Real options ambiguity risk and insurance

The decision making with competitionintroduces challenging problems.Villeneuve and Décamps examine the optimal investment policy for a constrained firm which has no access to external fina

REAL OPTIONS, AMBIGUITY, RISK AND INSURANCE

Studies in Probability, Optimization and Statistics

Volume 5

Published previously in this series Vol 4 H.K Koo (Ed.), New Trends in Financial Engineering – Works Under the Auspices

of the World Class University Program of Ajou University

Vol 3 A Bensoussan, Dynamic Programming and Inventory Control

Vol 2 J.W Cohen, Analysis of Random Walks

Vol 1 R Syski, Passage Times for Markov Chains

ISSN 0928-3986 (print) ISSN 2211-4602 (online)

Alain Ashbel S hool of Man Risk and Dec

U Distinguish

S Profes

ed Professor Mathematics,

Jae fessor of Fin

of Business

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Edited by

n Bensou Smith Chair P agement, the cision Analys hed Professo Shige Pen ssor of Mathe

of the Minis , Shandong U and eyoung Su nance, Depar Administrati Suwon, Kore

rlin • Tokyo •

biguity ance

n Financia olume Two

y ussan Professor

e University sis, City Univ

r, Ajou Univ

ng ematics try of Educa University, J

ung rtment of Fin ion, Ajou Un

ation of China Jinan, China

nancial Engin niversity

n, DC

k ring,

Dallas ong Kong

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neering

© 2013 The authors and IOS Press

All rights reserved No part of this book may be reproduced, stored in a retrieval system,

or transmitted, in any form or by any means, without prior written permission from the publisher ISBN 978-1-61499-237-0 (print)

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Alain BENSOUSSAN, Shige PENG and Jaeyoung SUNG1

This book is the second volume in the WCU financial engineering series by the financialengineering program of Ajou University, supported by the Korean Government underthe world-class-university (WCU) project grant Ajou University is the unique recipient

of the grant in Korea to establish a world class university in financial engineering Themain objective of the series is to disseminate, faster than textbooks, recent developments

of important issues in financial engineering to graduate students and researchers, ing surveys or pedagogical expositions of published important papers in broad perspec-tives, or analyses of recent important financial news on financial-engineering research,practices or regulations

provid-The first volume was published by the IOS press in 2011 under the title of “NewTrends in Financial Engineering”, containing articles to introduce recent topics in finan-cial engineering, contributed by WCU-project participants This volume focuses on im-portant topics in financial engineering such as ambiguity, real options, and credit risk andinsurance, and has 12 chapters organized in three parts These chapters are contributed

by globally recognized active researchers in mathematical finance mostly outside theWCU-project participants

Part I consists of five chapters Real options analysis addresses the issue of ment decisions in complex, innovative, risky projects This approach extends consid-erably the traditional NPV approach, much too limited to deal with the complexity ofreal situations In preparing the investment decision, a project manager should determinewhich project to choose, when to choose it, and in what scale He/She should incorpo-rate flexibility in order to benefit from acquiring later on important information about allaspects of uncertainties related to the investment Consequently, during the project life,the manager still faces further decisions on how to manage, contract, expand, or abandonand to meet industrial competition, not to mention performing basic managerial func-tions and making financial decisions Towards the end of the project, the manager facesclosure decisions such as sale, reorganization or liquidation Flexibility is not the uniquecharacteristics of real options One additional idea is to take advantage of valuation tech-niques developed in context of financial products, in order to define properly the value

invest-of industrial projects This is more and more possible in the context invest-of commodities with

an organized market The energy sector is an important example An important ence between real and financial options concerns the issue of competition For complexinvestment projects, there are generally few possible players For financial products, thenumber of players is very large and therefore each of them does not change dramati-

differ-1 Graduate Department of Financial Engineering, Ajou University The research culminated in this book was supported by WCU(World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-2009-000-20007-0).

Real Options, Ambiguity, Risk and Insurance

A Bensoussan et al (Eds.)

IOS Press, 2013

© 2013 The authors and IOS Press All rights reserved.

v

cally the context (it may be possible of course) The decision making with competitionintroduces challenging problems.

Villeneuve and Décamps examine the optimal investment policy for a constrained firm which has no access to external financing, and show that an increase

cash-in the volatility of the underlycash-ing asset can actually decrease the value of the growthoption value Huisman, Kort and Plasmans apply the real option theory to analyze a reallife case, and show that negative NPV projects are optimally undertaken (when discountrates are high and technology progresses fast) in the hope of new opportunities or growthoptions for the firm Thijssen enriches real options analysis by introducing industrialcompetition into standard real option problems and argues competition can be bad forwelfare in a dynamic setting Hugonnier and Morellec consider a real options problemfor a risk averse decision maker with undiversifiable risks and show that the risk aversioncan make him/her delay investment, reducing the (market) value of the project Finally,Bensoussan and Chevalier-Roignant consider capital budgeting decisions on not onlytiming but also scale of a project and show how optimal trigger policy integrates the twoaspects

Part II has three chapters on ambiguity We believe that the notion of ambiguity isone of major breakthroughs in the expected utility theory Ambiguity arises as uncer-tainties cannot be precisely described in the probability space The objective is to un-derstand rational decision making behaviors of an economic agent when his decisionmaking environment is subject to ambiguity Mathematics underlying those economicsproblems can be very challenging, imposing great obstacles to the economic analysis ofthe problems Chen, Tian and Zhao survey recent developments on problems of optimalstopping under ambiguity, and develop the theory of optimal stopping under ambiguity

in a general framework Ji and Wei review the principal-agent literature in continuoustime, and apply to the optimal insurance design problem in the presence of ambiguity.Shige Peng provides a survey of recent significant and systematic progress in the area

of G-expectations: new central limit theorems under sublinear expectations, Brownian

motions under ambiguity (G-Brownian motions), its related stochastic calculus of Itô’s

types and some typical pricing models He further shows that prices of contingent claims

in the world of ambiguity can be expressed asg-expectation (nonlinear expectation) of

future claims, and that the method of the nonlinear expectation turns out to be powerful

in characterizing these prices in general

In Part III, four chapters are devoted to risk and insurance In particular, this partcovers mutual insurance for non-traded risks, downside risk management, and credit risk

in fixed income markets Liu, Taksar and Yuan introduce mutual insurance which can

be viewed as a mutual reserve system for homogeneous mutual members, such as P&IClubs in marine mutual insurance and Federal Reserve reserve banks in the U.S., andexplain why many mutual insurance companies, which were once quite popular in thefinancial markets, are either disappeared or converted to non-mutual ones

The importance of downside risk minimization has attracted lots of attention fromboth practitioners and academics in light of recent experience of the Subprime MortgageCrisis Nagai discusses the large deviation estimates of the probability of falling below

a given target growth rate for controlled semi-martingales, in relation to certain ergodicrisk-sensitive stochastic control problems in the risk averse case Portfolio insurancetechniques are related to the downside risk minimization problem Sekine reviews sev-eral dynamic portfolio insurance techniques such as generalized CPPI (Constant Propor-

tion Portfolio Insurance) methods, American OBPI (Option-Based Portfolio Insurance)method, and DFP (Dynamic Fund Protection) method, and applies these techniques tosolve the long-term risk-sensitive growth rate maximization problem subject to the floorconstraint or the generalized drawdown constraint Credit risk is also an important topicfor both practitioners and academics, being particularly important to the determination

of subprime mortgage rates Ahn and Sung provide a pedagogical review of literaturefocusing on determinants of credit risk spreads with emphasis on methodological aspects

of structural models

This broad spectrum of concepts and methods shows the richness of the domain ofmathematical/engineering finance We hope this volume will be useful to both graduatestudents and researchers in understanding relatively new areas in economics and financeand challenging aspects of mathematics In this manner, we think contributing to theexpectations of the WCU project

vii

Acknowledgement

This book was supported by WCU(World Class University) program through the tional Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-2009-000-20007-0) We, together with the other contributing authors, are grateful for support from the staff of the IOS press, especially, Maureen Twaig, and Kim Willems and for assistance of Xiaoyan Chen, Sanghyun Cho and Gang Geun Lee

Na-Contents

Preface v Alain Bensoussan, Shige Peng and Jaeyoung Sung

Part 1 Real Options

Jean-Paul Décamps and Stéphane Villeneuve

3.3.3 φ as a Super Solution to HJB Equation (3.13) 17

Investment in High-Tech Industries: An Example from the LCD Industry 20

Kuno J.M Huisman, Peter M Kort and Joseph E.J Plasmans

Jacco J.J Thijssen

ix

Real Options and Risk Aversion 52

Julien Hugonnier and Erwan Morellec

4.1 Risk Aversion and the Option Value to Wait 58

Alain Bensoussan and Benoît Chevalier-Roignant

4.2 Example 2: Cobb-Douglas Production Function 82

Zengjing Chen, Weidong Tian and Guoqing Zhao

x

4 Markov Setting 108

Appendix A: Optimal Stopping Related to Reflected BSDEs 113

Appendix B: Proofs of Results on the Problem of Optimal Stopping 115

An Overview on the Principal-Agent Problems in Continuous Time 126

Shaolin Ji and Qingmeng Wei

2 Principal-Agent Problems Under Full Information 129

3 Principal-Agent Problems with Hidden Actions and Lump-Sum Payment 131

4 Principal-Agent Problems with Hidden Actions and Continuous Payment 133

5 Optimal Insurance Design Problem Under Knightian Uncertainty 134

5.3 Optimal Insurance Design from the Insured’s Perspective 136

Nonlinear Expectation Theory and Stochastic Calculus Under Knightian

Uncertainty 144 Shige Peng

2 BSDE and g-Expectation 148

2.1 Recall: SDE and Related Itô’s Stochastic Calculus 148

2.2 BSDE: Existence, Uniqueness and Comparison Theorem 150

3.1 g-Expectation and g-Martingales 155

3.2 Inverse Problem: Is an Expectation E a g-Expectation? 155

4 Nonlinear Expectations and Nonlinear Distributions 158

5 Central Limit Theorem and Law of Large Numbers 161

5.1 Normal Distributions Under a Sublinear Expectation 161

5.2 Central Limit Theorem and Law of Large Numbers 164

6 Brownian Motion Under a Sublinear Expectation 167

6.1 Brownian Motion Under a Sublinear Expectation 167

6.3 G-Brownian Motion in a Complete Sublinear Expectation Space 169

xi

6.6 Quadratic Variation Process of G-Brownian Motion 171

6.9 Brownian Motions, Martingales Under Nonlinear Expectation 172

Part 3 Risk and Insurance

John Liu, Michael Taksar and Jiguang Yuan

2.3 Variational Inequalities for the Optimal Value Function 191

3.1 The HJB Equation in the Continuation Region 192

3.2 The Optimal Value Function for the Original Problem 198

6 Long-Term Risk-Sensitized Growth-Rate Maximization 249

6.1 Long-Term Optimality with Floor Constraint 251

6.2 Long-Term Optimality with Generalized Drawdown Constraint 252

xii

Credit Risk Models: A Review 255 Cheonghee Ahn and Jaeyoung Sung

2.5 Chen, Collin-Dufresne and Goldstein Model 274

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Part 1 Real Options

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Optimal Investment Under Liquidity

variable is the observable value of an investment project that could be undertaken at a

fixed cost The real option literature has emphasized the ability of firms to delay theirirreversible investment decisions In the presence of sunk costs, this flexibility in thetiming of investment is valuable because it gives firms the option to wait for new infor-mation As a result, optimal investment policies are mathematically determined as thesolution to optimal stopping problems and prescribe to invest above the point at whichexpected discounted cash-flows cover sunk costs, in contrast with the usual net presentvalue rule The pioneered model is due to McDonald and Siegel [13] and has been ex-tended in various ways by many authors (see for instance Dixit and Pyndick [8] for anoverview of this literature) An important common feature of this literature is to assumethat the investment decision can be made independently of the financing of the sunk cost.This amounts to consider that capital markets are perfect so that any project with posi-tive net present value will find a funding (Modigliani and Miller [14]) However, capitalmarkets are not perfect, external financing is costly and firms accumulate cash to coverinvestment needs without resorting to the market.4Despite strong empirical evidences,the real option literature has somewhat neglected market imperfections and, typically,the role of cash holdings in the firms’ investment decision Very few papers focus on thelevel of self-financing that a firm should optimally decide in a dynamic setting A firstattempt in that direction is Boyle and Guthrie [3] More recently, Asvanunt, Broadie andSundaresan [2] develop a corporate model with interactions between cash reserves andinvestment opportunity when the firm has some outstanding debt Hugonnier, Malamudand Morellec [10] considers the interactions between cash holdings, dividend distribu-tion and capacity expansion when firms face uncertainty regarding their ability to raiseexternal funds and have to pay a search cost to meet outside investors

In this note, we try to merge the real option and the corporate finance literature byfocusing on the optimal investment policy for a cash-constrained firm More precisely,

1 Financial support from the European Research Council (249415-RMAC) is gratefully acknowledged.

2 CRM-IDEI, Université de Toulouse 1, Manufacture des Tabacs, 21, Allée de Brienne, 31000 Toulouse E-mail: jean-paul.decamps@univ-tlse1.fr.

3 CRM-IDEI, Université de Toulouse 1, Manufacture des Tabacs, 21, Allée de Brienne, 31000 Toulouse E-mail: stephane.villeneuve@univ-tlse1.fr.

4 Among many other papers, see for instance Bates, Kahle and Stulz [4].

Real Options, Ambiguity, Risk and Insurance

A Bensoussan et al (Eds.)

IOS Press, 2013

© 2013 The authors and IOS Press All rights reserved.

doi:10.3233/978-1-61499-238-7-3

3

we make the strong assumption that the firm has no access to capital markets As a sequence, the cash reserves of the firm must always remains non-negative to meet op-erating costs and the firm value is computed as the expected value of dividends pay-ment In this framework, when facing an investment opportunity, shareholders have both

con-a profitcon-ability concern (the optimcon-al time to undertcon-ake con-a growth opportunity) con-and con-a ity concern (the risk to be forced to liquidate a profitable project) The model presentedabove takes the result of Décamps and Villeneuve [6] and studies the consequences ofliquidity constraints on the decision to invest in a new project

liquid-2 Optimal investment in perfect capital markets

2.1 The benchmark model

As a benchmark, we begin with the seminal model of McDonald and Siegel [13] Westart with a probability space (Ω,F ,P), a filtration (Ft)t≥0 and a Brownian Motion

W = (W t)t≥0with respect toF t We assume that a decision maker continuously observes

the instantaneous cash-flow X of a project where X = (X t)t≥0 is a Geometric BrownianMotion with driftμand volatilityσ,

dX t=μX t dt +σX t dW t

We denote by r the constant risk-free rate and we assume that μ< r The decision

maker’s problem is to decide when to invest in this project at a fixed cost I After the

investment is made, the firm generates cash-flow forever As a result, the sum of the

discounted expected future cash-flows if investment is made at time t is

lem Because X is the only state variable, the set of admissible strategies is the set of

stopping times adapted toF tdenoted byT That is, the value function associated to the

investment opportunity is defined as

V (x) = sup τ∈T Ex

Under this formulation, it is easy to prove that the optimal investment strategy belongs

to the set of threshold strategies T y where T y is the hitting time of y by the process X.

Specifically, the investment option should be exercised the first time that the value of theinvestment project exceeds a critical threshold, the so-called optimal exercise boundary.The exercise boundary can be explicitly computed using a standard verification theorembased on the smooth-fit principle (see for instance Dixit and Pyndick [8], Part III) This

leads to an explicit expression for V ,

where L(x, y) =Ex (e −rT y ) and where x ∗represents the optimal exercise boundary or thelevel of cash-flow above which it is optimal to invest We have,

x ∗= ξ

ξ− 1 I(r −μ)with

2.2 Discussion

It is worth to mention that the seminal model of McDonald and Siegel is based on eral implicit assumptions, two of which are particulary strong: perfect information onthe future cash-flows and perfect capital markets In particular, it is not necessary to

sev-assume that the decision maker has the possibility to self-finance the sunk cost I In a

perfect capital market, she has the possibility to access to outside financing by issuingshares Let us describe a possible financing contract between the decision maker and an

outside investor Because X is perfectly observable, the expected profitΠ if the

invest-ment is made at time t is denoted by Π(X t) withΠ(x) = x

r−μ Therefore, the outside

financier may propose the following contract: if the decision maker invests at a level y,

the investor will ask for a proportion δ of shares that satisfiesδΠ(y) = I Along this

contract, the expected payoff for the decision maker will be(1−δ)Π(y) = Π(y) − I

Therefore, the decision maker has to choose the optimal level y that maximizes her profit,

that is,

max

y L(x, y)( Π(y) − I),

which is equivalent to the decision problem of the benchmark model Therefore, a

de-cision maker that cannot afford to self-finance I invests optimally at the same level

of investment x ∗ if she signs the contract described above As a consequence, der the assumption of perfect capital markets, the investment decision is made in-dependently of the financing decision The objective of this note is to relax the as-sumption of perfect capital markets and to illustrate the consequences of liquid-ity constraints on the investment decisions Before developing our model, we em-phasize that taking into account costly external financing may lead to challengingstopping problems Let us consider a decision maker who needs to finance the in-

un-vestment cost I Assume that banks are in perfect competition and offer consol

bonds with the following covenant: if the borrower is unable to pay the coupon,

the firm is forced to default As a consequence, the market price at time t of the

α

1{X t ≥c}

J.-P Décamps and S Villeneuve / Optimal Investment Under Liquidity Constraints 5

whereθt is the shift operator5and where c has to be computed so that D(x, c) ≥ I,

oth-erwise the debtholders refuse to lend I Because the competition between banks is sumed to be perfect, we must have D(X t , c) = I if the investment is made at time t The

as-participation constraint D(x, c) = I has several important consequences We observe that for a fixed x, the function D(x, ) defined on [0, x] is convex with D(x, 0) = D(x, x) = 0 thus, D(x, ) reaches a maximum at c ∗ (x) = (1 −α)−α1x and therefore the participation

constraint is satisfied if and only if

D(x, c ∗ (x)) = −α(1−α)−(α1+1)x

For any x satisfying the participation constraint (2.1), there are two levels c1, c2with

c1≤ c ∗ ≤ c2for which D(x, c i ) = I It is obvious that the decision-maker will choose the smallest coupon c1and thusΠ(X t) can be expressed as

flow X and thus control the payment process This is the topic of the next

sec-tion

5 See for instance Revuz and Yor [19] page 36 for the definition of the shift operator.

3 Optimal stopping for a cash-constrained firm

3.1 The model

We consider a firm with an activity in place that generates a cash-flow process The firmfaces liquidity constraints because it has no access to capital markets Consequently, thefirm defaults as soon as the cash process hits the threshold 0 The manager of the firmacts in the best interest of its shareholders and maximizes the expected present value ofdividends up to default At any time the firm has the option to invest in a real option thatincreases the drift of the cash generating process fromμ0toμ1>μ0without affectingits volatilityσ This growth opportunity requires a fixed investment cost I that must be

financed only by using the cash reserve

The mathematical formulation of our problem is as follows We start with a bility space(Ω,F ,P), a filtration (Ft)t≥0 and a Brownian Motion W = (W t)t≥0with re-spect toF t In the sequel,Z denotes the set of positive non-decreasing right continuous

proba-andF t-adapted processes andT , the set of F t-adapted stopping times A control policy

π = (Z tπ,τπ;t ≥ 0) models a dividend/investment policy and is said to be admissible if

Zπ

t belongs toZ and ifτπ belongs toT We denote the set of all admissible controls

byΠ The control component Zπ

t therefore corresponds to the total amount of dividends

paid out by the firm up to time t and the control componentτπ represents the investment

time in the growth opportunity A given control policy(Z tπ,τπ;t ≥ 0) fully characterizes

the associated investment process(I tπ)t≥0which belongs toZ and is defined by relation

I t = I1t≥τ π We denote by Xπ

t the cash reserve of the firm at time t under a control policy

π= (Z tπ,τπ;t ≥ 0) The dynamic of the cash process Xπ

singu-J.-P Décamps and S Villeneuve / Optimal Investment Under Liquidity Constraints 7

3.2 Value of the firm with no growth option

Assume for the moment that the firm has only access to one of the two technologies

(say, technology i= 0 for drift μ0and technology i= 1 for driftμ1) The cash process

X i = (X i,t)t≥0therefore satisfies

dX i,t=μi dt +σdW t − dZ i,t

We are back in the classical distribution problem studied in Jeanblanc and Shiryaev

[11], Radner and Shepp [16] or Asmussen and Taksar [1], the firm value is V i (X i,t∧τi,0)where

Computations are explicit and we have:

Proposition 3.1 (Jeanblanc and Shiryaev (1995))

(i) (Firm value)

– The firm value V i is given by:

i are the roots of the equation12σ2x +μi x − r = 0.

(ii) (Optimal policy)

r f i = 0 on [0, x i] whereL iis the infinitesimal generator of the drifted Brownian motion

μi t +σW t Remark also that V i is concave on[0, x i ] and linear above x i Finally, it is

also important to note that there is no obvious comparison between x and x (see for

instance Rochet and Villeneuve [18] Proposition 2) Coming back to our problem (3.3),

we deduce from these standard results that the strategies

π0

= (Z t0, 0) =

(x − x0)+1t=0 + L x0

t (μ0,W )1t>0 ,∞, (3.8)and

π1= (Z t1, 0) =

(x − I) − x1)+1t=0 + L x1

t (μ1,W )1t>0 , 0

(3.9)

lead to the inequalities V (x) ≥ V0(x) and V (x) ≥ V1(x − I) Strategy π0 corresponds

to the investment policy “never invest in the growth option (and follow the associated

optimal dividend policy)”, while strategyπ1corresponds to the investment policy “invest

immediately in the growth option (and follow the associated optimal dividend policy)”.

Finally, note that, because the inequality x − I ≤ 0 leads to immediate bankruptcy, the

firm value V1(x − I) is defined by:

3.3 Value of the firm with a growth option

The dynamic programming principle6gives the following representation for the valuefunction

where R = (R t)t≥0denotes the cash reserve process generated by the activity in place in

absence of dividend distribution, that is dR t=μ0dt +σdW t It results from the Markov

property that the process(e −r(t∧τ0π)V (R t∧τπ

0))t≥0 is a supermartingale which dominatesthe function max(V0(.),V1( − I)) Thus, according to optimal stopping theory, V domi-

nates the Snell envelope of the process(max(V0(R t ),V1(R t − I))) t≥0 Let us consider thestopping time problem with value function

Theorem 3.1 For all x ∈ [0,∞), V(x) =φ(x).

The rest of the note is devoted to the proof of Theorem 3.1

6 We refer to Décamps and Villeneuve [6] Proposition 3.1 for a proof.

J.-P Décamps and S Villeneuve / Optimal Investment Under Liquidity Constraints 9

3.3.1 A verification Theorem

Proving Theorem 3.1 amounts to show the reverse inequality V (x) ≤φ(x) This requires

a verification result for the Hamilton-Jacobi-Bellman (HJB) equation associated to lem (3.11) One indeed expects from the dynamic programming principle, the value func-tion to satisfy the HJB equation

prob-max(1− v  , L0v − rv, V1( − I) − v) = 0. (3.13)

The next proposition shows that any piecewise function C2which is a supersolution

to the HJB equation (3.13) is a majorant of the value function V

Proposition 3.2 (verification result for the HJB equation) Suppose we can find a positive

function ˜ V piecewise C2on (0, + ∞) with bounded first derivatives7and such that for all

x > 0,

(i) L0V˜− r ˜V ≤ 0 in the sense of distributions,

(ii) ˜ V (x) ≥ V1(x − I),

(iii) ˜ V  (x) ≥ 1,

with the initial condition ˜ V (0) = 0 then, ˜ V (x) ≥ V(x) for all x ∈ [0,∞).

Proof of Proposition 3.2 We have to prove that for any control policyπ= (Z tπ,τπ;t ≥

0), ˜V (x) ≥ Vπ(x) for all x > 0 Let us write the process Z tπ= Z t π,c + Z t π,d where Z π,c

Since ˜V satisfies (i), the second term of the right hand side is negative On the other hand,

the first derivative of ˜V being bounded, the third term is a square integrable martingale.

Taking expectations, we get

where assumptions(ii) and (iii) have been used for the second inequality

We know already that V ≥φ Thus, to complete the proof of Theorem 3.1, it remainssimply to verify thatφsatisfies the assumption of Proposition 3.2 This will clearly imply

the reverse inequality V (x) ≤φ(x) To achieve this goal we start by solving explicitly

optimal stopping problem (3.12)

3.3.2 Solution to optimal stopping problemφ

First, we have to know when V1( − I) dominates V0 According to Décamps and leneuve [6] Proposition 2.2, we have

Vil-Proposition 3.3 The following holds.

V (x) = V0(x) for all x ≥ 0 if and only if

ensuring that the growth opportunity is worthwhile Note that for all positive x, V (x) ≥

φ(x) ≥θ(x) whereθ is the value function of optimal stopping problem

for all positive x, the option valueθ(x) is larger than V0(x) then, we have the equalities

V (x) =φ(x) =θ(x) A crucial point will be to show that the inequalityθ(x) > V0(x) holds for all positive x, if and only if it is satisfied at the threshold x0that triggers dis-tribution of dividend when the firm is run under the technology in place (see Lemma3.8 hereinafter) In such a situation, the optimal dividend/investment policy will be topostpone dividend distribution, to invest at a certain threshold b in the growth opportu-

nity and to pay out any surplus above x1as dividend Next proposition specifies all thesepoints and derives the solution to optimal stopping problemφ

J.-P Décamps and S Villeneuve / Optimal Investment Under Liquidity Constraints 11

Proposition 3.4 The following holds.

(A) If condition (H1) is satisfied then,

(i) If θ(x0) > V0(x0) then, the value function φ satisfies for all positive x,

A, B, a, c are determined by the continuity and smooth-fit C1conditions at a and c:

φ(a) = V0(a),

φ(c) = V1(c − I),

φ (a) = V0 (a),

φ (c) = V1 (c − I).

(B) If condition (H1) is not satisfied then, for all positive x,φ(x) = V0(x).

Figures 1 and 2 illustrate cases (i) and (ii) of Proposition 3.4 We establish sition 3.4 through a series of lemmas The first one derives quasi explicitly the valuefunctionθ

Propo-Lemma 3.6 The value functionθ is defined by

Figure 1 θ(x0) > V0(x0 ).

Figure 2 θ(x0) < V0(x0 ).

The next Lemma characterizes the stopping region of optimal stopping problemφ

Lemma 3.7 The stopping region S of problemφsatisfies S = S0∪ S1with

S0={0 < x < ˜x|φ(x) = V0(x) }

J.-P Décamps and S Villeneuve / Optimal Investment Under Liquidity Constraints 13

S1={x > ˜x|φ(x) = V1(x − I)}, where ˜x is the unique crossing point of the value functions V0(.) and V1(x − ).

Proof of Lemma 3.7 According to Optimal Stopping Theory (see El Karoui [9],

The-orems 10.1.9 and 10.1.12 in Øksendal [15]), the stopping region S of problemφsatisfies

S = {x > 0|φ(x) = max(V0(x),V1(x − I))}.

Now, from Proposition 5.13 and Corollary 7.1 by Dayanik-Karatzas [5], the hitting time

τS= inf{t : R t ∈ S } is optimal and the optimal value function is C1on[0,∞) Moreover,

it follows from Lemma 4.3 from Villeneuve [20] that ˜x, defined as the unique crossing point of the value functions V0(.) and V1(x −.), does not belong to S Hence, the stopping

region can be decomposed into two subregions S = S0∪ S1with

S0={0 < x < ˜x|φ(x) = V0(x) },

and

We now obtain Assertion (i) of Proposition 3.4 as a byproduct of the next Lemma

Lemma 3.8 The following assertions are equivalent:

(i) θ(x0) > V0(x0).

(ii) θ(x) > V0(x) for all x > 0.

(iii) S0= /0.

Proof of Lemma 3.8 (i) = ⇒ (ii) We start with x ∈ (0,x0) Let us defineτx0= inf{t :

R t < x0} ∈ T The inequalityθ(x0) > V0(x0) together with the initial conditionθ(0) =

where the last inequality follows from (3.17) Thus,θ(x) > V0(x) for all 0 < x ≤ x0

As-sume now that x > x0 We distinguish two cases If b > x0, it follows from (3.5) and (3.15)that,θ(x) > V0(x) for x ≤ x0is equivalent toθ (x0) > 1 Then, the convexity properties

of f0yields toθ (x) > 1, for all x > 0 If, on the contrary, b ≤ x0then,θ(x) = V1(x − I)

for all x ≥ x0 Since V1 (x − I) ≥ 1 for all x ∈ [I,∞), the smooth fit principle implies

θ (x) ≥ 1 for all x ≥ x0 Therefore, the functionθ−V0is increasing for x ≥ x0whichends the proof

(ii) = ⇒ (iii) Simply remark that equations (3.14) and (3.12) giveφ≥θ Therefore,

we have,φ(x) ≥θ(x) > V0(x) for all x > 0 which implies the emptyness of S0

(iii) = ⇒ (i) Suppose S0= /0 and let us show thatθ =φ This will clearly ply θ(x0) =φ(x0) > V0(x0) and thus (i) From Optimal Stopping theory, the process

im-(e −r(t∧τ0∧τS)φ(X t∧τ 0∧τS))t≥0is a martingale Moreover, on the event{τS < t}, we have

φ(RτS ) = V1(RτS − I) a.s It results that

≤θ(x) +Ex e −rtφ(R t)

.

Now, it follows from (3.5), (3.10) that φ(x) ≤ Cx for some positive constant C This

impliesEx [e −rtφ(R t )] converges to 0 as t goes to infinity We therefore deduce thatφ≤θ

Assertion (ii) of Proposition 3.4 relies on the following lemma

Lemma 3.9 Assumeθ(x0)≤ V0(x0) then, there are two positive real numbers a ≥ x0and

c ≤ x1+ I such that

S0=]0, a] and S1= [c, +∞[.

Proof of Lemma 3.9 From the previous Lemma we know that the inequalityθ(x0)≤

V0(x0) implies S0 = /0 We start the proof with the shape of the subregion S0 Take x ∈ S0,

we have to prove that any y ≤ x belongs to S0 As a result, we will define a= sup{x <

˜x |x ∈ S0} Now, according to Proposition 5.13 by Dayanik and Karatzas [5], we have

Now, assuming that a < x0, (i.e.φ(x0) > V0(x0)) yields the contradiction:

where the second equality follows from the martingale property of the process

(e −r(t∧τx0 ∧τ 0 )V0(R t∧τx0 ∧τ 0))t≥0underPaand the last inequality follows from the martingale property of the process(e −r(t∧τ0 )φ(R t∧τ0))t≥0

super-The shape of the subregion S1is a direct consequence of Lemma 4.4 by Villeneuve

[20] The only difficulty is to prove that c ≤ x1+ I Let us consider x ∈ (a,c), and let us

introduce the stopping timesτa= inf{t : R t = a }, andτc= inf{t : R t = c }, we have:

r for x ∈ (a,c) We conclude

re-marking that, assuming the inequality c > x1+ I would yield to the contradiction

The equality V (x) =φ(x) follows then from Proposition 3.3

As a final remark note that, ifθ(x0) = V0(x0) then, we have that a = x0, c = b and the

value functionsφandθcoincide Indeed, using same argument than in the first part of the

proof of Lemma 3.8, we easily deduce fromθ(x0) = V0(x0) thatθ(x) = V0(x) =φ(x) for

x ≤ x0 Furthermore, (3.5) and (3.15) imply that,θ(x0) = V0(x0) is equivalent toθ (x0) =

V  (x0) = 1, which implies that a = x0 The equality c = b follows then from relations

(3.15) and (3.16) To summarize, ifθ(x0) = V0(x0) then,θis the lowest supermartingale

that majorizes e −r( τ∧τ0 )max(V0(R τ∧τ0),V1(R τ∧τ0− I)) from which it results thatθ=φ

3.3.3. φas a super solution to HJB equation (3.13)

We are now ready to prove thatφsatisfies the assumptions of Proposition 3.2 Formally,Proposition 3.5 φis a supersolution to HJB equation (3.13).

Proof of Proposition 3.5 The result clearly holds if, for all positive x,φ(x) = V0(x)

(that is, if condition (H1) is not satisfied ) Assume now that condition (H1) is satisfied.Two cases have to be considered

i) θ(x0) > V0(x0).

In this case,φ=θ according to part (i) of Proposition 3.4 It remains to checkthat the functionθ satisfies the assumptions of Proposition 3.2 But, according

to optimal stopping theory,θ ∈ C2[(0, ∞) \ b)], L0θ− rθ ≤ 0 and clearlyθ≥

V1( − I) Moreover, it is shown in the first part of the proof of Lemma 3.8 that

θ (x) ≥ 1 for all x > 0 Finally, let us check thatθ is bounded above in the

neighbourhood of zero Clearly we have that

θ(x) ≤ sup τ∈TEx



e −r( τ∧τ0 )V1(R τ∧τ0)



,

furthermore, the process(e −r(t∧τ0 )V1(R t∧τ0))t≥0is a supermartingale sinceμ1>

μ0 Thereforeθ≤ V1, the boundedness of the first derivative ofθ follows thenfrom Equation (3.10)

ii) θ(x0)≤ V0(x0).

In this case, the functionφ is characterized by part (ii) of Proposition 3.4 Thus,

φ= V0on(0, a),φ= V ( − I) on (c,+∞) andφ(x) = Aeα0+x + Beα0− xon(a, c).

Hence,φwill be a supersolution if we prove thatφ (x) ≥ 1 for all x > 0 In fact, it

is enough to prove thatφ (x) ≥ 1 for x ∈ (a,c) because V 

0≥ 1 and V 

1( − I) ≥ 1.

The smooth fit principle givesφ (a) = V0 (a) ≥ 1 and φ (c) = V1 (c − I) ≥ 1.

Clearly,φis convex in a right neighbourhood of a Therefore, ifφis convex on

(a, c), the proof is over If not, the second derivative ofφgiven by A(α+

0)2eα +

0x+

B(α−

0)2eα−

0xvanishes at most one time on(a, c), say in d Therefore,

1≤φ (a) ≤φ (x) ≤φ (d) for x ∈ (a,d),

4 Future works

While the real option literature has emerged and developed within the framework ofperfect capital markets, few papers have been interested in the financing of investmentcosts However, when the assumption of perfect capital markets is released, new issuesare emerging that have an interest both in Mathematics and Finance In particular, theinteractions between liquidity management and investment policies lead to the study ofmixed stochastic control problems that are relatively scarce in the applied probabilityliterature In the particular case where the firm have no access to external financing, thereal option problem associated to the optimal investment for a cash-constrained firm istackled by solving a stopping problem with a non linear payoff that exhibits interestingproperties in terms of investment decisions that are not predicted by the standard real op-tion theory In the standard real option literature as well as in the optimal dividend policyliterature, increasing the volatility of the cash process has an unambiguous effect: Greateruncertainty increases both the option value to invest (see McDonald and Siegel [13]),and the threshold that triggers distribution of dividend (see Rochet and Villeneuve [18])

An interesting feature of our model is that an increase of the volatility can kill the growth

option Because the difference x1− x0considered as a function of the volatilityσtends

to μ 1−μ 0

r whenσtends to infinity This implies that for large volatility, condition (H1) isnever satisfied and thus that the growth opportunity is worthless which is in contradictionwith the positive effect of uncertainty on the option value to invest in the standard model

of real option

The study can be extended in two directions From a mathematical viewpoint, itwould be interesting to know if the main result (Theorem 3.1) remains valid if one mod-els the dynamics of cash reserves with a more general class of regular diffusion From

a financial viewpoint, it would be natural and more realistic to release the liquidity straints by assuming that firms have access to outside financing In the state of our knowl-edge, this extension, if we focus on debt financing, leads to the same type of problemsthat the ones described in the discussion of Section 2

[4] Bates, T., Kahle, K.M and Stulz, R.M.: Why do U.S firms hold so much more cash than theyused to? Journal of Finance 64, 1985–2021 (2009)

[5] Dayanik, S and Karatzas, I.: On the optimal stopping problem for one-dimensional sions Stochastic Procresses and Their Application, 107, 173–212 (2003)

diffu-[6] Décamps, J.P and Villeneuve, S.: Optimal dividend policy and growth option Finance andStochastics, 11(1), 3–27 (2007)

[7] Dellacherie, C and Meyer, P.A.: Probabilité et potentiel Théorie des martingales, Hermann,Paris 1980

[8] Dixit, A.K and Pindyck, R.S.: Investment Under Uncertainty Princeton Univ Press 1994

[9] El Karoui, N.: Les aspects probabilistes du contrôle stochastique Lecture Notes in matics, 876, 74–239, Springer, Berlin 1981.

Mathe-[10] Hugonnier, J., Malamud, S and Morellec, E.: Capital supply uncertainty, cash holdings andInvestment, Swiss Finance Institute Research Paper No 11–44 (2011)

[11] Jeanblanc-Picqué, M and Shiryaev, A.N.: Optimization of the flow of dividends RussianMathematics Surveys, 50, 257–277 (1995)

[12] Karatzas, I and Shreve, S.: Brownian Motion and Stochastic Calculus, Springer, New York1988

[13] McDonald, R and Siegel, D.: The value of waiting to invest Quarterly Journal of Economics,

Investment in High-tech Industries: an

Kuno J.M HUISMANa,b, Peter M KORTa,cand Joseph E.J PLASMANSa,c

aCentER, Department of Econometrics and Operations Research,Tilburg University

bASML Netherlands B.V.

cDepartment of Economics, University of Antwerp

Abstract This chapter considers a representative firm taking investment decisions

in a high-tech environment where different generations of production facilities are

invented over time First, we develop a general real options investment model for

high-tech industries in which, according to standard practice, the sales price and the

unit production cost both satisfy a geometric Brownian motion (GBM) process.

Second, we use the developed model to analyze actual investment decisions in the

LCD industry Real life data is used to fit the parameters of the model and to discuss

the actual investments of the two largest companies in the LCD industry: Samsung

Display and LG Display We conclude that their investments in the 8th generation

LCD production facilities are have negative NPVs We present several reasons how

these investments can be justified.

Keywords High-tech Investment, Investment under Uncertainty, Product Innovation,

Real Options, Vector Autoregressive Model

1 Introduction

Due to the very advanced technology involved, investments in high-tech industries ally require significant irreversible investments In a special report on Samsung Electron-ics in The Economist (January 15th, 2005, p 60) it is stated that

usu-"Capital spending is more than $5 billion The company is building the world’s mostadvanced factory for making giant liquid crystal displays (LCDs), and between now and

2010 intends to spend around $24 billion on new chipmaking facilities, despite fallingchip prices."

1 The authors thank Pauline ’t Hart, whose master thesis provides a basis for this research, Ruslan Lukach for his computational assistance, Bertrand Melenberg, and participants of the Recent Topics in Investment under Uncertainty workshop in Dublin (April 2006), the 10th Annual International Conference on Real Options in New York (June 2006), the workshop on Real Options: Theory and Applications in Rimini (April 2007), the Ninth Workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics in Montréal (May 2007), the 3rd RODEO Research Forum on Real Option Applications in Utrecht (December 2007), the Ajou Workshop

in Financial Economics and Mathematics in Suwon (July 2012), and the 17th European Conference on Mathematics for Industry in Lund (July 2012) for their constructive comments.

A Bensoussan et al (Eds.)

IOS Press, 2013

© 2013 The authors and IOS Press All rights reserved.

doi:10.3233/978-1-61499-238-7-20

This chapter analyzes investment decisions of firms in high-tech industries Typicalexamples of high-tech industries are industries for electronic (consumer) products such

as dvd players, LCD television sets, personal computers, MP3 players, photo cameras,mobile phones, and personal digital assistants Prices for personal computers droppedvery fast during the last decades Delaying a purchase decision with one year thus impliesthat the same or even a better personal computer will be available for less money Thesame holds for other products, as confirmed in the article on Samsung Electronics (TheEconomist, January 15th, 2005, p 60):

"While electronic gadgets such as digital cameras, mobile phones and flat-screentelevisions remain as popular as ever, prices are falling."

Another feature of this kind of industries is that high-tech products become lete more quickly, i.e the economic lifetime of these products becomes shorter as timepasses As an example think of the quick increase in the number of megapixels in a digi-tal photo camera Every new generation of this product has more megapixels, which re-duces demand for previous generations From the production side it is known that there

obso-is considerable learning in the production process, implying that production costs aredecreasing over time We conclude that high-tech firms face sharply decreasing prices,rapid product changes, and decreasing production costs

In addition, in a lot of high-tech industries output prices are oscillating over time Inthe LCD industry this is called the crystal cycle (see, e.g [13]) Another example is thesemiconductor industry, about which [1, page 29] state:

" DRAM prices have not fallen in a smooth fashion but rather have oscillated inperiodic cycles around a declining trend."

During periods of high demand, firms invest heavily in expensive new plants Thisdrives prices and profits down, where the investments increase the total quantity pro-duced As demand grows the process repeats itself [13] shows that in the period 1990 to

2003 there have been five of such crystal cycles in the LCD industry The crystal cyclephenomenon is nicely illustrated in an article on the LCD industry in The Economist(July 24th, 2004, p 53):

"But with record spending this year on new and more efficient LCD productionplants, a surplus of capacity could emerge next year "There is no doubt that pricingpressure will intensify as new factories come on line," says Katsuhiko Machida thepresident of Japan’s Sharp But price cuts could help to boost demand further Increaseddemand and more efficient plants could mean that profit margins start to recover in 2006-but that could tempt firms to invest in still more LCD plants "

Real options theory is the appropriate tool to analyze investment decisions underuncertainty (see, e.g., [7], [17]) This theory stresses the irreversibility of most invest-

K.J.M Huisman et al / Investment in High-Tech Industries: An Example from the LCD Industry 21

ment decisions, and the ongoing uncertainty of the economic environment in which thesedecisions are made Real options theory recognizes the option value of waiting for bet-ter (but never complete) information It exploits an analogy with the theory of options

in financial markets, which permits a much richer dynamic framework than is possiblewith other capital budgeting techniques In the most recent capital budgeting literature,real options is accepted as the main tool to analyze a firm’s capital investment decisions(see, e.g [6], [4], [8], [3], [15]) Real options theory extends the net present value (NPV)method in such a way that it takes into account the possibility to delay the investmentdecision Especially if discount rates are high and technology progresses fast, which im-plies that the present is more important and less uncertain than the future (as is the case

in the industry that we study), NPV and real options theory are suitable tools to analyzefirm investment

In most real options models uncertainty is incorporated via a geometric Brownianmotion (GBM ) process (see, e.g., [16] and [5]) Departing from this theory, this chapter

analyzes the investment decisions of high-tech firms After that we confront this retical framework with real life data Econometric analysis justifies the GBM processes.The results are used in an industry analysis that focusses on two real life investmentsfrom the past

theo-This chapter is organized as follows Besides this introduction there are four tions Section 2 employs the standard real options approach, thus uncertainty modeledaccording toGBM , to analyze a high-tech investment decision In Section 3 we con-

sec-front this standard real options approach with real life data taken from quarterly cial reports of LG Displays Section 4 the model with fitted data is used for an industryanalysis, while the last section concludes

finan-2 The Investment Model with Geometric Brownian Motion

Consider a firm that can undertake an irreversible investment by paying a sunk cost

I (> 0), which is a quite common assumption in real options theory (see, e.g [7]) After

the investment the firm can produceQ units of the product per time period The price of

the product at timet equals P (t) Let P (t) follow a GBM :

variancedt Denoting the correlation coefficient between the two Wiener processes by

Proposition 1 Define the markup ratio as

The proof of the proposition can be found in Appendix A

Proposition 2 The value of the option to invest equals

Section 6.5], the optimal investment decision is completely governed byτ This implies

that a threshold valueτ ∗exists so that, whenever the price-cost ratio exceedsτ ∗ , it is

optimal for the firm to invest immediately Otherwise, it is optimal for the firm to waitwith investment

K.J.M Huisman et al / Investment in High-Tech Industries: An Example from the LCD Industry 23

As is standard in real options theory (cf [7]), the threshold valueτ ∗ can be found

by employing the value matching and smooth pasting conditions, which can be obtainedfrom (8) and (9):

This section applies the model of the previous section to the LCD industry In particular,

we investigate investment decisions of Samsung Display and LG Displays, which arethe largest two producers of LCD screens As we argued in the Introduction, in such

an industry the typical long run features are decreasing production costs and even morestrongly decreasing prices

Section 3.1 shortly discusses the industry After that we describe the productionprocess of such a company in Section 3.2 The data is presented and used for estimatingthe parameters in Section 3.3

3.1 Industry

We focus on the industry of TFT-LCD2panel production The companies that are active

in this industry sell their products, i.e LCD panels, to other companies (or other divisions

of the same company) These other companies integrate the LCD panels into productslike for example mobile phones, notebooks, monitors, and television sets

Japanese firms (NEC, Sharp, Toshiba) started the LCD industry in the late 1980s Inthe early 1990s South Korean firms (Samsung and Goldstar Inc., where the latter is the

2 TFT is the abbreviation for Thin Film Transistor TFT-LCD screens are a subset of all LCD screens Other types of LCD screens are DSTN (Dualscan Super Twisted Nematics) and STN (Super Twisted Nematic) screens, for example In the remainder of the chapter we write LCD instead of TFT-LCD when there is no confusion possible.

predecessor of LG Display) entered the market, followed by Taiwanese companies in thelate 1990s (AU Optronics (AUO), Chi Mei Optoelectronics (CMO), Chunghwa PictureTubes (CPT), Quanta Display Inc (QDI), where the latter merged with AUO in the fall

of 2006) LG Displays was formed as a joint venture between Korean LG Electronicsand Dutch Philips Electronics in 1999, the company was named LG.Displays LCD Inlate 2008 Philips sold all its shares and the company changed its name to LG Displays.Samsung divested its LCD activities in 2012 into the company that is called SamsungDisplay These two Korean companies currently account for more than 50% of the totalLCD production

3.2 Production Process

The most important characteristic of an LCD production facility is the size of the motherglass The size of the mother glass, or substrate, determines the so-called generation ofthe production facility For example, the 4th generation has a substrate size of 68 cm by

88 cm and was first operated by LG Display in 2000 In 2005 Sharp announced that itplans to build an 8th generation LCD plant with a substrate size of 220 cm by 240 cm

As the LCD panels are cut out of the substrate, the substrate on the one hand determineswhich panel sizes can be produced and on the other hand how efficient each possiblepanel size can be produced In this sense, every investment in a new generation implies

a process and a product innovation We have a process innovation, because a larger glass

area provides a more efficient solution of the cutting problem, and thus cheaper costs in

the production process Product innovation arises, because the larger area of the substratemakes it possible to produce larger screens

The substrate size that a company selects, heavily depends on the expectations thatthe company has about the prevailing standard sizes in the market For example, Samsungand Sony are using a 7th generation plant with a substrate size of 187 cm by 220 cm,because they expect that 40 inch and 46 inch television screens will become the standardsizes At the same time, LG Display and Chi Mei Optoelectronics are aiming at 42 inchand 47 inch television sets with their 7th generation production facility of 195 cm by 225cm

3.3 Data and Estimations

The dataset is taken from LG Display, which has been a listed company since 2004 For

31 quarters (from 2004Q33 up to and including 2012Q1) we analyzed the quarterly ports of the company In these reports they state the area sold in squared kilometers, therevenues generated by these sales, and the operating profits or losses that resulted Weset the costs of sales equal to revenues minus the operating profit The average price persquared meter LCD sold is then calculated as revenues divided by the area sold and theaverage cost per squared meter LCD is equal to the costs over the area sold In Figure 1the dataset is presented The cost increases that arise are due to the presence of learning

re-in the production process, which can be explare-ined as follows The LCD re-industry ences a so-called ramp up time (time needed to start a production line), with a stronglyincreasing yield (amount of good products relative to the total amount of products) in thefirst quarters after the start of production This makes that costs are at their highest level

experi-3 We denote by 2004Q3 the third quarter of 2004.

K.J.M Huisman et al / Investment in High-Tech Industries: An Example from the LCD Industry 25

in the industry that we study), NPV and real options theory are suitable tools to analyzefirm investment

In most real options. .. demand further Increaseddemand and more efficient plants could mean that profit margins start to recover in 2006-but that could tempt firms to invest in still more LCD plants "

Real options. .. First, we develop a general real options investment model for

high-tech industries in which, according to standard practice, the sales price and the

unit