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Foreword; Preface; Contents; Contributors; Part I Modelling; 1 Nonlinear Parabolic Equations Arising in Mathematical Finance; 1.1 Nonlinear Generalization of the Black-Scholes Equation for Pricing Financial Instruments; 1.2 Nonlinear Hamilton-Jacobi-Bellman Equation and Optimal Allocation Problems; 1.3 Existence of Classical Solutions, Comparison Principle; 1.4 Numerical Full Space-Time Discretization Scheme for Solving the Gamma Equation; 1.5 Numerical Results for the Nonlinear Model with Variable Transaction Costs; References; 2 Modeling of Herding and Wealth Distribution in Large Markets
2.1 Introduction2.2 A Cross-Diffusion Herding Model; 2.2.1 Existence of Solutions; 2.2.2 Bifurcation Analysis; 2.3 A Kinetic Model with Irrationality and Herding; 2.3.1 Public Information and Herding; 2.3.2 Grazing Collision Limit; 2.3.3 Existence of Weak Solutions; 2.3.4 Numerical Simulations; 2.4 A Kinetic Model with Wealth and Knowledge Exchanges; 2.4.1 Existence of Solutions; 2.4.2 Numerical Simulations; References; 3 Indifference Pricing in a Market with Transaction Costsand Jumps; 3.1 Introduction; 3.2 The Model; 3.2.1 Portfolio Dynamics; 3.2.2 Utility Maximization
3.2.3 Indifference Price3.2.4 Variable Reduction; 3.3 The Algorithm; 3.4 Numerical Results; 3.4.1 Brownian Motion; 3.4.2 Variance Gamma; References; 4 Negative Rates: New Market Practice; 4.1 Introduction; 4.1.1 Bachelier Model; 4.1.2 Displaced Diffusion Model: Shifted Lognormal Model; 4.2 The Free Boundary SABR Model; 4.2.1 The Parameters; 4.2.2 Applicability; 4.3 Approximation Formulae; 4.3.1 Approximation 1; 4.3.2 Approximation 2; 4.4 Numerical Results; 4.4.1 Approximations vs Integration; 4.4.2 Calibration; 4.5 Conclusions; References; 5 Accurate Vega Calculation for Bermudan Swaptions
5.1 Financial Models and Algorithmic Differentiation5.1.1 Financial Models and Sensitivities; 5.1.1.1 Evaluating Sensitivities; 5.1.2 Algorithmic Differentiation at a Glance; 5.2 Pricing Bermudan Swaptions with a Hull White Model; 5.2.1 Market Formulas for European Swaptions; 5.2.2 Analytical Pricing Formulas for the Hull White Model; 5.2.3 Pricing Bermudan Swaptions; 5.3 Pricing and Vega Calculation Example; 5.3.1 Implementation and Computational Costs; References; 6 Modelling and Calibration of Stochastic Correlation in Finance; 6.1 Introduction; 6.2 Stochastic Correlation Models
6.3 A General Stochastic Correlation Process6.3.1 The Transformed Mean-Reverting Process; 6.3.2 The van Emmerich's Correlation Model; 6.3.3 The Transformed Modified Ornstein-Uhlenbeck Process; 6.4 Calibration Via Density Function; 6.4.1 Transition Density Function; 6.4.2 Calibration; 6.5 Pricing Quantos with Stochastic Correlation; 6.6 Numerical Results; 6.7 Conclusions; References; Part II Analysis; 7 Lie Group Analysis of Nonlinear Black-Scholes Models; 7.1 Economical Setting of the Optimization Problem for a Portfolio with an Illiquid Asset with a Given Liquidation Time Distribution
2.1 Introduction2.2 A Cross-Diffusion Herding Model; 2.2.1 Existence of Solutions; 2.2.2 Bifurcation Analysis; 2.3 A Kinetic Model with Irrationality and Herding; 2.3.1 Public Information and Herding; 2.3.2 Grazing Collision Limit; 2.3.3 Existence of Weak Solutions; 2.3.4 Numerical Simulations; 2.4 A Kinetic Model with Wealth and Knowledge Exchanges; 2.4.1 Existence of Solutions; 2.4.2 Numerical Simulations; References; 3 Indifference Pricing in a Market with Transaction Costsand Jumps; 3.1 Introduction; 3.2 The Model; 3.2.1 Portfolio Dynamics; 3.2.2 Utility Maximization
3.2.3 Indifference Price3.2.4 Variable Reduction; 3.3 The Algorithm; 3.4 Numerical Results; 3.4.1 Brownian Motion; 3.4.2 Variance Gamma; References; 4 Negative Rates: New Market Practice; 4.1 Introduction; 4.1.1 Bachelier Model; 4.1.2 Displaced Diffusion Model: Shifted Lognormal Model; 4.2 The Free Boundary SABR Model; 4.2.1 The Parameters; 4.2.2 Applicability; 4.3 Approximation Formulae; 4.3.1 Approximation 1; 4.3.2 Approximation 2; 4.4 Numerical Results; 4.4.1 Approximations vs Integration; 4.4.2 Calibration; 4.5 Conclusions; References; 5 Accurate Vega Calculation for Bermudan Swaptions
5.1 Financial Models and Algorithmic Differentiation5.1.1 Financial Models and Sensitivities; 5.1.1.1 Evaluating Sensitivities; 5.1.2 Algorithmic Differentiation at a Glance; 5.2 Pricing Bermudan Swaptions with a Hull White Model; 5.2.1 Market Formulas for European Swaptions; 5.2.2 Analytical Pricing Formulas for the Hull White Model; 5.2.3 Pricing Bermudan Swaptions; 5.3 Pricing and Vega Calculation Example; 5.3.1 Implementation and Computational Costs; References; 6 Modelling and Calibration of Stochastic Correlation in Finance; 6.1 Introduction; 6.2 Stochastic Correlation Models
6.3 A General Stochastic Correlation Process6.3.1 The Transformed Mean-Reverting Process; 6.3.2 The van Emmerich's Correlation Model; 6.3.3 The Transformed Modified Ornstein-Uhlenbeck Process; 6.4 Calibration Via Density Function; 6.4.1 Transition Density Function; 6.4.2 Calibration; 6.5 Pricing Quantos with Stochastic Correlation; 6.6 Numerical Results; 6.7 Conclusions; References; Part II Analysis; 7 Lie Group Analysis of Nonlinear Black-Scholes Models; 7.1 Economical Setting of the Optimization Problem for a Portfolio with an Illiquid Asset with a Given Liquidation Time Distribution