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Mathematics of the Bond Market: A Lévy Processes Approach [Kõva köide]

(Polish Academy of Sciences), (Uniwersytet Warszawski, Poland)
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Mathematics of the Bond Market: A Lévy Processes ApproachSuurem pilt
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Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by Lévy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book's key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

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Analyses bond market models with Lévy stochastic factors, suitable for graduates and researchers in probability and mathematical finance.
Preface xiii
The Field xiii
Levy Modelling xiv
Aims of the Book xv
Structure of the Book xv
Acknowledgements xvii
Introduction 1(6)
1.1 Bonds
1(1)
1.2 Models
2(3)
1.3 Content of the Book
5(2)
PART I BOND MARKET IN DISCRETE TIME
7(98)
1 Elements of the Bond Market
9(14)
1.1 Prices and Rates
9(3)
1.2 Models of the Bond Market
12(1)
1.3 Portfolios and Strategies
13(3)
1.4 Contingent Claims
16(2)
1.5 Arbitrage
18(5)
2 Arbitrage-Free Bond Markets
23(42)
2.1 Martingale Modelling
23(1)
2.2 Martingale Measures for HJM Models
24(7)
2.2.1 Existence of Martingale Measures
24(3)
2.2.2 Uniqueness of the Martingale Measure
27(4)
2.3 Martingale Measures and Martingale Representation Property
31(13)
2.3.1 Martingale Representation Property
32(3)
2.3.2 Generalized Martingale Representation Property
35(2)
2.3.3 Girsanov's Theorems
37(4)
2.3.4 Application to HJM Models
41(3)
2.4 Markovian Models under the Martingale Measure
44(21)
2.4.1 Models with Markovian Trace
45(3)
2.4.2 Affine Models
48(4)
2.4.3 Dynamics of the Short Rate in Affine Models
52(6)
2.4.4 Shape of Forward Curves in Affine Models
58(3)
2.4.5 Factor Models
61(4)
3 Completeness
65(40)
3.1 Concepts of Completeness
65(3)
3.2 Necessary Conditions for Completeness
68(2)
3.3 Sufficient Conditions for Completeness
70(4)
3.4 Approximate Completeness
74(8)
3.4.1 General Characterization
75(2)
3.4.2 Bond Curves in a Finite Dimensional Space
77(1)
3.4.3 Bond Curves in Hilbert Spaces
78(4)
3.5 Models with Martingale Prices
82(13)
3.5.1 HJM Models
83(5)
3.5.2 Multiplicative Factor Model
88(4)
3.5.3 Affine Models
92(3)
3.6 Replication with Finite Portfolios
95(5)
3.7 Completeness and Martingale Measures
100(5)
PART II FUNDAMENTALS OF STOCHASTIC ANALYSIS
105(46)
4 Stochastic Preliminaries
107(19)
4.1 Generalities
107(2)
4.2 Doob-Meyer Decomposition
109(5)
4.2.1 Predictable Quadratic Variation of Square Integrable Martingales
111(1)
4.2.2 Compensators of Finite Variation Processes
112(2)
4.3 Semimartingales
114(3)
4.4 Stochastic Integration
117(9)
4.4.1 Bounded Variation Integrators
117(1)
4.4.2 Square Integrable Martingales as Integrators
118(3)
4.4.3 Integration over Random Measures
121(2)
4.4.4 Ito's Formula
123(3)
5 Levy Processes
126(16)
5.1 Basics on Levy Processes
126(2)
5.2 Levy-Ito Decomposition
128(3)
5.3 Special Classes
131(5)
5.3.1 Finite Variation Processes
131(2)
5.3.2 Subordinators
133(1)
5.3.3 Levy Martingales
134(2)
5.4 Stochastic Integration
136(6)
5.4.1 Square Integrable Integrators
137(1)
5.4.2 Integration over Compensated Jump Measures
138(2)
5.4.3 Stochastic Fubini's Theorem
140(1)
5.4.4 Ito's Formula for Levy Processes
141(1)
6 Martingale Representation and Girsanov's Theorems
142(9)
6.1 Martingale Representation Theorem
142(1)
6.2 Girsanov's Theorem and Equivalent Measures
143(8)
PART III BOND MARKET IN CONTINUOUS TIME
151(142)
7 Fundamentals
153(31)
7.1 Prices and Rates
153(8)
7.1.1 Bank Account and Discounted Bond Prices
155(2)
7.1.2 Prices and Rates in Function Spaces
157(4)
7.2 Portfolios and Strategies
161(10)
7.2.1 Portfolios
161(1)
7.2.2 Strategies and the Wealth Process
162(4)
7.2.3 Wealth Process as Stochastic Integral
166(5)
7.3 Non-arbitrage, Claims and Their Prices
171(2)
7.4 HJM Modelling
173(9)
7.4.1 Bond Prices Formula
177(3)
7.4.2 Forward Curves in Function Spaces
180(2)
7.5 Factor Models and the Musiela Parametrization
182(2)
8 Arbitrage-Free HJM Markets
184(23)
8.1 Heath--Jarrow--Morton Conditions
184(7)
8.1.1 Proof of Theorem 8.1.1
188(3)
8.2 Martingale Measures
191(16)
8.2.1 Specification of Drift
193(1)
8.2.2 Models with No Martingale Measures
194(3)
8.2.3 Invariance of Levy Noise
197(3)
8.2.4 Volatility-Based Models
200(3)
8.2.5 Uniqueness of the Martingale Measure
203(4)
9 Arbitrage-Free Forward Curves Models
207(13)
9.1 Term Structure Equation
207(13)
9.1.1 Markov Chain and CIR as Factor Processes
210(2)
9.1.2 Multiplicative Factor Process
212(2)
9.1.3 Affine Term Structure Model
214(2)
9.1.4 Ornstein--Uhlenbeck Factors
216(4)
10 Arbitrage-Free Affine Term Structure
220(32)
10.1 Preliminary Model Requirements
220(1)
10.2 Jump Diffusion Short Rate
221(17)
10.2.1 Analytical HJM Condition
222(4)
10.2.2 Generalized CIR Equations
226(8)
10.2.3 Exploding Short Rates
234(2)
10.2.4 Multidimensional Noise
236(2)
10.3 General Markovian Short Rate
238(14)
10.3.1 Filipovic's Theorems
238(2)
10.3.2 Comments on Filipovic's Theorems
240(4)
10.3.3 Examples
244(1)
10.3.4 Back to Short-Rate Equations
245(7)
11 Completeness
252(41)
11.1 Problem of Completeness
252(1)
11.2 Representation of Discounted Bond Prices
253(4)
11.3 Admissible Strategies
257(3)
11.4 Hedging Equation
260(1)
11.5 Completeness for the HJM Model
261(14)
11.5.1 Levy Measure with Finite Support
261(3)
11.5.2 Proofs of Theorems 11.5.1--11.5.3
264(5)
11.5.3 Incomplete Markets
269(6)
11.6 Completeness for Affine Models
275(2)
11.7 Completeness for Factor Models
277(3)
11.8 Approximate Completeness
280(13)
11.8.1 HJM Model
283(5)
11.8.2 Factor Model
288(1)
11.8.3 Affine Model
289(4)
PART IV STOCHASTIC EQUATIONS IN THE BOND MARKET
293(49)
12 Stochastic Equations for Forward Rates
295(5)
12.1 Heath-Jarrow-Morton Equation
295(1)
12.2 Morton's Equation
296(1)
12.3 The Equations in the Musiela Parametrization
297(3)
13 Analysis of the HJMM Equation
300(12)
13.1 Existence of Solutions to the HJMM Equation
300(12)
13.1.1 Local Solutions
302(5)
13.1.2 Global Solutions
307(2)
13.1.3 Applications to the Morton--Musiela Equation
309(3)
14 Analysis of Morton's Equation
312(20)
14.1 Results
312(3)
14.1.1 Comments on Assumptions (A1)--(A3)
314(1)
14.2 Applications of the Main Theorems
315(7)
14.3 Proof of Theorem 14.1.1
322(8)
14.3.1 Outline of the Proof
322(1)
14.3.2 Equivalence of Equations (14.1.1) and (14.1.9)
323(1)
14.3.3 Auxiliary Results
324(5)
14.3.4 Conclusion of the Proof
329(1)
14.4 Proof of Theorem 14.1.2
330(2)
15 Analysis of the Morton--Musiela Equation
332(10)
15.1 Formulation and Comments on the Results
332(2)
15.1.1 Comments on the Results
333(1)
15.2 Proofs of Theorems 15.1.1 and 15.1.2
334(8)
15.2.1 Equivalence Results
334(1)
15.2.2 Proof of Theorem 15.1.1
335(2)
15.2.3 Proof of Theorem 15.1.2
337(5)
Appendix A
342(18)
A.1 Martingale Representation for Jump Levy Processes
342(18)
A.1.1 Multiple Chaos Processes
343(4)
A.1.2 Representation of Chaoses
347(3)
A.1.3 Chaos Expansion Theorem
350(2)
A.1.4 Representation of Square Integrable Martingales
352(2)
A.1.5 Representations of Local Martingales
354(6)
Appendix B
360(7)
B.1 Semigroups and Generators
360(7)
B.1.1 Generators for Equations with Levy Noise
361(6)
Appendix C
367(6)
C.1 General Evolution Equations
367(6)
References 373(6)
Index 379
Micha Barski is Professor of Mathematics at the University of Warsaw. His interests include mathematical finance, especially bond market and risk measures. In the years 20112016 he held the position of Junior-Professor in Stochastic Processes and their Applications in Finance at the University of Leipzig. Jerzy Zabczyk is Professor Emeritus in the Institute of Mathematics at the Polish Academy of Sciences. His research interests include stochastic processes, evolution equations, control theory and mathematical finance. He published over ninety research papers. He is the author or co-author of seven books including Stochastic Equations in Infinite Dimensions (Cambridge, 1992, 2008, 2014), Stochastic Partial Differential Equations with Lévy Noise (Cambridge, 2007) and Mathematical Control Theory: An Introduction (1992, 1996, 2020).