Notations
Notations for Real Matrices
Real Valued Random Variables (Tables 1 and 2)
Miscellaneous
Chapter 1 Real Valued Affine Diffusions
1.1 The Ornstein-Uhlenbeck Process
1.2 The Cox-Ingersoll-Ross Process
1.2.1 Definition and Existence
1.2.2 Characteristic and Probability Density Functions
The Characteristic Function
The Probability Density Function of Xxt
1.2.3 A Nice Connection Between Ornstein-Uhlenbeck and Cox-Ingersoll-Ross Processes
1.2.4 The Feller Condition
1.3 Definition and Characterization of Affine Diffusions
1.4 Application to Interest Rate Modelling
1.4.1 Short Rate Models and Interest Rates Contractsin a Nutshell
1.4.2 The Vasicek Model
1.4.3 The Cox-Ingersoll-Ross Model
Chapter 2 An Introduction to Simulation Schemes for SDEs
2.1 The Euler-Maruyama Scheme
2.1.1 The Strong Error
2.1.2 The Weak Error
2.1.3 Beyond the Euler Scheme: Strong and Weak High Order Approximations
2.2 Strong Approximations
2.3 Weak Approximations
2.3.1 The Weak Error Analysis
The Euler-Maruyama Scheme
2.3.2 Composition of Approximation Schemes and Operator Splitting
2.3.3 The Ninomiya-Victoir Scheme
Further Developments on the Ninomiya and Victoir Scheme
Chapter 3 Simulation of the CIR Process
3.1 Exact Simulation Methods
3.2 Discretization Schemes
3.2.1 Implicit Euler Schemes
3.2.2 Modified Explicit Euler Schemes
3.3 Weak Order Schemes
3.3.1 A Second Order Scheme
3.3.2 The Quadratic-Exponential (QE) Scheme
3.3.3 A Third Order Scheme
3.3.4 A Second Order Scheme for the CIR Process with Time-Dependent Parameters
3.3.5 Study of the Cauchy Problem for the CIR
3.4 Numerical Results
Chapter 4 The Heston Model and Multidimensional Affine Diffusions
4.1 Definition and Properties of Affine Diffusions
4.2 The Heston Model
4.2.1 The Characteristic Function
4.2.2 Pricing Formulas for the European Options
4.2.3 Pricing with the Fast Fourier Transform
4.2.4 Simulation Schemes for the Heston Model
A Potential Second Order Scheme for (log(St),Vt)
Simulations with the Potential Second Order Scheme for the Heston Model
4.2.5 Pricing and Simulation with PREMIA
4.2.6 The Exact Simulation Method and Derivative Schemes
The Broadie and Kaya BroadieKaya Method
The Glasserman and Kim Method GlassermanKim
4.3 Affine Term Structure Short Rate Models (ATSM)
4.3.1 The Dai and Singleton Parametrization
4.3.2 A Potential Second Order Scheme
Chapter 5 Wishart Processes and Affine Diffusions on Positive Semidefinite Matrices
5.1 Existence and Uniqueness Results
5.1.1 ItΓ΄ Calculus on Matrices
5.1.2 The Infinitesimal Generator on Md(R) and Sd(R)
5.1.3 Strong Existence and Uniqueness Results
5.1.4 Weak Existence and Uniqueness
5.2 The Characteristic Function
5.3 Some Useful Identities in Law
5.3.1 A Connection with Matrix-Valued Ornstein-Uhlenbeck Processes
5.4 Financial Modelling with Wishart Processes
5.5 Exact Simulation of Wishart Processes
5.5.1 A Remarkable Splitting for WISd(x,a,0,Ind)
5.5.2 Exact Simulation for WISd(x,a,0,e1d;t)
The Case d =2
The General Case
5.5.3 Exact Simulation for Wishart Processes
5.5.4 The Bartlett's Decomposition Revisited
5.6 High Order Discretization Schemes for Wishart and Semidefinite Positive Affine Processes
5.6.1 Study of the Cauchy Problem
5.6.2 High Order Schemes for Wishart Processes
5.6.3 Second Order Schemes for Affine Diffusions on Sd+(R)
5.6.4 A Faster Second Order Scheme for AFFd(x,a,B,a) When -da a Sd+(R)
5.7 Numerical Results on the Simulation Methods
5.7.1 Time Comparison Between the Different Algorithms
5.7.2 Numerical Results on the Convergence
5.7.3 An Application in Finance to the Gourieroux and Sufana Model
5.8 Technical Proofs
5.8.1 Proof of Theorem 5.6.3
5.8.2 Proof of Proposition 5.6.4
Chapter 6 Processes of Wright-Fisher Type
6.1 Wright-Fisher Processes
6.1.1 Affine Transformations
6.1.2 Moments and Density Transition
6.1.3 Connection with the CIR Process
6.1.4 Complementary Results on Squared Bessel Processes
6.1.5 A Second-Order Scheme
6.2 A Mean-Reverting Process on Correlation Matrices: Definition and First Properties
6.2.1 The Infinitesimal Generator
6.2.2 Calculation of Moments and the Ergodic Law
6.3 MRC and Wishart Processes
6.3.1 The Connection Between Elementary Processes
6.3.2 A Remarkable Splitting of the Infinitesimal Generator
6.3.3 A Link with the Multi-allele Wright-Fisher Model
6.4 Existence and Uniqueness Results for MRC Processes
6.4.1 Strong Existence and Uniqueness
6.4.2 Linear ODEs on Correlation Matrices
6.4.3 Weak Existence and Uniqueness
6.5 Second Order Discretization Schemes for MRC Processes
6.5.1 A Second-Order Scheme for MRC Processes
6.5.2 A Faster Second-Order Scheme for MRC Processes Under Assumption (6.66)
6.5.3 Numerical Experiments on the Discretization Schemes
Appendix A Some Results on Matrices
A.1 Some Basic Results
A.2 The Extended Cholesky Decomposition
A.3 Some Algebraic Results on Correlation Matrices
A.4 Matrix Riccati Differential Equations
Appendix B Simulation of a Gamma Random Variable
References
Index