastic Model 38

Finally I compare the value of the markets where no switches occur and

the value of the markets where farmers may switch to harvest the favourable

crops. The value of the grain and maize market iswhere farmer can switch:

V(m)t = N

(m)t (P(m)t MCm)

V(g)t = N

(g)t (P(g)t MCg),

where N(j)

t is the number of farmers in jth market at time t. To compare the

value of the markets if a farmer cannot switch, I use for the number of farmers

a constant.

Extensions are possible. Not only for the type of competition7

, butalso for the type of demand shock and correlation of the demand shock with

other parameters. Other variables such as wage rate w and the purchase price

of capital Km in Eq. (4.2) could be random for example and they could even be

correlated with demand shift parameter Xt to capture macroeconomic shocks.

Caballero and Pindyck (1996) uses two different Brownian motions, one for the

output produced and one for the demand shock.

On could follow Ungern-Sternberg (1990) and assume that one type of

farmers are specialised in producing grain, others in producing maize. Or the

other type is specialised in easy switching between markets when one market

turns out to be favourable. So the marginal costs of the first type is significantly

lower producing one crop and can stay longer in the market at a lower price, but

the other type of farmer switch easily and can profit from a higher price level

(but has on average higher marginal costs). So there could be farmers who are

specialised in switching: they have higher marginal costs than the specialised

firms, but on average lower costs on both markets.

4.2 Simulation Results of the Competitive Industry Equilibrium

The simulation of the Stochastic model under exogenous competition is pro-

grammed in Matlab, see appendix C. Figure 4.1 shows the preliminary results

of this simulation where basic parameters are as in Table 4.1.

7See e.g. Grenadier (2002) derives equilibrium investment strategies in a Cournot-Nashframework.


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0

5

10

15

0510

15

20

Grainprice

P

0

5

10

15

010

20

30

40

M

aizeprice

P

0

5

10

15

100

10

20

30

Switchpoints

P

0

5

10

15

050

100

Num

beroffarmers

Units

0

5

10

15

0

500

1000

1500

2000

Ma

rketValue(withswitch)

t(years)

Value

0

5

10

15

investing in the competitive agricultural market

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