yhongl12发书必是精品！

Real Options Valuation
The Importance of Interest Rate Modelling
in Theory and Practice

Author
Dr. Marcus Schulmerich, CFA, FRM
Vice President
Allianz Global Investors Group
Nymphenburger Strasse 112-116
80636 Munich/Germany
marcus. schulmerich? alum.mit.edu

Introduction 1
1.1 Motivation for the Thesis 1
1.2 Contents and Structure of the Thesis 12
1.3 Main Results of the Thesis 15
Real Options in Theory and Practice 19
2.1 Introduction 19
2.2 Basics of Real Options 20
2.3 Diffusion Processes in Classical Probability Theory 27
2.3.1 Definition: Diffusion Process - 1st Version 27
2.3.2 Definitions: Infinitesimal Expectation and Variance .... 28
2.3.3 Denominations for Diffusion Processes 29
2.3.4 Stationary Distribution of a Diffusion Process 31
2.3.5 Example: One-Dimensional Brownian Motion 32
2.4 Introduction to the Ito Calculus 33
2.4.1 Definition: Ito Integral 33
2.4.2 Properties of an Ito Integral 34
2.4.3 Definition: A Broader Class of Ito Integrals 35
2.4.4 Definition: One-Dimensional Ito Process 36
2.4.5 Theorem: Existence and Uniqueness of the Solution of
a Stochastic Differential Equation 37
2.4.6 Coefficients of an Ito Process 38
2.4.7 Definition: Diffusion Process - 2nd Version 38
2.4.8 Remark: Relationship of the two Definitions of a
Diffusion Process 38
2.4.9 Theorem: Solution of a One-Dimensional, Linear
Stochastic Differential Equation 39
2.4.10 Definition: Mean Reversion Process 40
2.4.11 Example: Ornstein-Uhlenbeck Process 40
2.5 Discretization of Continuous-Time Stochastic Processes 41
2.5.1 Mathematical Methods to Generate Random Variables . 41
2.5.2 Euler Scheme 42
2.5.3 Milstein Scheme 43

2.5.4 Taylor 1.5 Scheme 43
2.5.5 Strong Convergence of Order K 44
2.6 Evolution of the Real Options Theory and Models in the
Literature 45
2.6.1 Decision-Tree Analysis 46
2.6.2 Contingent-Claims Analysis 50
2.6.3 Categorization of Real Options Valuation Methods .... 57
Analytical Methods 57
Numerical Methods 59
2.6.4 Flexibility due to Interest Rate Uncertainty 64
2.7 Summary 67
Stochastic Models for the Term Structure of Interest Rates 69
3.1 Introduction 69
3.2 The Term Structure of Interest Rates 70
3.2.1 Introduction to the Term Structure of Interest Rates
and Short Rate Models 70
3.2.2 Yield Curve Interpolation with Cubic Splines 75
Definition: Cubic Spline Function 76
Theorem: Basis of the Set SA 76
Algorithm: Cubic Spline Construction 80
3.2.3 General Overview of Stochastic Interest Rate Models .. 83
3.3 Stochastic Interest Rate Models 94
3.3.1 The Mean Reversion Feature in Short Rate Models .... 94
3.3.2 One-Factor Term Structure Models 95
Vasicek Model 95
Cox-Ingersoll-Ross Model 99
Ho-Lee Model 103
Hull-White One-Factor Model 106
Heath-Jarrow-Morton One-Factor Model 109
3.3.3 Two-Factor Term Structure Models 115
Hull-White Two-Factor Model 116
Heath-Jarrow-Morton Two-Factor Model 122
3.4 Summary 127
Real Options Valuation Tools in Corporate Finance 131
4.1 Introduction 131
4.2 Numerical Methods for Real Options Pricing, Constant
Interest Rates 134
4.2.1 Lattice Methods 134
Cox-Ross-Rubinstein Binomial Tree 134
Trigeorgis Log-Transformed Binomial Tree 141
4.2.2 Finite Difference Methods 152
4.3 Schwartz-Moon Model 163
4.4 Real Options Pricing with Stochastic Interest Rates 169

Contents xv
4.4.1 IngersoU-Ross Model 173
4.4.2 A Modification of the Cox-Ross-Rubinstein Binomial
Tree 177
4.4.3 A Modification of the Trigeorgis Log-Transformed
Binomial Tree 184
4.5 Summary 189
Analysis of Various Real Options in Simulations and
Backtesting 191
5.1 Introduction 191
5.2 Calibration of Stochastic Interest Rate Models 193
5.2.1 Calibration Procedure for the Vasicek and the
Cox-Ingersoll-Ross Models 194
5.2.2 Calibration Procedure for the Ho-Lee Model 194
5.3 Description of the Test Strategy 196
5.4 Test Situation 1: The Schwartz-Moon Model with a Deferred
Project Start 209
5.5 Test Situation 2: Prehminary Tests for Real Options Valuation 216
5.5.1 Analysis for Test Situation 2 in 11 Consecutive Steps .. 217
5.5.2 Recapitulation of the Main Results in Test Situation 2 . 241
5.6 Test Situation 3: Real Options Valuation Using Equilibrium
Models 242
5.6.1 The Influence of the Salvage Factor and the Expand
Factor 243
5.6.2 Analysis for the Vasicek Model 258
5.6.3 Analysis for the Cox-IngersoU-Ross Model 273
5.6.4 Recapitulation of the Main Results in Test Situation 3 . 283
5.7 Test Situation 4: Real Options Valuation Using No-Arbitrage
Models 284
5.7.1 Ho-Lee Model 284
5.7.2 Hull-White One-Factor Model versus Hull-White
Two-Factor Model 292
5.7.3 Ho-Lee Model versus Hull-White One-Factor Model 305
5.7.4 Recapitulation of the Main Results in Test Situation 4 . 307
5.8 Test Situation 5: Real Options Valuation in Historical
Backtesting 308
5.8.1 Analysis for Case 1 314
5.8.2 Analysis for Case 2 318
5.8.3 Analysis for Case 3 322
5.8.4 Recapitulation of the Main Results in Test Situation 5 . 326
5.9 Summary 328
Summary and Outlook 331
6.1 Purpose of the Research 331
6.2 Summary of the Research Results 333

xvi Contents
6.3 Economic Implications 339
6.4 Outlook and Future Areas of Research 340
List of Abbreviations and Symbols 343
References 345
Index 355

Foreword
Managerial decision-making during the lifetime of a project can have im-
portant implications on project handling and its contribution to shareholder
value. Traditional capital budgeting methods (in particular methods based
on net present value) fail to capture the role of managerial degrees of free-
dom and therefore tend to lead to a systematic undervaluation of the project.
In contrast, the real options approach to investment analysis characterizes
decision-making flexibility in terms of (real) option rights which can be eval-
uated analogously to financial options using contingent-claims pricing tech-
niques widely used in capital markets.
The research carried out by Marcus Schulmerich analyzes real options for non-
constant and stochastic interest rates versus constant interest rates. Analyzing
stochastic interest rates in the context of real options valuation is of particular
relevance given their long time to maturity which makes them more vulnera-
ble to interest rate risk than short-term financial options. To date, there has
not been a comprehensive review of this issue in the academic literature. The
fact that interest rates have fiuctuated widely over the recent years further
highlights the need for studying this issue.
This study incorporates variable and stochastic interest rates into numerical
approaches to real options valuation and analyzes the implications for the
efficiency of these numerical methods. The author starts out by providing
a critical review of the approaches taken in the literature to value complex
real option rights and adopts a pragmatic approach in expanding them. He is
specifically interested in assessing to what extent the assumption of a constant
discount rate frequently observed in corporate practice leads to wrong invest-
ment decisions. Although capital market experts would at no point assume
interest rates to be constant, this issue has only been marginally addressed
in the real options literature. However, as the author points out, unexpected
shifts in the interest rate curve will often exert a lasting influence on the
value-oriented control of projects.

viii Foreword
The main part of the study presents extensive numerical simulations and
the historical backtests for various complex real option rights by combining
standard numerical modelling techniques for real options and interest rate
risk. Following introductory assessments on the efficiency of standard nu-
merical valuation methods and the possibility of extending these methods
to non-constant interest rates, the author looks at various equilibrium and
no-arbitrage models for interest-rate modelling in real options valuation. A
concluding section examines the additional benefit achieved by including non-
constant interest rates in real options valuation through historical backtest-
ing. Alongside stochastic interest rates, models with constant interest rates
or implicit forward interest rates are analyzed. The simulation results pro-
vide important numerical findings for the first time indicating the extent to
which the common assumption of constant interest rates in valuation practice
can actually be justified. It turned out that it is important to adjust for the
shape of the term structure in real options valuation, even if interest rates
are modelled stochastically. In methodological terms, stochastic interest rate
models should be preferred, although their additional benefit over using im-
plicit forward rates is not verifiable. In fact, the use of imphcit forward rates is
overall preferable, especially given the ease of applying these models in corpo-
rate practice. Along the same lines, the assumption of constant interest rates
should be rejected on principle.
This research study by Marcus Schulmerich generates new knowledge for cap-
ital market research and corporate valuation practice and develops guidelines
for their practical implementation. The book closes an important gap in the
literature and represents a valuable contribution to answering the question
how real options insights can effectively be employed to improve the quality
of valuation exercises and thereby real investment decisions. We hope that
the study will be widely disseminated and that it will receive the attention it
deserves by academics as well as practitioners.
May 2005

Cambridge, MA, USA Prof. Stewart C. Myers, Ph.D.
Oestrich-Winkel, Germany Prof. Ulrich Hommel, Ph.D.