A Multistage Investment Game in Real Option Analysis

1
A Multi-stage Investment Game in Real Option
Analysis
Real Options -Theory meets Practice 9th Annual
International Conference, June 22-25, 2005,
Paris, France

• Junichi IMAI, Tohoku University,
  JAPANjimai_at_econ.tohoku.ac.jp
• Takahiro WATANABE, Tokyo Metropolitan
  University, JAPANforward0_at_nabenavi.net

• Recently, we revised the paper. If you want a
  new version, please download from
  http//www.nabenavi.net/paper/download/multi6com.p
  df
• We acknowledge the financial support by a
  Grant-in-Aid for Scientific Research from the
  Ministry of Education, Culture, Sorts, Science
  and Technology of Japan.

2
Recent Studies

• The concern with intersection between real option
  analysis and game theory has been growing.
• There are two types of models
• In discrete times with two or three stages
• Smit and Ankum (1993), Smit and Trigeorgis (2004)
• Imai and Watanabe (2004)
• In continuous time with an infinite horizon
• Grenadier (1996), Huisman(2001), Huisman and Kort
  (2003)
• These studies give some answers between
  managerial flexibility and strategic interaction.
• But, all studies are descriptive and qualitative
  approach.

3
Our Model

• In this paper, we investigate a simple option
  exercise game with many stages in discrete time.
• The model can be regarded as a model in
  continuous time with a finite horizon,
  approximately.
• Numerical and quantitative approach

Motivation

• To develop an numerical evaluation method of the
  investment under competition and uncertainty
  toward engineering
• To apply a more general and realistic stochastic
  process --- mean-reverting process, non-Gaussian
  process

4
Interests of the Research

• How the values of the project change ?
• Whether a firm invests immediately at initial
  time or not ?
• How boundaries of the investment change on time ?
• depending on...

• (1) Real Option's Factors
• volatility
• the investment cost
• initial demand
• (2) Competitive Factors
• sequential or simultaneous decision
• position of a firm under competition (leader or
  follower)

5
Definitions of the Model

• We consider a simple timing option game with many
  stages.

• Two firms called firm L and firm F
• Initially, each firm gains a profit with an old
  technology (or from a present project).
• Each firm has an option to adapt a new
  technology (or to invest in a new project).
• Each firm has an opportunity to invest at most
  once within a finite horizon T.
• The cost of the investment is denoted by I.
• The cash flow from the investment is uncertain
  and the investment is irreversible.

6
Firm's Decision

• At each stage, each firm decides whether the
  investment should be done or not, if the firm has
  not invested yet.
• We consider two situations
• (1) Sequential Decision
• At each stage, firm L makes his decision of the
  investment, and after observing the decision firm
  F does.
• (2) Simultaneous Decision
• At each stage, both firms make their decision
  simultaneously.
• But, if there exists two equilibria in the stage
  game, one equilibrium is selected randomly by the
  equal probability.
• Once a firm has invested, he does not make his
  decision anymore. (only one opportunity to
  invest)

7
Profit Flow from the Project

• At time t the profit flow of each firm i ( i L,
  F) is denoted by
• Y(t) Djk dt (j,k0,1)
• Y(t) demand of the project at time t
• Djk profit flow per unit of demand
• D00 neither of firms have invested
• D10 own firm i has invested and the rival firm
  has not
• D01 own firm i has not invested and the rival
  firm has
• D11 both firms have invested
• We assume that
• D10 gt D11 gt D00 gt D01 (strategic substitute)
• D10 - D00 gt D11 - D01 (first mover advantage)

8
A Lattice Model

• The demand Y(t) fluctuates stochastically over
  time.
• We construct a lattice model.

At each stage both firms play a sub-game for the
investment (if both have not invested yet).
case for sequential decision

• By choosing parameters carefully, the lattice
  model converges to a stochastic process in
  continuous time.

9
Basic Example

• First, we assume that the demand Y(t) follows a
  geometric Brownian motion.
• For simplicity we assume risk neutrality, so that
  the investors are risk neutral for the risk free
  rate r and the diffusion process of the demand
  Y(t) under the risk neutral probability measure
  is given by
• dY(t) r Y(t) dt ? Y(t) dz

Parameters for a Basic Numerical Example of a
Project D002.0, D011.0, D105.0, D113.0Y(0)10
0, ? 0.3, r0.05, T1
10
Results (1) Values of the Project with Cost
(sequential decisions)

• Values do not decrease monotonically with cost.
• The range of the cost is partitioned into three
  parts.

Values of the project
350
Only firm L invests at initial time
300
250
200
150
100
50
Investment Cost I
0
0
50
100
150
200
250
300
350
400
450
500
11
Results (2) Values of the Project with Initial
Demand Y(0)
I200, sequential decisions

• Similarly, the values do not increase
  monotonically and the range partitioned into
  three areas.

Firm F
Firm L
Values of the project
450
400
350
300
250
200
150
100
50
Initial Demand Y(0)
0
0
50
100
150
200
12
Results (3) Values of the Project with
Volatility ?
I200, sequential decisions

• The values of firm L also do not monotonically
  increase with respect to the volatility.

Value
300

• When the volatility is small, the value of firm
  L is much larger than the value of firm F.
• When the volatility becomes larger, the
  difference of the values becomes smaller.
• In smallest area, the calculation of L's value
  is not stable.

250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Volatility ?
13
Results (4) Boundaries of Demand

• We find that the boundary of the investment when
  the the rival firm has not invested is equal to
  one when both of firms have not.

• The boundary of the investment when the the
  rival firm has not invested is less than the
  boundary when the rival firm has.
• The boundaries increase as time goes by.
• Especially, they geometrically increase near the
  end of time.

Boundary Demand
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time t
14
Simultaneous Decision

• If there exists two equilibria (like a chicken
  game), one equilibrium is selected by the equal
  probability.

Value

• The value is the average of firm F and firm L in
  sequential decision.

350
300
250
200
150
100
50
Investment Cost I
0
0
50
100
150
200
250
300
350
400
450
500
15
Another Stochastic Process
fat tail, I200

• We apply a non-Gaussian process with non-zero
  skewness and kurtosis, which reflects the
  stylized fact that the returns of many securities
  have rather so called fat tail distribution.

boundary demand
fat tail
GBM
500
The boundary of demand of firm L in a fat tail is
larger than a geometric Brownian motion.
400
300
200
100
time t
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
16
Concluding Remarks

• We construct a multi-stage investment game with
  two firms under demand uncertainty.
• The values of the project do not change
  monotonically with cost, initial demand and
  volatility by the strategic effect.
• The situation is partitioned into the following
  three parts
• (1) both firms immediately invest at initial
  time.
• (2) one firm invests immediately and the firm
  prevents the other firm from his investment
• (3) both firms defer to invest at initial time
• We calculate boundaries to enter the project.
• We apply a general stochastic process to a model.

17
Implications and Future Works

• Which factor is important for the investment?
• In the middle range of the cost and demand, large
  volatility, real option factors are important.
• In the large or small area of the cost and
  demand, small volatility, competitive factors are
  important. (We have to take ''the firm L's
  position''.)

Future Works

• Case studies for the evaluation are needed.
• What parameters should be used ? (market share,
  elasticity of demand etc...) , especially, to
  describe the competitive factors.

• A revised version can be downloaded from
  http//www.nabenavi.net/paper/download/multi6com.p
  df

18
Thank you

• Recently, we revised the paper. If you want a
  new version, please download from
  http//www.nabenavi.net/paper/download/multi6com.p
  df
• We acknowledge the financial support by a
  Grant-in-Aid for Scientific Research from the
  Ministry of Education, Culture, Sorts, Science
  and Technology of Japan.