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E-raamat: Financial Modelling with Jump Processes

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Financial Modelling with Jump ProcessesSuurem pilt
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WINNER of a Riskbook.com Best of 2004 Book Award!

During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Lévy processes are beyond their reach.

Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by their use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations.

Topics covered in this book include: jump-diffusion models, Lévy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms.

This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable new set of tools for modelling market fluctuations.

Arvustused

"Pardon the pun, but I jumped at the opportunity to endorse this book. This book is the first complete treatment of markets rendered incomplete by the reality of jumps in prices and volatilities. If I were you, I would pounce." -Dr. Peter Carr, Head of Quantitative Research, Bloomberg LP and Director of Masters Program in Mathematical Finance, NYU

"This book is an extremely rich source of informationthe content speaks for itself" -ISI Short Book Reviews

"This book is an extremely rich source of information for recent developments in the use of jump processes in financial modelling, in particular the use of Levy processes. The authors work at a comfortable mathematical pace choosing carefully which proofs to include and exclude and never losing sight of financial interpretation and application.

"The authors conclude the main body of their text by saying: 'We hope that the present volume will encourage more researchers and practitioners to contribute to this topic and improve on our understanding of theoretical, numerical and practical issues related to financial modelling with jump processes'. I am quite convinced that this goal will be achieved." -Dr. Andreas E. Kyprianou, International Statistics Institute book reviews

"What makes this book attractive is its comprehensiveness. this is an excellent book. Read it. You will learn much." -Glyn A. Holton, Contingency Analysis

"One of the first texts which is entirely devoted to option pricing with non-continuous jump-type stochastic processes an easygoing presentation where the basic problems of jump models are not additionally obscured by technicalities." -Journal of the Royal Statistics

"I love this book. It will be required reading for students entering Levy finance. My judgment is that it will be useful both within academia, particularly to people in stochastics, econometrics, and other fields wanting to develop an interest in finance, and to practitioners." -N.H. Bingham, Journal of the American Statistical Association

Financial modelling beyond Brownian motion
1(17)
Models in the light of empirical facts
5(2)
Evidence from option markets
7(4)
Implied volatility smiles and skews
8(2)
Short-term options
10(1)
Hedging and risk management
11(2)
Objectives
13(4)
I Mathematical tools
17(152)
Basic tools
19(48)
Measure theory
20(6)
σ-algebras and measures
20(4)
Measures meet functions: integration
24(1)
Absolute continuity and densities
25(1)
Random variables
26(6)
Random variables and probability spaces
26(2)
What is (Ω, F, P) anyway?
28(1)
Characteristic functions
29(2)
Moment generating function
31(1)
Cumulant generating function
31(1)
Convergence of random variables
32(4)
Almost-sure convergence
32(2)
Convergence in probability
34(1)
Convergence in distribution
35(1)
Stochastic processes
36(8)
Stochastic processes as random functions
37(2)
Filtrations and histories
39(1)
Random times
40(1)
Martingales
41(2)
Predictable processes (*)
43(1)
The Poisson process
44(11)
Exponential random variables
44(3)
The Poisson distribution
47(1)
The Poisson process: definition and properties
48(4)
Compensated Poisson processes
52(1)
Counting processes
53(2)
Random measures and point processes
55(12)
Poisson random measures
57(2)
Compensated Poisson random measure
59(1)
Building jump processes from Poisson random measures
59(1)
Marked point processes (*)
60(7)
Levy processes: definitions and properties
67(36)
From random walks to Levy processes
68(2)
Compound Poisson processes
70(5)
Jump measures of compound Poisson processes
75(3)
Infinite activity Levy processes
78(7)
Pathwise properties of Levy processes
85(5)
Distributional properties
90(2)
Stable laws and processes
92(3)
Levy processes as Markov processes
95(2)
Levy processes and martingales
97(6)
Building Levy processes
103(28)
Model building with Levy processes
103(2)
``Jump-diffusions'' vs. infinite activity Levy processes
103(2)
Building new Levy processes from known ones
105(6)
Linear transformations
105(3)
Subordination
108(2)
Tilting and tempering the Levy measure
110(1)
Models of jump-diffusion type
111(2)
Building Levy processes by Brownian subordination
113(6)
General results
113(2)
Subordinating processes
115(1)
Models based on subordinated Brownian motion
116(3)
Tempered stable process
119(4)
Generalized hyperbolic model
123(8)
Multidimensional models with jumps
131(38)
Multivariate modelling via Brownian subordination
133(2)
Building multivariate models from common Poisson shocks
135(1)
Copulas for random variables
136(7)
Dependence concepts for Levy processes
143(2)
Copulas for Levy processes with positive jumps
145(11)
Copulas for general Levy processes
156(6)
Building multivariate models using Levy copulas
162(3)
Summary
165(4)
II Simulation and estimation
169(76)
Simulating Levy processes
171(36)
Simulation of compound Poisson processes
172(6)
Exact simulation of increments
178(6)
Approximation of an infinite activity Levy process by a compound Poisson process
184(4)
Approximation of small jumps by Brownian motion
188(4)
Series representations of Levy processes (*)
192(7)
Simulation of multidimensional Levy processes
199(8)
Modelling financial time series with Levy processes
207(38)
Empirical properties of asset returns
209(3)
Statistical estimation methods and their pitfalls
212(8)
Maximum likelihood estimation
212(5)
Generalized method of moments
217(2)
Discussion
219(1)
The distribution of returns: a tale of heavy tails
220(6)
How heavy tailed is the distribution of returns?
221(5)
Time aggregation and scaling
226(6)
Self-similarity
226(4)
Are financial returns self-similar?
230(2)
Realized variance and ``stochastic volatility''
232(4)
Pathwise properties of price trajectories (*)
236(5)
Holder regularity and singularity spectra
237(2)
Estimating singularity spectra
239(2)
Summary: advantages and shortcomings of Levy processes
241(4)
III Option pricing in models with jumps
245(206)
Stochastic calculus for jump processes
247(44)
Trading strategies and stochastic integrals
248(15)
Semimartingales
253(3)
Stochastic integrals for caglad processes
256(1)
Stochastic integrals with respect to Brownian motion
257(2)
Stochastic integrals with respect to Poisson random measures
259(4)
Quadratic variation
263(6)
Realized volatility and quadratic variation
263(4)
Quadratic covariation
267(2)
The Ito formula
269(13)
Pathwise calculus for finite activity jump processes
270(4)
Ito formula for diffusions with jumps
274(1)
Ito formula for Levy processes
275(4)
Ito formula for semimartingales
279(3)
Stochastic exponentials vs. ordinary exponentials
282(9)
Exponential of a Levy process
283(1)
Stochastic (Doleans-Dade) exponential
284(2)
Relation between ordinary and stochastic exponential
286(5)
Measure transformations for Levy processes
291(28)
Pricing rules and martingale measures
293(6)
Arbitrage-free pricing rules and martingale measures
296(3)
Market completeness
299(4)
Equivalence of measures for Levy processes: simple cases
303(4)
Equivalence of measures for Levy processes: general results
307(3)
The Esscher transform
310(2)
Relative entropy for Levy processes (*)
312(4)
Summary
316(3)
Pricing and hedging in incomplete markets
319(34)
Merton's approach
321(3)
Superhedging
324(3)
Utility maximization
327(4)
Certainty equivalent
327(1)
Utility indifference price
328(1)
The case of exponential utility
329(1)
On the applicability of indifference pricing
330(1)
Quadratic hedging
331(12)
Mean-variance hedging: martingale case
331(2)
Mean-variance hedging in exponential-Levy models
333(6)
Global vs. local risk minimization (*)
339(4)
``Optimal'' martingale measures
343(3)
Minimal entropy martingale measure
343(3)
Other martingale measures
346(1)
Hedging with options and model calibration
346(2)
Summary
348(5)
Risk-neutral modelling with exponential Levy processes
353(28)
European options in exp-Levy models
355(12)
Call options
356(1)
Implied volatility
357(4)
Fourier transform methods for option pricing
361(6)
Forward start options
367(1)
Barrier options
368(5)
American options
373(1)
Multi-asset options
374(7)
Integro-differential equations and numerical methods
381(50)
Partial integro-differential equations for computing option prices
382(12)
European options
384(5)
Barrier options
389(3)
American options
392(2)
Second order integro-differential equations
394(14)
Nonlocality and its consequences
396(1)
The Fourier view: pseudo-differential operators
396(2)
Classical solutions and Feynman-Kac representations
398(4)
Viscosity solutions (*)
402(6)
Trees and Markov chain methods
408(3)
Multinomial trees
409(1)
Multinomial trees as finite difference schemes
410(1)
Finite difference methods: theory and implementation
411(10)
Localization to a bounded domain
412(2)
Discretization in space
414(1)
An explicit-implicit finite difference method
415(3)
Convergence
418(3)
Analytic method of lines
421(1)
Galerkin methods
422(5)
Variational formulation and the Galerkin method
423(2)
Choice of a basis
425(2)
A comparison of numerical methods for PIDEs
427(4)
Inverse problems and model calibration
431(20)
Integrating prior views and option prices
434(2)
Nonlinear least squares
436(3)
Regularization using relative entropy
439(4)
Numerical implementation
443(3)
Numerical results
446(5)
Tests on simulated data
447(1)
Empirical result: single maturity
448(1)
Empirical results: several maturities
448(3)
IV Beyond Levy processes
451(48)
Time inhomogeneous jump processes
453(16)
Additive processes
454(6)
Exponential additive models
460(5)
Option pricing in risk-neutral exp-additive models
461(2)
Calibration to option prices
463(2)
Exp-additive models vs. local volatility models
465(4)
Stochastic volatility models with jumps
469(30)
Stochastic volatility models without jumps
471(6)
Implied volatility smiles
473(1)
The square root process
474(3)
A stochastic volatility model with jumps: the Bates model
477(3)
Non-Gaussian Ornstein-Uhlenbeck processes
480(8)
Definition and properties
481(3)
Stationary distributions of OU processes (*)
484(3)
Positive Ornstein-Uhlenbeck processes
487(1)
Ornstein-Uhlenbeck stochastic volatility models
488(2)
Time changed Levy processes
490(4)
Do we need stochastic volatility and jumps?
494(5)
A Modified Bessel functions 499(2)
References 501(28)
Subject index 529
Rama Cont, Peter Tankov
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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