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Financial Mathematics: A Comprehensive Treatment in Continuous Time Volume II [Kõva köide]

(Wilfrid Laurier University, Waterloo, Ontario, Canada), (Wilfrid Laurier University, Waterloo, Ontario, Canada)
  • Formaat: Hardback, 492 pages, kõrgus x laius: 254x178 mm, kaal: 1100 g, 36 Line drawings, black and white; 36 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 21-Dec-2022
  • Kirjastus: CRC Press
  • ISBN-10: 1138603635
  • ISBN-13: 9781138603639
Teised raamatud teemal:
Financial Mathematics: A Comprehensive Treatment in Continuous Time Volume IISuurem pilt
  • Formaat: Hardback, 492 pages, kõrgus x laius: 254x178 mm, kaal: 1100 g, 36 Line drawings, black and white; 36 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 21-Dec-2022
  • Kirjastus: CRC Press
  • ISBN-10: 1138603635
  • ISBN-13: 9781138603639
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781138603639.html
Märksõnad:
The book has been tested and refined through years of classroom teaching experience. With an abundance of examples, problems, and fully worked out solutions, the text introduces the financial theory and relevant mathematical methods in a mathematically rigorous yet engaging way.

This textbook provides complete coverage of continuous-time financial models that form the cornerstones of financial derivative pricing theory. Unlike similar texts in the field, this one presents multiple problem-solving approaches, linking related comprehensive techniques for pricing different types of financial derivatives.

Key features:











In-depth coverage of continuous-time theory and methodology





Numerous, fully worked out examples and exercises in every chapter





Mathematically rigorous and consistent, yet bridging various basic and more advanced concepts





Judicious balance of financial theory and mathematical methods





Guide to Material

This revision contains:











Almost 150 pages worth of new material in all chapters





A appendix on probability theory





An expanded set of solved problems and additional exercises





Answers to all exercises

This book is a comprehensive, self-contained, and unified treatment of the main theory and application of mathematical methods behind modern-day financial mathematics.

The text complements Financial Mathematics: A Comprehensive Treatment in Discrete Time, by the same authors, also published by CRC Press.
List of Figures
xi
Preface xiii
Authors xvii
I Stochastic Calculus with Brownian Motion
1(168)
1 One-Dimensional Brownian Motion and Related Processes
3(56)
1.1 Multivariate Normal Distributions
3(2)
1.1.1 Multivariate Normal Distribution
3(1)
1.1.2 Conditional Normal Distributions
4(1)
1.2 Standard Brownian Motion
5(23)
1.2.1 One-Dimensional Symmetric Random Walk
5(6)
1.2.2 Formal Definition and Basic Properties of Brownian Motion
11(3)
1.2.3 Multivariate Distribution of Brownian Motion
14(3)
1.2.4 The Markov Property and the Transition PDF
17(8)
1.2.5 Quadratic Variation and Nondifferentiability of Paths
25(3)
1.3 Some Processes Derived from Brownian Motion
28(9)
1.3.1 Drifted Brownian Motion
28(1)
1.3.2 Geometric Brownian Motion
29(2)
1.3.3 Processes related by a monotonic mapping
31(1)
1.3.4 Brownian Bridge
32(3)
1.3.5 Gaussian Processes
35(2)
1.4 First Hitting Times and Maximum and Minimum of Brownian Motion
37(17)
1.4.1 The Reflection Principle: Standard Brownian Motion
37(8)
1.4.2 Translated and Scaled Driftless Brownian Motion
45(2)
1.4.3 Brownian Motion with Drift
47(7)
1.5 Exercises
54(5)
2 Introduction to Continuous-Time Stochastic Calculus
59(110)
2.1 The Riemann Integral of Brownian Motion
59(3)
2.1.1 The Riemann Integral
59(1)
2.1.2 The Integral of a Brownian Path
59(3)
2.2 The Riemann-Stieltjes Integral of Brownian Motion
62(4)
2.2.1 The Riemann-Stieltjes Integral
62(2)
2.2.2 Integrals w.r.t. Brownian Motion: Preliminary Discussion
64(2)
2.3 The Ito Integral and Its Basic Properties
66(17)
2.3.1 The Ito Integral for Simple Processes
66(8)
2.3.2 The Ito Integral for General Processes
74(9)
2.4 Ito Processes and Their Properties
83(7)
2.4.1 Gaussian Processes Generated by Ito Integrals
83(1)
2.4.2 Ito Processes
84(2)
2.4.3 Quadratic and Co-Variation of Ito Processes
86(4)
2.5 Ito's Formula for Functions of BM and Ito Processes
90(8)
2.5.1 Ito's Formula for Functions of BM
90(3)
2.5.2 An "Antiderivative" Formula for Evaluating Ito Integrals
93(2)
2.5.3 Ito's Formula for Ito Processes
95(3)
2.6 Stochastic Differential Equations
98(9)
2.6.1 Solutions to Linear SDEs
99(7)
2.6.2 Existence and Uniqueness of a Strong Solution to an SDE
106(1)
2.7 The Markov Property, Martingales, Feynman-Kac Formulae, and Transition CDFs and PDFs
107(17)
2.7.1 Forward Kolmogorov PDE
121(2)
2.7.2 Transition CDF/PDF for Time-Homogeneous Diffusions
123(1)
2.8 Radon-Nikodym Derivative Process and Girsanov's Theorem
124(10)
2.8.1 Some Applications of Girsanov's Theorem
130(4)
2.9 Brownian Martingale Representation Theorem
134(1)
2.10 Stochastic Calculus for Multidimensional BM
135(25)
2.10.1 The ltd Integral and Ito's Formula for Multiple Processes on Multidimensional BM
135(12)
2.10.2 Multidimensional SDEs, Feynman-Kac Formulae, and Transition CDFs and PDFs
147(10)
2.10.3 Girsanov's Theorem for Multidimensional BM
157(3)
2.10.4 Martingale Representation Theorem for Multidimensional BM
160(1)
2.11 Exercises
160(9)
II Continuous-Time Modelling
169(246)
3 Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock
171(90)
3.1 From the CRR Model to the BSM Model
172(8)
3.1.1 Portfolio Strategies in the Binomial Tree Model
174(3)
3.1.2 The Cox-Ross-Rubinstein Model and its Continuous-Time Limit
177(3)
3.2 Replication (Hedging) and Derivative Pricing in the Simplest Black-Scholes Economy
180(36)
3.2.1 Pricing Standard European Calls and Puts
187(3)
3.2.2 Hedging Standard European Calls and Puts
190(6)
3.2.3 Europeans with Piecewise Linear Payoffs
196(5)
3.2.4 Power Options
201(3)
3.2.5 Dividend Paying Stock
204(7)
3.2.6 Option Pricing with the Stock Numeraire
211(5)
3.3 Forward Starting, Chooser, and Compound Options
216(9)
3.4 Some European-Style Path-Dependent Derivatives
225(20)
3.4.1 Risk-Neutral Pricing under GBM
228(4)
3.4.2 Pricing Single Barrier Options
232(6)
3.4.3 Pricing Lookback Options
238(7)
3.5 Structural Credit Risk Models
245(4)
3.5.1 The Merton Model
246(1)
3.5.2 The Black-Cox Model
247(2)
3.6 Exercises
249(12)
4 Risk-Neutral Pricing in a Multi-Asset Economy
261(46)
4.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing
262(7)
4.2 Equivalent Martingale Measures: Derivative Pricing with General Numeraire Assets
269(5)
4.3 Black-Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives
274(26)
4.3.1 Standard European Option Pricing for Multi-Stock GBM
277(3)
4.3.2 Explicit Pricing Formulae for the GBM Model
280(9)
4.3.3 Cross-Currency Option Valuation
289(6)
4.3.4 Option Valuation with General Numeraire Assets
295(5)
4.4 Exercises
300(7)
5 American Options
307(24)
5.1 Basic Properties of Early-Exercise Options
307(4)
5.2 Arbitrage-Free Pricing of American Options
311(7)
5.2.1 Optimal Stopping Formulation and Early-Exercise Boundary
311(3)
5.2.2 The Smooth Pasting Condition
314(2)
5.2.3 Put-Call Symmetry Relation
316(1)
5.2.4 Dynamic Programming Approach for Bermudan Options
317(1)
5.3 Perpetual American Options
318(4)
5.3.1 Pricing a Perpetual Put Option
319(2)
5.3.2 Pricing a Perpetual Call Option
321(1)
5.4 Finite-Expiration American Options
322(5)
5.4.1 The PDE Formulation
322(3)
5.4.2 The Integral Equation Formulation
325(2)
5.5 Exercises
327(4)
6 Interest-Rate Modelling and Derivative Pricing
331(40)
6.1 Basic Fixed Income Instruments
331(5)
6.1.1 Bonds
331(1)
6.1.2 Forward Rates
332(1)
6.1.3 Arbitrage-Free Pricing
333(1)
6.1.4 Fixed Income Derivatives
334(2)
6.2 Single-Factor Models
336(10)
6.2.1 Diffusion Models for the Short Rate Process
337(1)
6.2.2 PDE for the Zero-Coupon Bond Value
338(2)
6.2.3 Affine Term Structure Models
340(1)
6.2.4 The Ho-Lee Model
341(1)
6.2.5 The Vasicek Model
342(2)
6.2.6 The Cox-Ingersoll-Ross Model
344(2)
6.3 Heath-Jarrow-Morton Formulation
346(7)
6.3.1 HJM under Risk-Neutral Measure
347(3)
6.3.2 Relationship between HJM and Affine Yield Models
350(3)
6.4 Multifactor Affine Term Structure Models
353(3)
6.4.1 Gaussian Multifactor Models
354(1)
6.4.2 Equivalent Classes of Affine Models
355(1)
6.5 Pricing Derivatives under Forward Measures
356(7)
6.5.1 Forward Measures
356(2)
6.5.2 Pricing Stock Options under Stochastic Interest Rates
358(2)
6.5.3 Pricing Options on Zero-Coupon Bonds
360(3)
6.6 LIBOR Model
363(4)
6.6.1 LIBOR Rates
363(1)
6.6.2 The Brace-Gatarek-Musiela Model of LIBOR Rates
364(1)
6.6.3 Pricing Caplets, Caps, and Swaps
365(2)
6.7 Exercises
367(4)
7 Alternative Models of Asset Price Dynamics
371(44)
7.1 Characteristic Functions
371(8)
7.1.1 Definition and Properties
371(4)
7.1.2 Recovering the Distribution Function
375(1)
7.1.3 Pricing Standard European Options
375(3)
7.1.4 The Carr-Madan Method for Pricing Vanilla Options
378(1)
7.2 Stochastic Volatility Diffusion Models
379(8)
7.2.1 Local Volatility Models
379(3)
7.2.2 Constant Elasticity of Variance Model
382(5)
7.3 The Heston model
387(7)
7.3.1 Solution to the Ricatti Equation
391(2)
7.3.2 Implied Volatility for the Heston Model
393(1)
7.4 Models with Jumps
394(16)
7.4.1 The Poisson Process
394(4)
7.4.2 Jump Diffusion Models with a Compound Poisson Component
398(4)
7.4.3 The Merton Jump Diffusion Model
402(1)
7.4.4 Characteristic Function for a Jump Diffusion Process
403(2)
7.4.5 Change of Measure for Jump Diffusion Processes
405(4)
7.4.6 The Variance Gamma Model
409(1)
7.5 Exercises
410(5)
A Essentials of General Probability Theory
415(34)
A.1 Random Variables and Lebesgue Integration
415(14)
A.2 Multidimensional Lebesgue Integration
429(2)
A.3 Multiple Random Variables and Joint Distributions
431(10)
A.4 Conditioning
441(5)
A.5 Changing Probability Measures
446(3)
B Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables
449(4)
C Answers and Hints to Exercises
453(26)
C.1
Chapter 1
453(3)
C.2
Chapter 2
456(3)
C.3
Chapter 3
459(9)
C.4
Chapter 4
468(6)
C.5
Chapter 5
474(2)
C.6
Chapter 6
476(2)
C.7
Chapter 7
478(1)
D Glossary of Symbols and Abbreviations
479(6)
References 485(4)
Index 489
Giuseppe Campolieti is Professor of Mathematics at Wilfrid Laurier University in Waterloo, Canada. He has been Natural Sciences and Engineering Research Council postdoctoral research fellow and university research fellow at the University of Toronto. In 1998, he joined the Masters in Mathematical Finance as an instructor and later as an adjunct professor in financial mathematics until 2002. Dr. Campolieti also founded a financial software and consulting company in 1998. He joined Laurier in 2002 as Associate Professor of Mathematics and as SHARCNET Chair in Financial Mathematics.

Roman N. Makarov is Associate Professor and Chair of Mathematics at Wilfrid Laurier University. Prior to joining Laurier in 2003, he was an Assistant Professor of Mathematics at Siberian State University of Telecommunications and Informatics and a senior research fellow at the Laboratory of Monte Carlo Methods at the Institute of Computational Mathematics and Mathematical Geophysics in Novosibirsk, Russia.