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E-grāmata: Arbitrage Theory in Continuous Time

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(Professor of Mathematical Finance, Department of Finance, Stockholm School of Economics)
  • Formāts: 592 pages
  • Sērija : Oxford Finance Series
  • Izdošanas datums: 05-Dec-2019
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780192592446
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Arbitrage Theory in Continuous TimePalielināt
  • Formāts: 592 pages
  • Sērija : Oxford Finance Series
  • Izdošanas datums: 05-Dec-2019
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780192592446
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The fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications.

Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments.

In the substantially extended fourth edition Tomas Björk has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. This edition includes an entirely new section presenting dynamic equilibrium theory, covering unit net supply endowments models and the Cox-Ingersoll-Ross equilibrium factor model.



Providing two full treatments of arbitrage theory-the classical delta hedging approach and the modern martingale approach-this book is written so that these approaches can be studied independently of each other, thus providing the less mathematically-oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action.

This textbook is a natural choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in the market.

Recenzijas

Review from previous edition This book is one of the best of a large number of new books on mathematical and probabilistic models in finance, positioned between the books by Hull and Duffie on a mathematical scale...This is a highly reasonable book and strikes a balance between mathematical development and intuitive explanation. * Short Book Reviews *

1 Introduction
1(6)
1.1 Problem Formulation
1(6)
Part I Discrete Time Models
2 The Binomial Model
7(20)
2.1 The One Period Model
7(9)
2.1.1 Model Description
7(1)
2.1.2 Portfolios and Arbitrage
8(3)
2.1.3 Contingent Claims
11(2)
2.1.4 Risk Neutral Valuation
13(3)
2.2 The Multiperiod Model
16(10)
2.2.1 Portfolios and Arbitrage
16(3)
2.2.2 Contingent Claims
19(7)
2.3 Exercises
26(1)
2.4 Notes
26(1)
3 A More General One Period Model
27(16)
3.1 The Model
27(1)
3.2 Absence of Arbitrage
28(5)
3.3 Martingale Measures
33(1)
3.4 Martingale Pricing
34(2)
3.5 Completeness
36(2)
3.6 Stochastic Discount Factors
38(1)
3.7 Exercises
39(4)
Part II Stochastic Calculus
4 Stochastic Integrals
43(24)
4.1 Introduction
43(2)
4.1.1 The Wiener Process
43(2)
4.2 Information
45(2)
4.3 Stochastic Integrals
47(2)
4.4 Martingales
49(2)
4.5 Stochastic Calculus and the Ito Formula
51(5)
4.6 Examples
56(4)
4.7 The Multidimensional Ito Formula
60(2)
4.8 Correlated Wiener Processes
62(2)
4.9 Exercises
64(2)
4.10 Notes
66(1)
5 Stochastic Differential Equations
67(20)
5.1 Stochastic Differential Equations
67(1)
5.2 Geometric Brownian Motion
68(3)
5.3 The Linear SDE
71(1)
5.4 The Infinitesimal Operator
72(1)
5.5 Partial Differential Equations
73(3)
5.6 The Kolmogorov Equations
76(3)
5.7 Exercises
79(4)
5.8 Notes
83(4)
Part III Arbitrage Theory
6 Portfolio Dynamics
87(8)
6.1 Introduction
87(1)
6.2 Self-financing Portfolios in Discrete Time
87(4)
6.2.1 Basic Definitions
87(1)
6.2.2 Self-financing Portfolios
88(2)
6.2.3 The Cumulative Dividend Process
90(1)
6.3 Self-financing Portfolios in Continuous Time
91(2)
6.4 Portfolio Weights
93(2)
7 Arbitrage Pricing
95(24)
7.1 Introduction
95(1)
7.2 More on the Bank Account
96(2)
7.3 Contingent Claims and Arbitrage
98(5)
7.4 The Black-Scholes Equation
103(3)
7.5 Risk Neutral Valuation
106(2)
7.6 The Black-Scholes Formula
108(2)
7.7 Forward and Futures Contracts
110(2)
7.7.1 Forward Contracts
110(1)
7.7.2 Futures Contracts and the Black Formula
111(1)
7.8 Volatility
112(2)
7.8.1 Historic Volatility
113(1)
7.8.2 Implied Volatility
113(1)
7.9 American Options
114(2)
7.10 Exercises
116(2)
7.11 Notes
118(1)
8 Completeness and Hedging
119(9)
8.1 Introduction
119(1)
8.2 Completeness in the Black-Scholes Model
120(5)
8.3 Completeness-Absence of Arbitrage
125(1)
8.4 Exercises
126(1)
8.5 Notes
127(1)
9 A Primer on Incomplete Markets
128(10)
9.1 Introduction
128(1)
9.2 A Scalar Non-priced Underlying Asset
128(7)
9.3 Summing Up
135(2)
9.4 Exercises
137(1)
9.5 Notes
137(1)
10 Parity Relations and Delta Hedging
138(12)
10.1 Parity Relations
138(2)
10.2 The Greeks
140(4)
10.3 Delta and Gamma Hedging
144(3)
10.4 Exercises
147(3)
11 The Martingale Approach to Arbitrage Theory
150(21)
11.1 The Case of Zero Interest Rate
151(2)
11.2 Absence of Arbitrage and Martingale Measures
153(1)
11.3 A Rough Sketch of the Proof
154(5)
11.3.1 Existence of an EMM Implies Absence of Arbitrage
154(1)
11.3.2 Absence of Arbitrage Implies Existence of an EMM
155(4)
11.4 The General Case
159(3)
11.5 Completeness
162(2)
11.6 Pricing Contingent Claims
164(1)
11.7 Pricing by Replication
165(1)
11.8 Stochastic Discount Factors
166(1)
11.9 Summary for the Working Economist
167(3)
11.10 Notes
170(1)
12 The Mathematics of the Martingale Approach
171(14)
12.1 Stochastic Integral Representations
171(3)
12.2 The Girsanov Theorem: Heuristics
174(2)
12.3 The Girsanov Theorem
176(3)
12.4 The Converse of the Girsanov Theorem
179(1)
12.5 Girsanov Transformations and Stochastic Differentials
180(1)
12.6 Maximum Likelihood Estimation
181(2)
12.7 Exercises
183(1)
12.8 Notes
184(1)
13 Black-Scholes from a Martingale Point of View
185(6)
13.1 Absence of Arbitrage
185(2)
13.2 Pricing
187(1)
13.3 Completeness
187(4)
14 Multidimensional Models: Martingale Approach
191(11)
14.1 Absence of Arbitrage
192(1)
14.2 Completeness
193(1)
14.3 Hedging
194(2)
14.4 Pricing
196(1)
14.5 Markovian Models and PDEs
197(1)
14.6 Market Prices of Risk
198(1)
14.7 The Stochastic Discount Factor
199(1)
14.8 The Hansen-Jagannathan Bounds
200(1)
14.9 Exercises
201(1)
14.10 Notes
201(1)
15 Change of Numeraire
202(17)
15.1 Introduction
202(1)
15.2 Generalities
202(2)
15.3 Changing the Numeraire
204(3)
15.4 Some Examples
207(3)
15.5 Forward Measures
210(2)
15.5.1 Using the T-bond as Numeraire
211(1)
15.6 A General Option Pricing Formula
212(3)
15.6.1 General Theory
212(1)
15.6.2 The Case of Deterministic Volatility
213(2)
15.7 The Numeraire Portfolio
215(2)
15.7.1 General Theory
215(1)
15.7.2 The Objective Measure P as a Martingale Measure
216(1)
15.8 Exercises
217(1)
15.9 Notes
217(2)
16 Dividends
219(17)
16.1 Discrete Dividends
219(7)
16.1.1 Dividend Structure
219(1)
16.1.2 The Price Structure
220(1)
16.1.3 A Black-Scholes Model with a Discrete Dividend
221(1)
16.1.4 Option Pricing
222(1)
16.1.5 Risk Neutral Valuation
223(1)
16.1.6 An Example
224(2)
16.2 Continuous Dividends I: Classical Methods
226(3)
16.2.1 Continuous Dividend Yield
226(3)
16.3 Continuous Dividends II: Martingale Methods
229(6)
16.3.1 The Bank Account as Numeraire
230(1)
16.3.2 Continuous Dividend Yield Revisited
231(1)
16.3.3 An Arbitrary Numeraire
232(3)
16.4 Exercises
235(1)
17 Forward and Futures Contracts
236(8)
17.1 Forward Contracts
236(2)
17.2 Futures Contracts
238(3)
17.3 Futures Options and Black-76
241(2)
17.3.1 Generalities
241(1)
17.3.2 The Black-76 Formula
241(2)
17.4 Exercises
243(1)
17.5 Notes
243(1)
18 Currency Derivatives
244(13)
18.1 Pure Currency Contracts
244(3)
18.2 The Martingale Approach
247(3)
18.3 Domestic and Foreign Equity Markets
250(2)
18.4 An Extended Black-Scholes Model
252(1)
18.5 The Siegel Paradox
253(2)
18.6 Exercises
255(1)
18.7 Notes
256(1)
19 Bonds and Interest Rates
257(15)
19.1 Zero Coupon Bonds
257(1)
19.2 Interest Rates
258(7)
19.2.1 Definitions
258(2)
19.2.2 Relations between df (t,T), dp(t,T), and dr(t)
260(3)
19.2.3 An Expectation Hypothesis
263(1)
19.2.4 An Alternative View of the Money Account
264(1)
19.3 Coupon Bonds, Swaps, and Yields
265(5)
19.3.1 Fixed Coupon Bonds
266(1)
19.3.2 Floating Rate Bonds
266(1)
19.3.3 Interest Rate Swaps
267(2)
19.3.4 Yield and Duration
269(1)
19.4 Exercises
270(1)
19.5 Notes
271(1)
20 Short Rate Models
272(8)
20.1 Generalities
272(2)
20.2 The Term Structure Equation
274(2)
20.3 Martingale Analysis
276(2)
20.4 Exercises
278(1)
20.5 Notes
279(1)
21 Martingale Models for the Short Rate
280(16)
21.1 Q-Dynamics
280(1)
21.2 Properties of the Short Rate Models
281(3)
21.2.1 Models with Linear Dynamics
281(1)
21.2.2 Models with Mean Reversion
282(1)
21.2.3 Lognormal Models
283(1)
21.2.4 Square Root Models
283(1)
21.3 Inversion of the Yield Curve
284(1)
21.4 Affine Term Structures
285(3)
21.4.1 Definition and Existence
285(3)
21.5 Analytical Results for Some Standard Models
288(4)
21.5.1 The Vaskek Model
288(1)
21.5.2 The Ho-Lee Model
289(1)
21.5.3 The CIR Model
290(1)
21.5.4 The Hull-White Model
290(2)
21.6 Bond Options in the Hull-White Model
292(1)
21.7 Exercises
293(2)
21.8 Notes
295(1)
22 Forward Rate Models
296(9)
22.1 The Heath-Jarrow-Morton Framework
296(2)
22.2 Martingale Modeling
298(2)
22.3 The General Gaussian Model
300(1)
22.4 The Musiela Parameterization
301(1)
22.5 Exercises
302(2)
22.6 Notes
304(1)
23 LIBOR Market Models
305(12)
23.1 Caps: Definition and Market Practice
306(2)
23.2 The LIBOR Market Model
308(1)
23.3 Pricing Caps in the LIBOR Model
309(1)
23.4 Terminal Measure Dynamics and Existence
310(3)
23.5 Calibration and Simulation
313(1)
23.6 The Discrete Savings Account
314(2)
23.7 Notes
316(1)
24 Potentials and Positive Interest
317(16)
24.1 Generalities
317(1)
24.2 The Flesaker-Hughston Framework
318(3)
24.3 Changing Base Measure
321(1)
24.4 Decomposition of a Potential
322(1)
24.5 The Markov Potential Approach of Rogers
323(5)
24.6 Exercises
328(1)
24.7 Notes
329(4)
Part IV Optimal Control and Investment Theory
25 Stochastic Optimal Control
333(18)
25.1 An Example
333(1)
25.2 The Formal Problem
334(3)
25.3 Embedding the Problem
337(2)
25.4 Time Consistency and the Bellman Principle
339(1)
25.5 Deriving the Hamilton-Jacobi-Bellman Equation
340(5)
25.6 Handling the HJB Equation
345(2)
25.7 The Linear Regulator
347(2)
25.8 Exercises
349(1)
25.9 Notes
350(1)
26 Optimal Consumption and Investment
351(13)
26.1 A Generalization
351(1)
26.2 Optimal Consumption and Investment
352(3)
26.3 The Mutual Fund Theorems
355(6)
26.3.1 The Case with No Risk Free Asset
355(4)
26.3.2 The Case with a Risk Free Asset
359(2)
26.4 Exercises
361(2)
26.5 Notes
363(1)
27 The Martingale Approach to Optimal Investment
364(12)
27.1 Generalities
364(1)
27.2 The Basic Idea
365(1)
27.3 The Optimal Terminal Wealth
366(1)
27.4 The Optimal Wealth Process
367(1)
27.5 The Optimal Portfolio
368(1)
27.6 Log Utility
369(2)
27.6.1 The Optimal Terminal Wealth
369(1)
27.6.2 The Optimal Wealth Process
369(1)
27.6.3 The Optimal Portfolio
370(1)
27.7 Other Utility Functions
371(1)
27.8 Optimal Consumption Problems
371(3)
27.9 Exercises
374(1)
27.10 Notes
375(1)
28 Optimal Stopping Theory and American Options
376(23)
28.1 Introduction
376(1)
28.2 Generalities
376(1)
28.3 Some Simple Results
377(1)
28.4 Discrete Time
378(8)
28.4.1 The General Case
378(4)
28.4.2 Markovian Models
382(2)
28.4.3 Infinite Horizon
384(2)
28.5 Continuous Time
386(6)
28.5.1 General Theory
386(1)
28.5.2 Diffusion Models
387(5)
28.5.3 Connections to the General Theory
392(1)
28.6 American Options
392(3)
28.6.1 The American Call without Dividends
392(1)
28.6.2 The American Put Option
392(2)
28.6.3 The Perpetual American Put
394(1)
28.7 Exercises
395(1)
28.8 Notes
395(4)
Part V Incomplete Markets
29 Incomplete Markets
399(6)
29.1 Introduction
399(1)
29.2 A Markov Factor Model
400(1)
29.3 The Independent Factor Markov Model
401(2)
29.3.1 Absence of Arbitrage
402(1)
29.3.2 Incompleteness
402(1)
29.4 Methods to Handle Market Incompleteness
403(1)
29.5 Notes
404(1)
30 The Esscher Transform and the Minimal Martingale Measure
405(8)
30.1 The Esscher Transform
405(4)
30.1.1 The Standard Esscher Transform
405(1)
30.1.2 The Generalized Esscher Transform
406(2)
30.1.3 The Markov Factor Model
408(1)
30.1.4 The Independent Factor Markov Model
408(1)
30.2 The Minimal Martingale Measure
409(3)
30.2.1 Definition and Existence
410(1)
30.2.2 Basic Properties of QM
411(1)
30.2.3 Economic Interpretation of QM
411(1)
30.3 Notes
412(1)
31 Minimizing f-Divergence
413(13)
31.1 Definition and Basic Properties
413(1)
31.2 Minimal Reverse Entropy
414(1)
31.3 Minimal Entropy in a Factor Model
415(3)
31.4 Duality
418(7)
31.4.1 Utility Maximization of Financial Derivatives
419(2)
31.4.2 Minimax Measures
421(2)
31.4.3 Log Utility
423(1)
31.4.4 Exponential Utility
423(2)
31.5 Notes
425(1)
32 Portfolio Optimization in Incomplete Markets
426(10)
32.1 Setup
426(1)
32.2 The Complete Market Case
427(1)
32.3 Incomplete Market, Finite Ω
428(5)
32.4 Incomplete Market, General Ω
433(2)
32.5 Notes
435(1)
33 Utility Indifference Pricing and Other Topics
436(5)
33.1 Global Indifference Pricing
436(2)
33.2 Marginal Indifference Pricing
438(1)
33.3 Hedging
439(1)
33.4 Notes
440(1)
34 Good Deal Bounds
441(10)
34.1 General Ideas
441(2)
34.2 The Model
443(1)
34.3 The Good Deal Bounds
443(2)
34.4 The Embedded Optimization Problem
445(1)
34.5 Relations to the Minimal Martingale Measure
445(1)
34.6 An Option with Basis Risk
446(2)
34.7 Notes
448(3)
Part VI Dynamic Equilibrium Theory
35 Equilibrium Theory: A Simple Production Model
451(7)
35.1 The Model
451(2)
35.2 Equilibrium
453(2)
35.3 Introducing a Central Planner
455(2)
35.4 Exercises
457(1)
35.5 Notes
457(1)
36 The Cox-Ingersoll-Ross Factor Model
458(9)
36.1 The Model
458(1)
36.1.1 Exogenous Objects
458(1)
36.1.2 Endogenous Objects
458(1)
36.1.3 Economic Agents
459(1)
36.2 The Portfolio Problem
459(2)
36.2.1 Portfolio Dynamics
460(1)
36.2.2 The Control Problem and the HJB Equation
460(1)
36.3 Equilibrium
461(1)
36.4 The Short Rate and the Risk Premium for F
462(1)
36.5 The Equilibrium Stochastic Discount Factor
462(2)
36.6 Risk Neutral Valuation
464(1)
36.7 Introducing a Central Planner
464(1)
36.8 Exercises
465(1)
36.9 Notes
466(1)
37 The Cox-Ingersoll-Ross Interest Rate Model
467(3)
37.1 Exercises
469(1)
37.2 Notes
469(1)
38 Endowment Equilibrium: Unit Net Supply
470(11)
38.1 The Model
470(2)
38.1.1 Exogenous Objects
470(1)
38.1.2 Endogenous Objects
470(1)
38.1.3 Economic Agents
471(1)
38.1.4 Equilibrium Conditions
472(1)
38.2 The Martingale Approach
472(3)
38.2.1 The Control Problem
472(1)
38.2.2 Equilibrium
473(1)
38.2.3 Log Utility
474(1)
38.3 Extending the Model
475(3)
38.3.1 The General Scalar Case
475(1)
38.3.2 A Factor Model
476(2)
38.4 Several Endowment Processes
478(1)
38.5 Exercises
479(1)
38.6 Notes
480(1)
Appendices
A Measure and Integration
481(26)
A.1 Sets and Mappings
481(2)
A.2 Measures and Sigma-Algebras
483(2)
A.3 Integration
485(4)
A.4 Sigma-Algebras and Partitions
489(1)
A.5 Sets of Measure Zero
490(1)
A.6 The LP Spaces
491(1)
A.7 Hilbert Spaces
492(3)
A.8 Sigma-Algebras and Generators
495(4)
A.9 Product Measures
499(1)
A.10 The Lebesgue Integral
499(1)
A.11 The Radon-Nikodym Theorem
500(4)
A.12 Exercises
504(2)
A.13 Notes
506(1)
B Probability Theory
507(19)
B.1 Random Variables and Processes
507(3)
B.2 Partitions and Information
510(1)
B.3 Sigma-Algebras and Information
511(3)
B.4 Independence
514(2)
B.5 Conditional Expectations
516(6)
B.6 Equivalent Probability Measures
522(2)
B.7 Exercises
524(1)
B.8 Notes
525(1)
C Martingales and Stopping Times
526(10)
C.1 Martingales
526(3)
C.2 Discrete Stochastic Integrals
529(1)
C.3 Likelihood Processes
530(1)
C.4 Stopping Times
530(3)
C.5 Exercises
533(3)
D Convex Duality
536(3)
D.1 Conjugate Functions
536(1)
D.2 Lagrange Functions and Saddle Points
536(1)
D.3 An Envelope Theorem
537(2)
References 539(10)
Index 549
Tomas Björk is Professor Emeritus of Mathematical Finance at the Stockholm School of Economics. He has previously worked at the Mathematics Department of the Royal Institute of Technology, also in Stockholm. Tomas Björk has been president of the Bachelier Finance Society, co-editor of Mathematical Finance, and has been on the editorial board for Finance and Stochastics and other journals. He has published numerous journal articles on mathematical finance, and in particular is known for his research on point process driven forward rate models, consistent forward rate curves, general interest rate theory, finite dimensional realisations of infinite dimensional SDEs, good deal bounds, and time inconsistent control theory.
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