Copyright 2025
Hardback
ISBN 9781032636306
394 Pages 58 B/W Illustrations
Published December 23, 2024 by Chapman & Hall
Malliavin Calculus in Finance: Theory and Practice, Second Edition introduces the study of stochastic volatility (SV) models via Malliavin Calculus. Originally motivated by the study of the existence of smooth densities of certain random variables, Malliavin calculus has had a profound impact on stochastic analysis. In particular, it has been found to be an effective tool in quantitative finance, as in the computation of hedging strategies or the efficient estimation of the Greeks.
This book aims to bridge the gap between theory and practice and demonstrate the practical value of Malliavin calculus. It offers readers the chance to discover an easy-to-apply tool that allows us to recover, unify, and generalize several previous results in the literature on stochastic volatility modeling related to the vanilla, the forward, and the VIX implied volatility surfaces. It can be applied to local, stochastic, and also to rough volatilities (driven by a fractional Brownian motion) leading to simple and explicit results.
Intermediate-advanced level text on quantitative finance, oriented to practitioners with a basic background in stochastic analysis, which could also be useful for researchers and students in quantitative finance
Includes examples on concrete models such as the Heston, the SABR and rough volatilities, as well as several numerical experiments and the corresponding Python scripts
Covers applications on vanillas, forward start options, and options on the VIX.
The book also has a Github repository with the Python library corresponding to the numerical examples in the text. The library has been implemented so that the users can re-use the numerical code for building their examples. The repository can be accessed here: https://bit.ly/2KNex2Y.
Includes a new chapter to study implied volatility within the Bachelier framework.
Chapters 7 and 8 have been thoroughly updated to introduce a more detailed discussion on the relationship between implied and local volatilities, according to the new results in the literature.
I. A primer on option pricing and volatility modeling. 1. The option pricing problem. 1.1. Derivatives. 1.2. Non-arbitrage prices and the Black-Scholes formula. 1.3. The Black-Scholes model. 1.4. The Black-Scholes implied volatility and the non-constant volatility case. 1.5. Chapter's digest. 2. The volatility process. 2.1. The estimation of the integrated and the spot volatility. 2.2. Local volatilities. 2.3. Stochastic volatilities. 2.4. Stochastic-local volatilities 2.5. Models based on the fractional Brownian motion and rough volatilities. 2.6. Volatility derivatives. 2.7. Chapter’s Digest. II. Mathematical tools. 3. A primer on Malliavin Calculus. 3.1. Definitions and basic properties. 3.2. Computation of Malliavin Derivatives. 3.3. Malliavin derivatives for general SV models. 3.4. Chapter's digest. 4. Key tools in Malliavin Calculus. 4.1. The Clark-Ocone-Haussman formula. 4.2. The integration by parts formula. 4.3. The anticipating Ito's formula. 4.4. Chapter’s Digest. 5. Fractional Brownian motion and rough volatilities. 5.1. The fractional Brownian motion. 5.2. The Riemann-Liouville fractional Brownian motion. 5.3. Stochastic integration with respect to the fBm. 5.4. Simulation methods for the fBm and the RLfBm. 5.5. The fractional Brownian motion in finance. 5.6. The Malliavin derivative of fractional volatilities. 5.7. Chapter's digest. III. Applications of Malliavin Calculus to the study of the implied volatility surface. 6. The ATM short time level of the implied volatility. 6.1. Basic definitions and notation. 6.2. The classical Hull and White formula. 6.3. An extension of the Hull and White formula from the anticipating Ito's formula. 6.4. Decomposition formulas for implied volatilities. 6.5. The ATM short-time level of the implied volatility. 6.6. Chapter's digest. 7. The ATM short-time skew. 7.1. The term structure of the empirical implied volatility surface. 7.2. The main problem and notations. 7.3. The uncorrelated case. 7.4. The correlated case. 7.5. The short-time limit of implied volatility skew. 7.6. Applications. 7.7. Is the volatility long-memory, short memory, or both?. 7.8. A comparison with jump-diffusion models: the Bates model. 7.9. Chapter's digest. 8.0. The ATM short-time curvature. 8.1. Some empirical facts. 8.2. The uncorrelated case. 8.3. The correlated case. 8.4. Examples. 8.5. Chapter's digest. IV. The implied volatility of non-vanilla options. 9. Options with random strikes and the forward smile. 9.1. A decomposition formula for random strike options. 9.2. Forward start options as random strike options. 9.3. Forward-Start options and the decomposition formula. 9.4. The ATM short-time limit of the implied volatility. 9.5. At-the-money skew. 9.6. At-the-money curvature. 9.7. Chapter's digest. 10. Options on the VIX. 10.1. The ATM short time level and skew of the implied volatility. 10.2. VIX options. 10.3. Chapter's digest. Section V Non log-normal models. 11. The Bachelier implied volatility. 11.1. Bachelier-type Models. 11.2. A Decomposition formula for option prices. 11.3. A Decomposition formula for implied volitality. 11.4. The Bachelier ATM skew. 11.5. Chapter's digest. Bibliography. Index.
Copyright 2025
Paperback
ISBN 9781032899732
266 Pages
Published December 19, 2024 by Chapman & Hall
Hardback
Quantum calculus is the modern name for the investigation of calculus without limits. Quantum calculus, or q-calculus, began with F.H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by renowned mathematicians Euler and Jacobi.
Lately, quantum calculus has aroused a great amount of interest due to the high demand of mathematics that model quantum computing. The q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other quantum theory sciences, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize quantum calculus has been defined.
General Quantum Variational Calculus is specially designed for those who wish to understand this important mathematical concept, as the text encompasses recent developments of general quantum variational calculus. The material is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques.
This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It can be used as a textbook at the graduate level and as a reference for several disciplines.
1. Elements of the Multimensional General Quantum Calculus
1.1 The Multidimensional General Quantum Calculus
1.2 Line Integrals
1.3 The Green Formula
1.4 Advanced Practical Problems
2. β-Differential Systems
2.1 Structure of β-Differential Systems
2.2 β-Matrix Exponential Function
2.3 The β-Liouville Theorem
2.4 Constant Coefficients
2.5 Nonlinear Systems
2.6 Advanced Practical Problems
3. Functionals
3.1 Definition for Functionals
3.2 Self-Adjoint Second Order Matrix Equations
3.3 The Jacobi Condition
3.4 Sturmian Theory
4. Linear Hamiltonian Dynamic Systems
4.1 Linear Symplectic Dynamic Systems
4.2 Hamiltonian Systems
4.3 Conjoined Bases
4.4 Riccati Equations
4.5 The Picone Identity
4.6 ”Big” Linear Hamiltonian Systems
4.7 Positivity of Quadratic Functionals
5. The First Variation
5.1 The Dubois-Reymond Lemma
5.2 The Variational Problem
5.3 The Euler-Lagrange Equation
5.4 The Legendre Condition
5.5 The Jacobi Condition
5.6 Advanced Practical Problems
6. Higher Order Calculus of Variations
6.1 Statement of the Variational Problem
6.2 The Euler Equation
6.3 Advanced Practical Problems
7. Double Integral Calculus of Variations
7.1 Statement of the Variational Problem
7.2 First and Second Variation
7.3 The Euler Condition
7.4 Advanced Practical Problems
8. The Noether Second Theorem
8.1 Invariance under Transformations
8.2 The Noether Second Theorem without Transformations of Time
8.3 The Noether Second Theorem with Transformations of Time
8.4 The Noether Second Theorem-Double Delta Integral Case
References
Index
Copyright 2025
Hardback
ISBN 9781032587165
350 Pages 2 B/W Illustrations
Published December 10, 2024 by Chapman & Hall
Variational-Hemivariational Inequalities with Applications, Second Edition represents the outcome of the cross-fertilization of nonlinear functional analysis and mathematical modelling, demonstrating its application to solid and contact mechanics. Based on authors’ original results, the book illustrates the use of various functional methods (including monotonicity, pseudomonotonicity, compactness, penalty and fixed-point methods) in the study of various nonlinear problems in analysis and mechanics. The classes of history-dependent operators and almost history-dependent operators are exposed in a large generality. A systematic and unified presentation contains a carefully selected collection of new results on variational-hemivariational inequalities with or without unilateral constraints. A wide spectrum of static, quasistatic, dynamic contact problems for elastic, viscoelastic and viscoplastic materials illustrates the applicability of these theoretical results.
Written for mathematicians, applied mathematicians, engineers and scientists, this book is also a valuable tool for graduate students and researchers in nonlinear analysis, mathematical modelling, mechanics of solids, and contact mechanics.
Convergence and well-posedness results for elliptic and history-dependent variational-hemivariational inequalities
Existence results on various optimal control problems with applications in solid and contact mechanics
Existence, uniqueness and stability results for evolutionary and differential variational-hemivariational inequalities with unilateral constraints
Modelling and analysis of static and quasistatic contact problems for elastic and viscoelastic materials with looking effect
Modelling and analysis of viscoelastic and viscoplastic dynamic contact problems with unilateral constraints.
I. Variational Problems in Solid Mechanics. 1. Elliptic Variational Inequalities. 1.1. Background on functional analysis. 1.2. Existence and uniqueness results. 1.3. Convergence results. 1.4. Optimal control. 1.5. Well-posedness results. 2. History-Dependent Operators. 2.1. Spaces of continuous functions. 2.2. Definitions and basic properties. 2.3. Fixed point properties. 2.4. History-dependent equations in Hilbert spaces. 2.5. Nonlinear implicit equations in Banach spaces. 2.6. History-dependent variational inequalities. 2.7. Relevant particular cases. 3. Displacement-Traction Problems in Solid Mechanics. 3.1. Modeling of displacement-traction problems. 3.2. A displacement-traction problem with locking materials. 3.3. One-dimensional elastic examples. 3.4. Two viscoelastic problems. 3.5. One-dimensional examples. 3.6. A viscoplastic problem. II. Variational-Hemivariational Inequalities. 4. Elements of Nonsmooth Analysis. 4.1. Monotone and pseudomonotone operators. 4.2. Bochner-Lebesgue spaces. 4.3. Subgradient of convex functions. 4.4. Subgradient in the sense of Clarke. 4.5. Mixed equilibrium problem. 4.6. Miscellaneous results. 5. Elliptic Variational-Hemivariational Inequalities. 5.1. An existence and uniqueness result. 5.2. Convergence results. 5.3. Optimal control. 5.4. Penalty methods. 5.5. Well-posedness results. 5.6. Relevant particular cases. 6. History-Dependent Variational-Hemivariational Inequalities. 6.1. An existence and uniqueness result. 6.2. Convergence results. 6.3. Optimal control. 6.4. A penalty method. 6.5. A well-posedness result. 6.6. Relevant particular cases. 7. Evolutionary Variational-Hemivariational Inequalities. 7.1. A class of inclusions with history-dependent operators. 7.2. History-dependent inequalities with unilateral constraints. 7.3. Constrainted differential variational-hemivariational inequalities. 7.4. Relevant particular cases. III. Applications to Contact Mechanics. 8. Static Contact Problems. 8.1. Modeling of static contact problems. 8.2. A contact problem with normal compliance. 8.3. A contact problem with unilateral constraints. 8.4. Convergence and optimal control results. 8.5. A contact problem for locking materials. 8.6. Convergence and optimal control results. 8.7. Penalty methods. 9. Time-Dependent and Quasistatic Contact Problems. 9.1. Physical setting and mathematical models. 9.2. Two time-dependent elastic contact problems. 9.3. A quasistatic viscoplastic contact problem. 9.4. A time-dependent viscoelastic contact problem. 9.5. Convergence and optimal control results. 9.6. A frictional viscoelastic contact problem. 9.7. A quasistatic contact problem with locking materials. 10. Dynamic Contact Problems. 10.1. Mathematical models of dynamic contact. 10.2. A viscoelastic contact problem with normal damped response. 10.3. A unilateral viscoelastic frictional contact problem. 10.4. A unilateral viscoplastic frictionless contact problem.
Copyright 2025
Paperback
Hardback
￡120.00
ISBN 9781032602257
504 Pages 32 B/W Illustrations
Published December 24, 2024 by Chapman & Hall
Updated to reflect current research and extended to cover more advanced topics as well as the basics, Algebraic Number Theory and Fermat’s Last Theorem, Fifth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics?the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers, initially from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work.
Pell's Equation x^2-dy^2=1: all solutions can be obtained from a single `fundamental' solution, which can be found using continued fractions.
Galois theory of number field extensions, relating the field structure to that of the group of automorphisms.
More material on cyclotomic fields, and some results on cubic fields.
Advanced properties of prime ideals, including the valuation of a fractional ideal relative to a prime ideal, localisation at a prime ideal, and discrete valuation rings.
Ramification theory, which discusses how a prime ideal factorises when the number field is extended to a larger one.
A short proof of the Quadratic Reciprocity Law based on properties of cyclotomic fields. This
Valuations and p-adic numbers. Topology of the p-adic integers.
Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.
I. Algebraic Methods. 1. Algebraic Background. 2. Algebraic Numbers. 3. Quadratic and Cyclotomic Fields. 4. Pell's Equation. 5. Factorisation into Irreducibles. 6. Ideals. II. Geometric Methods. 7. Lattices. 8. Minkowski's Theorem. 9. Geometric Representation of Algebraic Numbers. 10. Dirichlet's Units Theorem. 11. Class-Group and Class-Number. III. Number-Theoretic Applications. 12. Computational Methods. 13. Kummer's Special Case of Fermat's Last Theorem. IV. Elliptic Curves and Elliptic Functions. 14. Elliptic Curves. 15. Elliptic Functions. V. Wiles's Proof of Fermat's Last Theorem. 16. The Path to the Final Breakthrough. 17. Wiles's Strategy and Subsequent Developments. VI. Galois Theory and Other Topics. 18. Extensions and Galois Theory. 19. Cyclotomic and Cubic Fields. 20. Prime Ideals Revisited. 21. Ramification Theory. 22. Quadratic Reciprocity. 23. Valuations and p-adic Numbers.
Copyright 2025
Hardback
ISBN 9781032778914
556 Pages 72 B/W Illustrations
April 7, 2025 by Chapman & Hall
CONTEMPORARY ABSTRACT ALGEBRA, ELEVENTH EDITION is intended for a course whose main purpose is to enable students to do computations and write proofs. This text stresses the importance of obtaining a solid introduction to the traditional topics, while at the same time presenting abstract algebra as a contemporary and very much active subject which is currently being used by working physicists, chemists, and computer scientists.
For more than three decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging.
The author presents the concepts and methodologies of used by working mathematicians, computer scientists, physicists, and chemists. Students will learn how to do computations and to write proofs. A unique feature of the book are exercises that build the skill of generalizing, a skill that students should develop but rarely do.
This new edition is streamlined. The 10th edition had 26 new examples, 330 new exercises, a few new theorems, and a substantial, number of minor modifications to the explanatory material, discussion text, and proofs. We have omitted suggested readings, references, biographies, etc that tally to 56 pages less. A number of corrections were also made for this edition.
Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. The hallmark features of previous editions of the book are enhanced in this edition. These include:
A good mixture of approximately 1900 exercises.
Approximately 300 worked-out examples
Many applications from scientific and computing fields and everyday life.
Historical notes and biographies that spotlight people and events.
Motivational and humorous quotations.
Numerous connections to number theory and geometry.
While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructor’s Solutions Manual offers solutions for all the exercises. The Student Solution Manual has comprehensive solutions for all odd-numbered exercises and many even-numbered exercises and is well-loved for alternative solutions as well.
1 Introduction to Groups
2 Groups
3 Finite Groups; Subgroups
4 Cyclic Groups
5 Permutation Groups
6 Ismorphisms
7 Cosets and Lagrange's Theorem
8 External Direct Products
9 Normal Subgroups and Factor Groups
10 Group Homomorphisms
11 Fundamental Theorem of Finite Abelian Groups
12 Introduction to Rings
13 Integral Domains
14 Ideals and Factor Rings
15 Ring Homomorphisms
16 Polynomial Rings
17 Factorization of Polynomials
18 Divisibilty in Integral Domains
19 Extension Fields
20 Algebraic Extensions
21 Finite Fields
22 Geometric Constructions
23 Sylow Theorems
24 Finite Simple Groups
25 Generators and Relations
26 Symmetry Groups
27 Symmetry and Counting
28 Cayley Digraphs of Groups
29 Introduction to Algebraic Coding Theory
30 An Introduction to Galois Theory
31 Cyclotomic Extensions