Telegraph Processes and Option Pricing

Description

This book provides an extensive, systematic overview of the modern theory of telegraph processes and their multidimensional counterparts, together with numerous fruitful applications in financial modelling. Focusing on stochastic processes of bounded variation instead of classical diffusion, or more generally, Lévy processes, has two obvious benefits. First, the mathematical technique is much simpler, which helps to concentrate on the key problems of stochastic analysis and applications, including financial market modelling. Second, this approach overcomes some shortcomings of the (parabolic) nature of classical diffusions that contradict physical intuition, such as infinite propagation velocity and infinite total variation of paths. In this second edition, some sections of the previous text are included without any changes, while most others have been expanded and significantly revised. These are supplemented by predominantly new results concerning piecewise linear processes with arbitrary sequences of velocities, jump amplitudes, and switching intensities. The chapter on functionals of the telegraph process has been significantly expanded by adding sections on exponential functionals, telegraph meanders and running extrema, the times of the first passages of telegraph processes with alternating random jumps, and distribution of the Euclidean distance between two independent telegraph processes. A new chapter on the multidimensional counterparts of the telegraph processes is also included. The book is intended for graduate students in mathematics, probability, statistics and quantitative finance, and for researchers working at academic institutions, in industry and engineering. It can also be used by university lecturers and professionals in various applied areas. Preface to the Second Edition Preface to the First Edition Contents About the Authors 1 Preliminaries 1.1 Hypergeometric Functions 1.2 Modified Bessel Functions 1.3 Generalised Functions and Integral Transforms 1.4 One-Dimensional Markov Processes 1.5 Brownian Motion and Diffusion on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 1.6 Stochastic Integrals and Itô's Formula 1.7 Poisson Process, Exponential, and HypoexponentialDistributions 2 Symmetric Telegraph Process on the Line 2.1 Definition of Process and the Structure of Distribution 2.2 Kolmogorov Equations 2.3 Telegraph Equation 2.4 Characteristic Function 2.5 Transition Density 2.6 Probability Distribution Function 2.7 Convergence to Brownian Motion 2.8 Laplace Transforms Notes 3 Asymmetric Jump-Telegraph Processes 3.1 Asymmetric Continuous Telegraph Process 3.1.1 Generator and Transition Densities 3.1.2 Moment Generating Functions 3.1.3 Moments 3.2 Self-Exciting Piecewise Linear Continuous Processes 3.2.1 Piecewise Constant Process 3.2.2 Piecewise Linear Process 3.2.3 First Passage Time 3.3 Telegraph Processes with Alternating Deterministic Jumps 3.3.1 Transition Densities 3.3.2 Expectations and Variances, Jump-TelegraphMartingales 3.3.3 Change of Measure for Jump-Telegraph Processes 3.4 Telegraph Processes with Alternating Random Jumps 3.5 Short Memory Telegraph Processes with Random Jumps 3.5.1 Jump-Telegraph Processes with Parameters Depending on the Past 3.5.2 Martingales 3.6 Piecewise Linear Process with Renewal Starting Points 3.6.1 Distributions of X(t) 3.6.2 First Passage Time 3.6.2.1 Positive Velocities 3.6.2.2 Velocities of Opposite Signs 3.7 Double Telegraph Processes 3.7.1 Doubly Stochastic Poisson Process 3.7.2 Doubly Stochastic Telegraph Process with Jumps 3.7.3 Girsanov Transformation 3.8 Jump-Telegraph Processes with Poisson-Modulated Exponential Switching Times 3.8.1 Building a Model 3.8.2 Poisson-Modulated Exponential Distribution 3.8.3 Example: Poisson-Modulated Exponential Distributions with Linearly Increasing Switching Intensities 3.8.4 Piecewise Linear Process with Two Alternating Poisson Modulated Patterns and a Double Jump Component 3.9 Piecewise Deterministic Processes Following Two Alternating Patterns 3.9.1 Piecewise Linear Processes in the Linear NormedSpace 3.9.2 Time-Homogeneous Piecewise Deterministic Process 3.9.3 Examples 3.9.3.1 Squared Telegraph Process 3.9.3.2 A Random Motion in a Plane and Polar Coordinates 3.9.4 Self-Similarity 4 Jump-Diffusion Processes with Regime Switching 4.1 The Basic Model: Markov Modulated Jump-DiffusionProcesses 4.2 Generalised Jump-Telegraph-Diffusion Processes, Distributions, and Expectations 4.2.1 The Definition of Jump-Telegraph-Diffusion Process 4.2.2 The Distribution and Mean of X(t) 4.3 Girsanov's Transformation 4.3.1 Martingality Criterion 4.3.2 Girsanov's Theorem 4.3.3 Relative Entropy 4.4 Equivalent Martingale Measure 4.5 Esscher Transform and Minimum Entropy Martingale Measure 5 Functionals of Telegraph Process 5.1 Motions with Barriers 5.1.1 Telegraph Process with Reflecting Barrier 5.1.2 Telegraph Process with Absorbing Barrier 5.2 Occupation Time Distributions 5.2.1 Feynman–Kac Connection 5.2.2 Statement of the Main Result 5.2.3 Proof of Theorems 5.6 and 5.7 5.3 Exponential Functional 5.4 First Passage Time for the Continuous Telegraph Process 5.5 Telegraphic Meanders and Running Extrema 5.6 First Passage Times for Telegraph Processes with Alternating Random Jumps 5.6.1 Negative Jumps 5.6.2 Double Exponential Jumps 5.6.3 Numerical Analysis 5.6.4 Examples with Symmetric Parameters 5.7 Distance Between Two Telegraph Processes 5.7.1 Probability Distribution Function 5.7.2 Numerical Example Notes 6 Telegraph-Type Processes in Higher Dimensions 6.1 Definition of the Process and Structure of Distribution 6.2 Integral Transforms of the Distribution 6.2.1 Recurrent Relations 6.2.2 Conditional Characteristic Functions 6.2.3 Volterra Integral Equation for Characteristic Function 6.2.4 Limit Theorem 6.2.5 Asymptotic Relation for Transition Density 6.2.6 Integral Equation for Transition Density 6.2.7 Hyperparabolic Operators 6.3 Stochastic Motion in the Plane R2 6.4 Stochastic Motion in the Space R3 6.5 Stochastic Motion in the Space R4 6.6 Stochastic Motion in the Space R6 7 Financial Modelling Based on Telegraph Processes 7.1 Hedging Strategies and Option Pricing 7.1.1 Option Pricing, Hedging, and Martingales 7.1.2 Black–Scholes Model and Girsanov's Theorem 7.2 Market Model Based on Jump-Telegraph Processes 7.3 Diffusion Rescaling and Natural Volatility 7.4 Fundamental Equation and Perfect Hedging 7.5 Pricing Call Options 7.6 Historical and Implied Volatilities in the Jump-Telegraph Model 7.6.1 Historical Volatility 7.6.2 Implied Volatility and Numerical Results 7.7 Pricing Exotic Options 7.8 Short Memory Telegraph Processes with Random Jumps and the Market Model 7.9 Double Telegraph Processes and Complete Market Models 7.9.1 Martingale Measures 7.9.2 The Double Telegraph Market Model 7.10 Self-Exciting Piecewise Linear Processes with Jumps 7.10.1 Jumps Are Added 7.10.2 Processes of Two Alternating Self-Exciting Patterns and with a Double Jump Component 7.10.3 Market Model Sketch 7.11 Market model based on jump-telegraph processes with Poisson-modulated exponential switching times 7.12 Diffusion-Telegraph Processes and Market Models 7.12.1 The Market Model 7.12.2 Girsanov's Theorem, Relative Entropy 7.12.3 The Option Pricing 7.12.4 The Two-State Markov Model and Numerical Study References Index