Format:
Online-Ressource (XII, 128 p. 5 illus, online resource)
ISBN:
9783642405266
Series Statement:
SpringerBriefs in Statistics
Content:
Preface -- 1.Preliminaries -- 2.Telegraph Process on the Line -- 3.Functionals of Telegraph Process -- 4.Asymmetric Jump-Telegraph Processes -- 5.Financial Modelling and Option Pricing -- Index. .
Content:
The telegraph process is a useful mathematical model for describing the stochastic motion of a particle that moves with finite speed on the real line and alternates between two possible directions of motion at random time instants. That is why it can be considered as the finite-velocity counterpart of the classical Einstein-Smoluchowski's model of the Brownian motion in which the infinite speed of motion and the infinite intensity of the alternating directions are assumed. The book will be interesting to specialists in the area of diffusion processes with finite speed of propagation and in financial modelling. It will also be useful for students and postgraduates who are taking their first steps in these intriguing and attractive fields.
Note:
Description based upon print version of record
,
Preface; Contents; 1 Preliminaries; 1.1 One-Dimensional Markov Processes; 1.2 Brownian Motion and Diffusion on R; 1.3 Stochastic Integrals and Itô's Formula; 1.4 Poisson Process; 1.5 Modified Bessel Functions; 1.6 Generalised Functions and Integral Transforms; References; 2 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 Convergence to Brownian Motion; 2.7 Laplace Transforms; References; 3 Functionals of Telegraph Process
,
3.1 Motions with Barriers3.1.1 Telegraph Process with Reflecting Barrier; 3.1.2 Telegraph Process with Absorbing Barrier; 3.2 Occupation Time Distributions; 3.2.1 Feynmann-Kac Connection; 3.2.2 Statement of the Main Result; 3.2.3 Proof of Theorems 3.6 and 3.7; 3.3 First Passage Time; References; 4 Asymmetric Jump-Telegraph Processes; 4.1 Generalised Jump-Telegraph Processes; 4.1.1 Transition Densities; 4.1.2 Expectations and Variances Jump-Telegraph Martingales; 4.1.3 Change of Measure for Jump-Telegraph Processes; 4.2 Moments; References; 5 Financial Modelling and Option Pricing
,
5.1 Hedging Strategies and Option Pricing5.1.1 Option Pricing, Hedging and Martingales; 5.1.2 Black-Scholes Model and Girsanov's Theorem; 5.2 Market Model Based on Jump-Telegraph Processes; 5.3 Diffusion Rescaling and Natural Volatility; 5.4 Fundamental Equation and Perfect Hedging; 5.5 Pricing Call Options; 5.6 Historical and Implied Volatilities in the Jump Telegraph Model; 5.6.1 Historical Volatility; 5.6.2 Implied Volatility and Numerical Results; 5.7 Pricing Exotic Options; References; Index
Additional Edition:
ISBN 9783642405259
Additional Edition:
Druckausg. Kolesnik, Alexander D. Telegraph processes and option pricing Heidelberg [u.a.] : Springer, 2013 ISBN 3642405258
Additional Edition:
ISBN 9783642405259
Language:
English
Subjects:
Economics
,
Mathematics
Keywords:
Optionspreistheorie
;
Telegrafenprozess
;
Stochastischer Prozess
;
Optionspreistheorie
DOI:
10.1007/978-3-642-40526-6
URL:
Volltext
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