Part Ⅰ Continuous Path Processes

1 Continuous-Path Random Processes:Mathematical Prerequisites

1.1 Some Definitions

1.1.1 Measurability

1.1.2 Monotone Class Theorem

1.1.3 Probability Measures

1.1.4 Filtration

1.1.5 Law of a Random Variable,Expectation

1.1.6 Independence

1.1.7 Equivalent Probabilities and Radon-Nikod？m Densities

1.1.8 Construction of Simple Probability Spaces

1.1.9 Conditional Expectation

1.1.10 Stochastic Processes

1.1.11 Convergence

1.1.12 Laplace Transform

1.1.13 Gaussian Processes

1.1.14 Markov Processes

1.1.15 Uniform Integrability

1.2 Martingales

1.2.1 Definition and Main Properties

1.2.2 Spaces of Martingales

1.2.3 Stopping Times

1.2.4 Local Martingales

1.3 Continuous Semi-martingales

1.3.1 Brackets of Continuous Local Martingales

1.3.2 Brackets of Continuous Semi-martingales

1.4 Brownian Motion

1.4.1 One-dimensional Brownian Motion

1.4.2 d-dimensional Brownian Motion

1.4.3 Correlated Brownian Motions

1.5 Stochastic Calculus

1.5.1 Stochastic Integration

1.5.2 Integration by Parts

1.5.3 It？'s Formula:The Fundamental Formula of Stochastic Calculus

1.5.4 Stochastic Differential Equations

1.5.5 Stochastic Differential Equations:The Onedimensional Case

1.5.6 Partial Differential Equations

1.5.7 Doléans-Dade Exponential

1.6 Predictable Representation Property

1.6.1 Brownian Motion Case

1.6.2 Towards a General Definition of the Predictable Representation Property

1.6.3 Dudley's Theorem

1.6.4 Backward Stochastic Differential Equations

1.7 Change of Probability and Girsanov's Theorem

1.7.1 Change of Probability

1.7.2 Decomposition of P-Martingales as Q-semi-martingales

1.7.3 Girsanov's Theorem:The One-dimensional Brownian Motion Case

1.7.4 Multidimensional Case

1.7.5 Absolute Continuity

1.7.6 Condition for Martingale Property of Exponential Local Martingales

1.7.7 Predictable Representation Property under a Change of Probability

1.7.8 An Example of Invariance of BM under Change of Measure

2 Basic Concepts and Examples in Finance

2.1 A Semi-martingale Framework

2.1.1 The Financial Market

2.1.2 Arbitrage Opportunities

2.1.3 Equivalent Martingale Measure

2.1.4 Admissible Strategies

2.1.5 Complete Market

2.2 A Diffusion Model

2.2.1 Absence of Arbitrage

2.2.2 Completeness of the Market

2.2.3 PDE Evaluation of Contingent Claims in a Complete Market

2.3 The Black and Scholes Model

2.3.1 The Model

2.3.2 European Call and Put Options

2.3.3 The Greeks

2.3.4 General Case

2.3.5 Dividend Paying Assets

2.3.6 R？le of Information

2.4 Change of Numéraire

2.4.1 Change of Numéraire and Black-Scholes Formula

2.4.2 Self-financing Strategy and Change of Numéraire

2.4.3 Change of Numéraire and Change of Probability

2.4.4 Forward Measure

2.4.5 Self-financing Strategies:Constrained Strategies

2.5 Feynman-Kac

2.5.1 Feynman-Kac Formula

2.5.2 Occupation Time for a Brownian Motion

2.5.3 Occupation Time for a Drifted Brownian Motion

2.5.4 Cumulative Options

2.5.5 Quantiles

2.6 Ornstein-Uhlenbeck Processes and Related Processes

2.6.1 Definition and Properties

2.6.2 Zero-coupon Bond

2.6.3 Absolute Continuity Relationship for Generalized Vasicek Processes

2.6.4 Square of a Generalized Vasicek Process

2.6.5 Powers of δ-Dimensional Radial OU Processes,Alias CIR Processes

2.7 Valuation of European Options

2.7.1 The Garman and Kohlhagen Model for Currency Options

2.7.2 Evaluation of an Exchange Option

2.7.3 Quanto Options

3 Hitting Times:A Mix of Mathematics and Finance

3.1 Hitting Times and the Law of the Maximum for Brownian Motion

3.1.1 The Law of the Pair of Random Variables(Wt,Mt)

3.1.2 Hitting Times Process

3.1.3 Law of the Maximum of a Brownian Motion over[0,t]

3.1.4 Laws of Hitting Times

3.1.5 Law of the Infimum

3.1.6 Laplace Transforms of Hitting Times

3.2 Hitting Times for a Drifted Brownian Motion

3.2.1 Joint Laws of (MX,X) and (mX,X) at Time t

3.2.2 Laws of Maximum,Minimum,and Hitting Times

3.2.3 Laplace Transforms

3.2.4 Computation of W(v)(ll{Ty(X)<t}e-λTy(X))

3.2.5 Normal Inverse Gaussian Law

3.3 Hitting Times for Geometric Brownian Motion

3.3.1 Laws of the Pairs(MSt,St)and(mSt,St)

3.3.2 Laplace Transforms

3.3.3 Computation of E(e-λTa(S)ll{Ta(S)<t})

3.4 Hitting Times in Other Cases

3.4.1 Ornstein-Uhlenbeck Processes

3.4.2 Deterministic Volatility and Nonconstant Barrier

3.5 Hitting Time of a Two-sided Barrier for BM and GBM

3.5.1 Brownian Case

3.5.2 Drifted Brownian Motion

3.6 Barrier Options

3.6.1 Put-Call Symmetry

3.6.2 Binary Options and △'s

3.6.3 Barrier Options:General Characteristics

3.6.4 Valuation and Hedging of a Regular Down-and-In Call Option When the Underlying is a Martingale

3.6.5 Mathematical Results Deduced from the Previous Approach

3.6.6 Valuation and Hedging of Regular Down-and-In Call Options:The General Case

3.6.7 Valuation and Hedging of Reverse Barrier Options

3.6.8 The Emerging Calls Method

3.6.9 Closed Form Expressions

3.7 Lookback Options

3.7.1 Using Binary Options

3.7.2 Traditional Approach

3.8 Double-barrier Options

3.9 Other Options

3.9.1 Options Involving a Hitting Time

3.9.2 Boost Options

3.9.3 Exponential Down Barrier Option

3.10 A Structural Approach to Default Risk

3.10.1 Merton's Model

3.10.2 First Passage Time Models

3.11 American Options

3.11.1 American Stock Options

3.11.2 American Currency Options

3.11.3 Perpetual American Currency Options

3.12 Real Options

3.12.1 Optimal Entry with Stochastic Investment Costs

3.12.2 Optimal Entry in the Presence of Competition

3.12.3 Optimal Entry and Optimal Exit

3.12.4 Optimal Exit and Optimal Entry in the Presence of Competition

3.12.5 Optimal Entry and Exit Decisions

4 Complements on Brownian Motion

4.1 Local Time

4.1.1 A Stochastic Fubini Theorem

4.1.2 Occupation Time Formula

4.1.3 An Approximation of Local Time

4.1.4 Local Times for Semi-martingales

4.1.5 Tanaka's Formula

4.1.6 The Balayage Formula

4.1.7 Skorokhod's Reflection Lemma

4.1.8 Local Time of a Semi-martingale

4.1.9 Generalized It？-Tanaka Formula

4.2 Applications

4.2.1 Dupire's Formula

4.2.2 Stop-Loss Strategy

4.2.3 Knock-out BOOST

4.2.4 Passport Options

4.3 Bridges,Excursions,and Meanders

4.3.1 Brownian Motion Zeros

4.3.2 Excursions

4.3.3 Laws of Tx,dt and gt

4.3.4 Laws of(Bt,gt,dt)

4.3.5 Brownian Bridge

4.3.6 Slow Brownian Filtrations

4.3.7 Meanders

4.3.8 The Azéma Martingale

4.3.9 Drifted Brownian Motion

4.4 Parisian Options

4.4.1 The Law of(G-,eD(W),WG-,eD)

4.4.2 Valuation of a Down-and-In Parisian Option

4.4.3 PDE Approach

4.4.4 American Parisian Options

5 Complements on Continuous Path Processes

5.1 Time Changes

5.1.1 Inverse of an Increasing Process

5.1.2 Time Changes and Stopping Times

5.1.3 Brownian Motion and Time Changes

5.2 Dual Predictable Projections

5.2.1 Definitions

5.2.2 Examples

5.3 Diffusions

5.3.1 (Time-homogeneous)Diffusions

5.3.2 Scale Function and Speed Measure

5.3.3 Boundary Points

5.3.4 Change of Time or Change of Space Variable

5.3.5Recurrence

5.3.6 Resolvent Kernel and Green Function

5.3.7 Examples

5.4 Non-homogeneous Diffusions

5.4.1 Kolmogorov's Equations

5.4.2 Application:Dupire's Formula

5.4.3 Fokker-Planck Equation

5.4.4 Valuation of Contingent Claims

5.5 Local Times for a Diffusion

5.5.1 Various Definitions of Local Times

5.5.2 Some Diffusions Involving Local Time

5.6 Last Passage Times

5.6.1 Notation and Basic Results

5.6.2 Last Passage Time of a Transient Diffusion

5.6.3 Last Passage Time Before Hitting a Level

5.6.4 Last Passage Time Before Maturity

5.6.5 Absolutely Continuous Compensator

5.6.6 Time When the Supremum is Reached

5.6.7 Last Passage Times for Particular Martingales

5.7 Pitman's Theorem about(2Mt-Wt)

5.7.1 Time Reversal of Brownian Motion

5.7.2 Pitman's Theorem

5.8 Filtrations

5.8.1 Strong and Weak Brownian Filtrations

5.8.2 Some Examples

5.9 Enlargements of Filtrations

5.9.1 Immersion of Filtrations

5.9.2 The Brownian Bridge as an Example of Initial Enlargement

5.9.3 Initial Enlargement:General Results

5.9.4 Progressive Enlargement

5.10 Filtering the Information

5.10.1 Independent Drift

5.10.2 Other Examples of Canonical Decomposition

5.10.3 Innovation Process

6 A Special Family of Diffusions:Bessel Processes

6.1 Definitions and First Properties

6.1.1 The Euclidean Norm of the n-Dimensional Brownian Motion

6.1.2 General Definitions

6.1.3 Path Properties

6.1.4 Infinitesimal Generator

6.1.5 Absolute Continuity

6.2 Properties

6.2.1 Additivity of BESQ's

6.2.2 Transition Densities

6.2.3 Hitting Times for Bessel Processes

6.2.4 Lamperti's Theorem

6.2.5 Laplace Transforms

6.2.6 BESQ Processes with Negative Dimensions

6.2.7 Squared Radial Ornstein-Uhlenbeck

6.3 Cox-Ingersoll-Ross Processes

6.3.1 CIR Processes and BESQ

6.3.2 Transition Probabilities for a CIR Process

6.3.3 CIR Processes as Spot Rate Models

6.3.4 Zero-coupon Bond

6.3.5 Inhomogeneous CIR Process

6.4 Constant Elasticity of Variance Process

6.4.1 Particular Case μ=0

6.4.2 CEV Processes and CIR Processes

6.4.3 CEV Processes and BESQ Processes

6.4.4 Properties

6.4.5 Scale Functions for CEV Processes

6.4.6 Option Pricing in a CEV Model

6.5 Some Computations on Bessel Bridges

6.5.1 Bessel Bridges

6.5.2 Bessel Bridges and Ornstein-Uhlenbeck Processes

6.5.3 European Bond Option

6.5.4 American Bond Options and the CIR Model

6.6 Asian Options

6.6.1 Parity and Symmetry Formulae

6.6.2 Laws of A(v)θ and A(v)t

6.6.3 The Moments of At

6.6.4 Laplace Transform Approach

6.6.5 PDE Approach

6.7 Stochastic Volatility

6.7.1 Black and Scholes Implied Volatility

6.7.2 A General Stochastic Volatility Model

6.7.3 Option Pricing in Presence of Non-normality of Returns:The Martingale Approach

6.7.4 Hull and White Model

6.7.5 Closed-form Solutions in Some Correlated Cases

6.7.6 PDE Approach

6.7.7 Heston's Model

6.7.8 Mellin Transform

Part Ⅱ Jump Processes

7 Default Risk:An Enlargement of Filtration Approach

7.1 A Toy Model

7.1.1 Defaultable Zero-coupon with Payment at Maturity

7.1.2 Defaultable Zero-coupon with Payment at Hit

7.2 Toy Model and Martingales

7.2.1 Key Lemma

7.2.2 The Fundamental Martingale

7.2.3 Hazard Function

7.2.4 Incompleteness of the Toy Model,non Arbitrage Prices

7.2.5 Predictable Representation Theorem

7.2.6 Risk-neutral Probability Measures

7.2.7 Partial Information:Duffie and Lando's Model

7.3 Default Times with a Given Stochastic Intensity

7.3.1 Construction of Default Time with a Given Stochastic Intensity

7.3.2 Conditional Expectation with Respect to Ft

7.3.3 Enlargements of Filtrations

7.3.4 Conditional Expectations with Respect to ？t

7.3.5 Conditional Expectations of F∞-Measurable Random Variables

7.3.6 Correlated Defaults:Copula Approach

7.3.7 Correlated Defaults:Jarrow and Yu's Model

7.4 Conditional Survival Probability Approach

7.4.1 Conditional Expectations

7.5 Conditional Survival Probability Approach and Immersion

7.5.1 (H)-Hypothesis and Arbitrages

7.5.2 Pricing Contingent Claims

7.5.3 Correlated Defaults:Kusuoka's Example

7.5.4 Stochastic Barrier

7.5.5 Predictable Representation Theorems

7.5.6 Hedging Contingent Claims with DZC

7.6 General Case:Without the(H)-Hypothesis

7.6.1 An Example of Partial Observation

7.6.2 Two Defaults,Trivial Reference Filtration

7.6.3 Initial Times

7.6.4 Explosive Defaults

7.7 Intensity Approach

7.7.1 Definition

7.7.2 Valuation Formula

7.8 Credit Default Swaps

7.8.1 Dynamics of the CDS's Price in a single name setting

7.8.2 Dynamics of the CDS's Price in a multi-name setting

7.9 PDE Approach for Hedging Defaultable Claims

7.9.1 Defaultable Asset with Total Default

7.9.2 PDE for Valuation

7.9.3 General Case

8 Poisson Processes and Ruin Theory

8.1 Counting Processes and Stochastic Integrals

8.2 Standard Poisson Process

8.2.1 Definition and First Properties

8.2.2 Martingale Properties

8.2.3 Infinitesimal Generator

8.2.4 Change of Probability Measure:An Example

8.2.5 Hitting Times

8.3 Inhomogeneous Poisson Processes

8.3.1 Definition

8.3.2 Martingale Properties

8.3.3 Watanabe's Characterization of Inhomogeneous Poisson Processes

8.3.4 Stochastic Calculus

8.3.5 Predictable Representation Property

8.3.6 Multidimensional Poisson Processes

8.4 Stochastic Intensity Processes

8.4.1 Doubly Stochastic Poisson Processes

8.4.2 Inhomogeneous Poisson Processes with Stochastic Intensity

8.4.3 It？'s Formula

8.4.4 Exponential Martingales

8.4.5 Change of Probability Measure

8.4.6 An Elementary Model of Prices Involving Jumps

8.5 Poisson Bridges

8.5.1 Definition of the Poisson Bridge

8.5.2 Harness Property

8.6 Compound Poisson Processes

8.6.1 Definition and Properties

8.6.2 Integration Formula

8.6.3 Martingales

8.6.4 It？'s Formula

8.6.5 Hitting Times

8.6.6 Change of Probability Measure

8.6.7 Price Process

8.6.8 Martingale Representation Theorem

8.6.9 Option Pricing

8.7 Ruin Process

8.7.1 Ruin Probability

8.7.2 Integral Equation

8.7.3 An Example

8.8 Marked Point Processes

8.8.1 Random Measure

8.8.2 Definition

8.8.3 An Integration Formula

8.8.4 Marked Point Processes with Intensity and Associated Martingales

8.8.5 Girsanov's Theorem

8.8.6 Predictable Representation Theorem

8.9 Poisson Point Processes

8.9.1 Poisson Measures

8.9.2 Point Processes

8.9.3 Poisson Point Processes

8.9.4 The It？ Measure of Brownian Excursions

9 General Processes:Mathematical Facts

9.1 Some Basic Facts about càdlàg Processes

9.1.1 An Illustrative Lemma

9.1.2 Finite Variation Processes,Pure Jump Processes

9.1.3 Some σ-algebras

9.2 Stochastic Integration for Square Integrable Martingales

9.2.1 Square Integrable Martingales

9.2.2 Stochastic Integral

9.3 Stochastic Integration for Semi-martingales

9.3.1 Local Martingales

9.3.2 Quadratic Covariation and Predictable Bracket of Two Local Martingales

9.3.3 Orthogonality

9.3.4 Semi-martingales

9.3.5 Stochastic Integration for Semi-martingales

9.3.6 Quadratic Covariation of Two Semi-martingales

9.3.7 Particular Cases

9.3.8 Predictable Bracket of Two Semi-martingales

9.4 It？'s Formula and Girsanov's Theorem

9.4.1 It？'s Formula:Optional and Predictable Forms

9.4.2 Semi-martingale Local Times

9.4.3 Exponential Semi-martingales

9.4.4 Change of Probability, Girsanov's Theorem

9.5 Existence and Uniqueness of the e.m.m

9.5.1 Predictable Representation Property

9.5.2 Necessary Conditions for Existence

9.5.3 Uniqueness Property

9.5.4 Examples

9.6 Self-financing Strategies and Integration by Parts

9.6.1 The Model

9.6.2 Self-financing Strategies and Change of Numéraire

9.7 Valuation in an Incomplete Market

9.7.1 Replication Criteria

9.7.2 Choice of an Equivalent Martingale Measure

9.7.3 Indifference Prices

10 Mixed Processes

10.1 Definition

10.2 It？'s Formula

10.2.1 Integration by Parts

10.2.2 It？'s Formula:One-dimensional Case

10.2.3 Multidimensional Case

10.2.4 Stochastic Differential Equations

10.2.5 Feynman-Kac Formula

10.2.6 Predictable Representation Theorem

10.3 Change of Probability

10.3.1 Exponential Local Martingales

10.3.2 Girsanov's Theorem

10.4 Mixed Processes in Finance

10.4.1 Computation of the Moments

10.4.2 Symmetry

10.4.3 Hitting Times

10.4.4 Affine Jump-Diffusion Model

10.4.5 General Jump-Diffusion Processes

10.5 Incompleteness

10.5.1 The Set of Risk-neutral Probability Measures

10.5.2 The Range of Prices for European Call Options

10.5.3 General Contingent Claims

10.6 Complete Markets with Jumps

10.6.1 A Three Assets Model

10.6.2 Structure Equations

10.7 Valuation of Options

10.7.1 The Valuation of European Options

10.7.2 American Option

11 Lévy Processes

11.1 Infinitely Divisible Random Variables

11.1.1 Definition

11.1.2 Self-decomposable Random Variables

11.1.3 Stable Random Variables

11.2 Lévy Processes

11.2.1 Definition and Main Properties

11.2.2 Poisson Point Processes,Lévy Measures

11.2.3 Lévy-Khintchine Formula for a Lévy Process

11.2.4 It？'s Formulae for a One-dimensional Lévy Process

11.2.5 It？'s Formula for Lévy-It？ Processes

11.2.6 Martingales

11.2.7 Harness Property

11.2.8 Representation Theorem of Martingales in a Lévy Setting

11.3 Absolutely Continuous Changes of Measures

11.3.1 Esscher Transform

11.3.2 Preserving the Lévy Property with Absolute Continuity

11.3.3 General Case

11.4 Fluctuation Theory

11.4.1 Maximum and Minimum

11.4.2 Pecherskii-Rogozin Identity

11.5 Spectrally Negative Lévy Processes

11.5.1 Two-sided Exit Times

11.5.2 Laplace Exponent of the Ladder Process

11.5.3 D.Kendall's Identity

11.6 Subordinators

11.6.1 Definition and Examples

11.6.2 Lévy Characteristics of a Subordinated Process

11.7 Exponential Lévy Processes as Stock Price Processes

11.7.1 Option Pricing with Esscher Transform

11.7.2 A Differential Equation for Option Pricing

11.7.3 Put-call Symmetry

11.7.4 Arbitrage and Completeness

11.8 Variance-Gamma Model

11.9 Valuation of Contingent Claims

11.9.1 Perpetual American Options

A List of Special Features,Probability Laws,and Functions

A.1 Main Formulae

A.1.1 Absolute Continuity Relationships

A.1.2 Bessel Processes

A.1.3 Brownian Motion

A.1.4 Diffusions

A.1.5 Finance

A.1.6 Girsanov's Theorem

A.1.7 Hitting Times

A.1.8 It？'s Formulae

A.1.9 Lévy Processes

A.1.10 Semi-martingales

A.2 Processes

A.3 Some Main Models

A.4 Some Important Probability Distributions

A.4.1 Laws with Density

A.4.2 Some Algebraic Properties for Special r.v.'s

A.4.3 Poisson Law

A.4.4 Gamma and Inverse Gaussian Law

A.4.5 Generalized Inverse Gaussian and Normal Inverse Gaussian

A.4.6 Variance Gamma VG(σ,ν,θ)

A.4.7 Tempered Stable TS(Y±,C±,M±)

A.5 Special Functions

A.5.1 Gamma and Beta Functions

A.5.2 Bessel Functions

A.5.3 Hermite Functions

A.5.4 Parabolic Cylinder Functions

A.5.5 Airy Function

A.5.6 Kummer Functions

A.5.7 Whittaker Functions

A.5.8 Some Laplace Transforms

References

B Some Papers and Books on Specific Subjects

B.1 Theory of Continuous Processes

B.1.1 Books

B.1.2 Stochastic Differential Equations

B.1.3 Backward SDE

B.1.4 Martingale Representation Theorems

B.1.5 Enlargement of Filtrations

B.1.6 Exponential Functionals

B.1.7 Uniform Integrability of Martingales

B.2 Particular Processes

B.2.1 Ornstein-Uhlenbeck Processes

B.2.2 CIR Processes

B.2.3 CEV Processes

B.2.4 Bessel Processes

B.3 Processes with Discontinuous Paths

B.3.1 Some Books

B.3.2 Survey Papers

B.4 Hitting Times

B.5 Lévy Processes

B.5.1 Books

B.5.2 Some Papers

B.6 Some Books on Finance

B.6.1 Discrete Time

B.6.2 Continuous Time

B.6.3 Collective Books

B.6.4 History

B.7 Arbitrage

B.8 Exotic Options

B.8.1 Books

B.8.2 Articles

Index of Authors

Index of Symbols

Subject Index