連続時間ファイナンスの経済学<br>The Economics of Continuous-Time Finance

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連続時間ファイナンスの経済学
The Economics of Continuous-Time Finance

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  • 製本 Hardcover:ハードカバー版/ページ数 621 p.
  • 言語 ENG,ENG
  • 商品コード 9780262036542
  • DDC分類 332.01519233

Full Description


An introduction to economic applications of the theory of continuous-time finance that strikes a balance between mathematical rigor and economic interpretation of financial market regularities.This book introduces the economic applications of the theory of continuous-time finance, with the goal of enabling the construction of realistic models, particularly those involving incomplete markets. Indeed, most recent applications of continuous-time finance aim to capture the imperfections and dysfunctions of financial markets-characteristics that became especially apparent during the market turmoil that started in 2008. The book begins by using discrete time to illustrate the basic mechanisms and introduce such notions as completeness, redundant pricing, and no arbitrage. It develops the continuous-time analog of those mechanisms and introduces the powerful tools of stochastic calculus. Going beyond other textbooks, the book then focuses on the study of markets in which some form of incompleteness, volatility, heterogeneity, friction, or behavioral subtlety arises. After presenting solutions methods for control problems and related partial differential equations, the text examines portfolio optimization and equilibrium in incomplete markets, interest rate and fixed-income modeling, and stochastic volatility. Finally, it presents models where investors form different beliefs or suffer frictions, form habits, or have recursive utilities, studying the effects not only on optimal portfolio choices but also on equilibrium, or the price of primitive securities. The book strikes a balance between mathematical rigor and the need for economic interpretation of financial market regularities, although with an emphasis on the latter.

Table of Contents

1 Introduction                                   1  (6)
1.1 Motivation 1 (2)
1.2 Outline 3 (1)
1.3 How to Use This Book 4 (1)
1.4 Apologies 5 (1)
1.5 Acknowledgments 5 (2)
I Discrete-Time Economies 7 (112)
2 Pricing of Redundant Securities 9 (44)
2.1 Single-Period Economies 10 (17)
2.1.1 The Most Elementary Problem in 10 (3)
Finance
2.1.2 Uncertainty 13 (1)
2.1.3 Securities Payoffs and Prices, and 14 (2)
Investors' Budget Set
2.1.4 Absence of Arbitrage and 16 (6)
Risk-Neutral Pricing
2.1.5 Complete versus Incomplete Markets 22 (1)
2.1.6 Complete Markets and State-Price 23 (1)
Uniqueness
2.1.7 Benchmark Example 24 (1)
2.1.8 Valuation of Redundant Securities 25 (2)
2.2 Multiperiod Economies 27 (23)
2.2.1 Information Arrival over Time and 28 (5)
Stochastic Processes
2.2.2 Self-Financing Constraint and 33 (2)
Redundant Securities
2.2.3 Arbitrage, No Arbitrage, and 35 (5)
Risk-Neutral Pricing
2.2.4 Valuation of Redundant Securities 40 (1)
2.2.5 Statically versus Dynamically 40 (2)
Complete Markets
2.2.6 Benchmark Example 42 (8)
2.3 Conclusion 50 (3)
3 Investor Optimality and Pricing in the Case 53 (38)
of Homogeneous Investors
3.1 One-Period Economies 54 (15)
3.1.1 Investor Optimality and Security 54 (3)
Pricing under Certainty
3.1.2 Investor Optimality and Security 57 (1)
Pricing under Uncertainty
3.1.3 Arrow-Debreu Securities 58 (5)
3.1.4 Complex or Real-World Securities 63 (3)
3.1.5 Relation with the No-Arbitrage 66 (2)
Approach
3.1.6 The Dual Problem 68 (1)
3.2 A Benchmark Example 69 (3)
3.2.1 Isoelastic Utility 69 (1)
3.2.2 Securities Pricing 70 (1)
3.2.3 From Security Prices to State 71 (1)
Prices, Risk-Neutral Probabilities, and
Stochastic Discount Factors
3.3 Multiperiod Model 72 (13)
3.3.1 Optimization Methods 74 (1)
3.3.2 Recursive Approach 74 (3)
3.3.3 Global Approach 77 (4)
3.3.4 Securities Pricing 81 (4)
3.4 Benchmark Example (continued) 85 (2)
3.5 Conclusion 87 (4)
4 Equilibrium and Pricing of Basic Securities 91 (28)
4.1 One-Period Economies 91 (3)
4.2 Competitive Equilibrium 94 (13)
4.2.1 Equalization of State Prices 99 (2)
4.2.2 Risk Sharing 101(2)
4.2.3 Security Pricing by the 103(3)
Representative Investor and the CAPM
4.2.4 The Benchmark Example (continued) 106(1)
4.3 Incomplete Market 107(3)
4.4 Multiple-Period Economies 110(7)
4.4.1 Radner Equilibrium 110(2)
4.4.2 State Prices and Representative 112(1)
Investor: From Radner to Arrow-Debreu
Equilibria
4.4.3 Securities Pricing 113(2)
4.4.4 Risk Sharing 115(1)
4.4.5 A Side Comment on Time-Additive 116(1)
Utility Functions
4.5 Conclusion 117(2)
II Pricing In Continuous Time 119(108)
5 Brownian Motion and Ito Processes 121(28)
5.1 Martingales and Markov Processes 121(2)
5.2 Continuity for Stochastic Processes and 123(2)
Diffusions
5.3 Brownian Motion 125(8)
5.3.1 Intuitive Construction 125(5)
5.3.2 A Financial Motivation 130(1)
5.3.3 Definition 131(2)
5.4 Ito Processes 133(2)
5.5 Benchmark Example (continued) 135(5)
5.5.1 The Black-Scholes Model 135(3)
5.5.2 Construction from Discrete-Time 138(2)
5.6 Ito's Lemma 140(5)
5.6.1 Interpretation 142(2)
5.6.2 Examples 144(1)
5.7 Dynkin Operator 145(1)
5.8 Conclusion 146(3)
6 Black-Scholes and Redundant Securities 149(26)
6.1 Replicating-Portfolio Argument 151(4)
6.1.1 Building the Black-Scholes PDE 151(3)
6.1.2 Solving the Black-Scholes PDE 154(1)
6.2 Martingale-Pricing Argument 155(2)
6.3 Hedging-Portfolio Argument 157(3)
6.3.1 Comparing the Arguments: Intuition 159(1)
6.4 Extensions: Dividends 160(3)
6.4.1 Dividend Paid on the Underlying 160(2)
6.4.2 Dividend Paid on the Option 162(1)
6.5 Extensions: A Partially Generalized 163(2)
Black-Scholes Model
6.5.1 Replicating-Portfolio Argument 163(2)
6.5.2 Martingale-Pricing Argument 165(1)
6.5.3 Hedging Argument 165(1)
6.6 Implied Probabilities 165(1)
6.7 The Price of Risk of a Derivative 166(2)
6.8 Benchmark Example (continued) 168(4)
6.9 Conclusion 172(3)
7 Portfolios, Stochastic Integrals, and 175(24)
Stochastic Differential Equations
7.1 Pathologies 176(2)
7.1.1 Doubling Strategies 176(1)
7.1.2 Local Martingales 177(1)
7.2 Stochastic Integrals 178(6)
7.3 Admissible Strategies 184(2)
7.4 Ito Processes and Stochastic 186(5)
Differential Equations
7.4.1 Ito Processes 186(1)
7.4.2 Stochastic Differential Equations 186(2)
7.4.3 When Are Ito Processes Markov 188(3)
Processes? When Are They Diffusions?
7.5 Bubbles 191(2)
7.6 Ito Processes and the 193(2)
Martingale-Representation Theorem
7.7 Benchmark Example (continued) 195(1)
7.8 Conclusion 196(3)
8 Pricing Redundant Securities 199(28)
8.1 Market Setup 200(2)
8.2 Changes of Measure 202(8)
8.2.1 Equivalent Measures and the 203(3)
Radon-Nikodym Derivative
8.2.2 Girsanov's Theorem: How to Shift to 206(3)
the Risk-Neutral Measure in Black-Scholes
Economies
8.2.3 Change of Measure, Stochastic 209(1)
Discount Factors, and State Prices
8.3 Fundamental Theorem of Security Pricing 210(2)
8.4 Market Completeness 212(6)
8.4.1 Necessary and Sufficient Condition 212(5)
8.4.2 Martingale Measure, Stochastic 217(1)
Discount Factor, and State-Price
Uniqueness
8.5 Asset-Specific Completeness 218(2)
8.6 Benchmark Example (continued) 220(3)
8.7 Conclusion 223(4)
III Individual Optimality In Continuous Time 227(50)
9 Dynamic Optimization and Portfolio Choice 229(30)
9.1 A Single Risky Security 230(5)
9.1.1 Budget Constraint 231(1)
9.1.2 HD Returns and Dynamic Programming 232(1)
Solution
9.1.3 The Marginality Condition 233(1)
9.1.4 Subcases and Examples 234(1)
9.2 A Single Risky Security with HD Returns 235(7)
and One Riskless Security
9.2.1 Myopic Portfolio Weights 237(1)
9.2.2 Examples Revisited 237(5)
9.3 Multiple, Correlated Risky Securities 242(1)
with IID Returns Plus One Riskless Security
9.4 Non-HD, Multiple Risky Securities, and 243(7)
a Riskless Security
9.4.1 Myopic and Hedging Portfolios 246(2)
9.4.2 Fund Interpretation 248(1)
9.4.3 Optimization and Nonlinear PDE 249(1)
9.5 Exploiting Market Completeness: 250(2)
Building a Bridge to Chapter 10
9.6 Benchmark Example (continued) 252(2)
9.7 Conclusion 254(1)
9.8 Appendix: The Link to Chapter 10 255(4)
10 Global Optimization and Portfolio Choice 259(18)
10.1 Model Setup 260(4)
10.1.1 Lifetime versus Dynamic Budget 262(2)
Constraint
10.2 Solution 264(4)
10.2.1 Optimal Wealth 266(1)
10.2.2 Portfolio Mix 267(1)
10.3 Properties of the Global Approach 268(1)
10.4 Non-negativity Constraints on 269(1)
Consumption and Wealth
10.5 The Growth-Optimal Portfolio 270(1)
10.6 Benchmark Example (continued) 271(1)
10.7 Conclusion 272(5)
IV Equilibrium In Continuous Time 277(42)
11 Equilibrium Restrictions and the CAPM 279(20)
11.1 Intertemporal CAPM and Betas 280(1)
11.2 Co-risk and Linearity 281(1)
11.3 Consumption-Based CAPM 282(3)
11.4 The Cox-Ingersoll-Ross Equilibrium 285(8)
11.4.1 Model Setup 285(3)
11.4.2 The Riskless Security 288(2)
11.4.3 The Risky Securities 290(3)
11.5 Benchmark Example (continued) 293(1)
11.6 Conclusion 294(1)
11.7 Appendix: Aggregation Leading to the 295(4)
CAPM
12 Equilibrium in Complete Markets 299(20)
12.1 Model Setup: Exogenous and Admissible 300(1)
Variables
12.2 Definition and Existence of Equilibrium 301(2)
12.3 Obtaining Equilibrium 303(3)
12.3.1 Direct Calculation 303(1)
12.3.2 The Representative Investor 304(2)
12.4 Asset Pricing in Equilibrium 306(3)
12.4.1 The Risldess Security 307(1)
12.4.2 The Market Prices of Risk: CAPM 308(1)
without Markovianity
12.4.3 The Risky Securities 308(1)
12.5 Diffusive and Markovian Equilibria 309(1)
12.6 The Empirical Relevance of State 310(2)
Variables
12.7 Benchmark Example (continued) 312(3)
12.7.1 Equilibria with Black-Scholes 313(1)
Securities
12.7.2 Equilibria with Heterogeneous 313(2)
Power-Utility Investors
12.8 Conclusion 315(4)
V Applications And Extensions 319(250)
13 Solution Techniques and Applications 321(28)
13.1 Probabilistic Methods 322(5)
13.1.1 Probabilities of Transitions as 322(3)
Solutions of PDEs
13.1.2 Integral Representation of the 325(1)
Solution (Characteristic Functions)
13.1.3 Other Uses of Integral 326(1)
Representations
13.2 Simulation Methods 327(6)
13.2.1 Euler-Matsuyama Scheme 328(1)
13.2.2 Mil'shtein's Scheme 328(2)
13.2.3 Stability 330(1)
13.2.4 The Doss or Nelson-and-Ramaswamy 330(1)
Transformation
13.2.5 The Use of "Variational Calculus" 331(2)
in Simulations
13.3 Analytical Methods 333(3)
13.3.1 Solutions of Linear PDEs 333(1)
13.3.2 The Affine Framework and Solutions 334(2)
of Riccati Equations
13.4 Approximate Analytical Method: 336(3)
Perturbation Method
13.5 Numerical Methods: Approximations of 339(5)
Continuous Systems
13.5.1 Lattice Approximations 339(2)
13.5.2 Finite-Difference Approximations 341(3)
13.6 Conclusion 344(5)
14 Portfolio Choice and Equilibrium 349(24)
Restrictions in Incomplete Markets
14.1 Single-Period Analysis 350(4)
14.1.1 Model Setup 350(1)
14.1.2 The Dual Reformulation and State 351(1)
Prices
14.1.3 Consumption and Portfolios 351(3)
14.2 The Dynamic Setting in Continuous Time 354(9)
14.2.1 Model Setup 354(1)
14.2.2 Dual 355(1)
14.2.3 Implied State Prices 356(3)
14.2.4 Consumption 359(1)
14.2.5 Portfolios 359(4)
14.3 Portfolio Constraints 363(2)
14.4 The Minimum-Squared Deviation Approach 365(3)
14.4.1 The One-Period Case 365(1)
14.4.2 Continuous Time: The "Minimal" 366(2)
Martingale Measure
14.5 Conclusion 368(1)
14.6 Appendix: Derivation of the Dual 369(4)
Problem 14.16
15 Incomplete-Market Equilibrium 373(30)
15.1 Various Concepts of Incomplete-Market 373(7)
Equilibrium, Existence, and Welfare
Properties
15.1.1 One-Good, Static Setting 373(2)
15.1.2 Problems in More General Settings 375(1)
15.1.3 Example: The Role of Idiosyncratic 376(2)
Risk
15.1.4 Incomplete Markets and Welfare: 378(2)
Equilibrium and Constrained Pareto
Optimality in the Static Setting
15.2 Obtaining Continuous-Time 380(5)
Incomplete-Market Equilibria
15.2.1 Direct Calculation 380(5)
15.2.2 Representative Investor 385(1)
15.3 Revisiting the Breeden CAPM: The 385(1)
Effect of Incompleteness on Risk Premia in
General Equilibrium
15.4 Benchmark Example: Restricted 386(6)
Participation
15.4.1 Endowment and Securities Markets 387(1)
15.4.2 Consumption Choices and State 388(4)
Prices
15.5 Bubbles in Equilibrium 392(6)
15.5.1 Benchmark Example (continued) 393(3)
15.5.2 Bubble Interpretation 396(2)
15.6 Conclusion 398(1)
15.7 Appendix: Idiosyncratic Risk Revisited 398(5)
16 Interest Rates and Bond Modeling 403(34)
16.1 Definitions: Short Rate, Yields, and 404(3)
Forward Rates
16.2 Examples of Markov Models 407(5)
16.2.1 Vasicek 407(3)
16.2.2 Modified Vasicek Models 410(1)
16.2.3 Cox, Ingersoll, and Ross 411(1)
16.3 Affine Models 412(2)
16.4 Various Ways of Specifying the 414(7)
Behavior of the Bond Market
16.4.1 Specifying the Behavior of Bond 415(1)
Prices
16.4.2 Specifying the Behavior of Forward 415(2)
Rates
16.4.3 Specifying the Behavior of the 417(1)
Short Rate
16.4.4 Condition for the Short Rate to Be 418(3)
Markovian
16.5 Effective versus Risk-Neutral Measures 421(2)
16.6 Application: Pricing of Redundant 423(2)
Assets
16.7 A Convenient Change of Numeraire 425(4)
16.7.1 Change of Discounting Asset as a 426(1)
Change of Measure
16.7.2 Using Bond Prices for Discounting: 427(2)
The Forward Measure
16.8 General Equilibrium Considerations 429(1)
16.9 Interpretation of Factors 430(1)
16.10 Conclusion 431(1)
16.11 Appendix: Proof of Proposition 16.3 432(5)
17 Stochastic Volatility 437(24)
17.1 Motivation 437(5)
17.1.1 Empirics 437(3)
17.1.2 Time-Varying Volatility with 440(2)
Market Completeness
17.2 Examples of Markov Models with 442(5)
Stochastic Volatility
17.2.1 Hull and White 442(2)
17.2.2 Heston 444(3)
17.3 Stochastic Volatility and Forward 447(6)
Variance
17.3.1 Static Arbitrage Restrictions 448(1)
17.3.2 Definition of Forward Variance 448(2)
17.3.3 Interpretation of Forward Variance 450(2)
17.3.4 Summary of the Option Valuation 452(1)
Procedure
17.4 VIX 453(1)
17.5 Stochastic Volatility in Equilibrium 454(1)
17.6 Conclusion 454(1)
17.7 Appendix: GARCH 455(6)
17.7.1 Parameter Specification and 456(1)
Estimate
17.7.2 GARCH versus Continuous-Time 457(4)
Stochastic Volatility
18 Heterogeneous Expectations 461(26)
18.1 Difference of Opinion 462(7)
18.1.1 Endowment and Securities Markets 463(2)
18.1.2 The Several Risk Premia 465(2)
18.1.3 Investor Optimization 467(1)
18.1.4 Comparative Analysis of a Change 468(1)
in Disagreement
18.2 The Value of Information 469(6)
18.2.1 Bayesian Updating and Disagreement 469(2)
between Investors
18.2.2 Information and Portfolio Choice 471(4)
18.3 Equilibrium 475(6)
18.3.1 Equilibrium Consumption and Price 476(2)
Parameters
18.3.2 Consensus Beliefs 478(3)
18.4 Sentiment Risk 481(3)
18.5 Conclusion 484(3)
19 Stopping, Regulation, Portfolio Selection, 487(34)
and Pricing under Trading Costs
19.1 Cost Functions and Mathematical Tools 490(2)
19.2 An Irreversible Decision: To Exercise 492(5)
or Not to Exercise
19.2.1 The American Put 494(3)
19.3 Reversible Decisions: How to Regulate 497(8)
19.3.1 Base Case: Calculating the Value 498(1)
Function
19.3.2 Boundary Conditions: Value Matching 499(1)
19.3.3 Optimizing the Regulator via 500(1)
Smooth-Pasting Boundary Conditions: The
Case of Impulse Control
19.3.4 Optimizing the Regulator via 501(2)
Smooth-Pasting Boundary Conditions: The
Case of Instantaneous Control
19.3.5 Why Not Classical Control? 503(2)
19.4 The Portfolio Problem under 505(7)
Proportional Trading Costs
19.4.1 A Semi-explicit Policy Case: Power 507(4)
Utility
19.4.2 Comment on Quadratic Costs 511(1)
19.5 The Portfolio Problem under Fixed or 512(2)
Quasi-fixed Trading Costs
19.6 Option Pricing under Trading Costs 514(3)
19.6.1 The Inadequacy of the Replication 514(1)
and Super-Replication Approaches
19.6.2 Option Pricing within a Portfolio 515(2)
Context
19.7 Equilibria and Other Open Problems 517(1)
19.8 Conclusion 518(3)
20 Portfolio Selection and Equilibrium with 521(18)
Habit Formation
20.1 Motivation: The Equity-Premium and 522(3)
Other Puzzles
20.2 Habit Formation 525(9)
20.2.1 Internal Habit 526(2)
20.2.2 External Habit 528(6)
20.3 Risk Aversion versus Elasticity of 534(2)
Intertemporal Substitution (EIS)
20.4 Conclusion 536(3)
21 Portfolio Selection and Equilibrium with 539(34)
Recursive Utility
21.1 Modeling Strategy 539(3)
21.1.1 The Restriction: Time Consistency 539(1)
21.1.2 The Motivation 540(2)
21.2 Recursive Utility: Definition in 542(1)
Discrete-Time
21.3 Recursive Utility in Discrete Time: 543(3)
Two Representations
21.3.1 Aggregator Representation 543(2)
21.3.2 Discount-Factor Representation 545(1)
21.4 Recursive Utility: Continuous Time 546(4)
21.4.1 Stochastic Differential Utility 546(1)
21.4.2 Variational Utility 547(3)
21.5 Individual Investor Optimality 550(4)
21.5.1 Choice of the Consumption Path in 550(2)
Complete Markets
21.5.2 Benchmark Example 552(2)
21.6 Equilibrium with Recursive Utility in 554(6)
Complete Markets
21.6.1 Direct Calculation of Equilibrium 554(2)
21.6.2 Calculating a Pareto Optimum 556(1)
Conveniently
21.6.3 The Markovian Case 557(1)
21.6.4 The Market Prices of Risk 558(2)
21.7 Back to the Puzzles: Pricing under 560(1)
Recursive Utility
21.8 Conclusion 561(1)
21.9 Appendix 1: Proof of the 561(2)
Giovannini-Weil Stochastic Discount Factor,
Equation (21.5)
21.10 Appendix 2: Preference for the Timing 563(6)
of Uncertainty Resolution
An Afterword 569(2)
Basic Notation 571(2)
Appendixes 573(22)
A A Review of Utility Theory and 573(10)
Mean-Variance Theory
A.1 Expected Utility 573(5)
A.1.1 Additivity and Time Impatience 575(1)
A.1.2 Risk Aversion and Prudence 575(2)
A.1.3 The HARA Class 577(1)
A.2 Mean-Variance Utility Theory 578(5)
A.2.1 The Mean-Variance Frontier 579(2)
A.2.2 The Mean-Variance CAPM 581(2)
B Global and Recursive Optimization 583(12)
B.1 Global Optimization 583(2)
B.2 Discrete-Time Recursive Optimality 585(3)
B.2.1 Statement of the Optimization 585(1)
Problem
B.2.2 Bellman's Principle in Discrete Time 586(2)
B.3 Continuous-Time Recursive Optimality: 588(4)
Classical Control
B.3.1 Statement of the Optimization 588(1)
Problem
B.3.2 Bellman's Principle and Its 589(1)
Verification Theorem
B.3.3 Perturbation Reasoning or 590(2)
Variational Calculus
B.4 Continuous-Time Optimality under 592(3)
Frictions: Singular Control
B.4.1 Bellman's Principle 593(2)
References 595(14)
Author Index 609(4)
Index 613