Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessibleintroduction to the technical literature. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references. The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website. UBBO WIERSEMA was educated in Applied Mathematics at Delft, in Operations Research at Berkeley, and in Financial Economics and Financial Mathematics at the London School of Economics. He joined The Business School for Financial Markets (the ICMA Centre) at the University of Reading, UK, in 1997, to develop and teach its curriculum in Quantitative Finance. Prior to that, he was a derivatives mathematician at the merchant bank Robert Fleming in the City of London. Before that his career was focused in Operations Research in the US and Europe. INDICE: Preface1 Brownian Motion1.1 Origins1.2 Brownian Motion Specification1.3 Use of Brownian Motion in Stock Price Dynamics1.4 Construction of Brownian Motion from a Symmetric Random Walk1.5 Covariance of Brownian Motion1.6 Correlated Brownian Motions 1.7 Successive Brownian Motion Increments 1.8 Features of a Brownian Motion Path 1.9 Exercises1.10 Summary2 Martingales2.1 Simple Example2.2 Filtration2.3 Conditional Expectation 2.4 Martingale Description2.5 Martingale Analysis Steps2.6 Examples of Martingale Analysis2.7 Process of Independent Increments2.8 Exercises2.9 Summary3 Itō Stochastic Integral 3.1 How a Stochastic Integral Arises3.2 Stochastic Integral for Non-Random Step-Functions3.3 Stochastic Integral for Non-Anticipating Random Step-Functions 3.4 Extension to Non-Anticipating General Random Integrands3.5 Properties of an Itō Stochastic Integral 3.6 Significance of Integrand Position3.7 Itō integral of Non-Random Integrand3.8 Area under a Brownian Motion Path3.9 Exercises3.10 Summary3.11 A Tribute to Kiyosi Itō Acknowledgment4 Itō Calculus4.1 Stochastic Differential Notation4.2 Taylor Expansion in Ordinary Calculus4.3 Itō’s Formula as a Set of Rules4.4 Illustrations of Itō’s Formula4.5 Lévy Characterization of Brownian Motion4.6 Combinations of Brownian Motions4.7 Multiple Correlated Brownian Motions4.8 Area under a Brownian Motion Path – Revisited4.9 Justification of Itō’s Formula4.10 Exercises4.11 Summary5 Stochastic Differential Equations5.1 Structure of a Stochastic Differential Equation5.2 Arithmetic Brownian Motion SDE 5.3 Geometric Brownian Motion SDE5.4 Ornstein–Uhlenbeck SDE5.5 Mean-Reversion SDE5.6 Mean-Reversion with Square-Root Diffusion SDE5.7 Expected Value of Square-Root Diffusion Process5.8 Coupled SDEs5.9 Checking the Solution of a SDE5.10 General Solution Methods for Linear SDEs5.11 Martingale Representation5.12 Exercises5.13 Summary6 Option Valuation6.1 Partial Differential Equation Method6.2 Martingale Method in One-Period Binomial Framework6.3 Martingale Method in Continuous-Time Framework6.4 Overview of Risk-Neutral Method 6.5 Martingale Method Valuation of Some European Options6.6 Links between Methods6.6.1 Feynman-Kač Link between PDE Method and Martingale Method6.6.2 Multi-Period Binomial Link to Continuous6.7 Exercise6.8 Summary 7 Change of Probability7.1 Change of Discrete Probability Mass7.2 Change of Normal Density 7.3 Change of Brownian Motion 7.4 Girsanov Transformation7.5 Use in Stock Price Dynamics – Revisited7.6 General Drift Change7.7 Use in Importance Sampling7.8 Use in Deriving Conditional Expectations7.9 Concept of Change of Probability7.10 Exercises7.11 Summary8 Numeraire8.1 Change of Numeraire8.2 Forward Price Dynamics8.3 Option Valuation under most Suitable Numeraire 8.4 Relating Change of Numeraire to Change of Probability 8.5 Change of Numeraire for Geometric Brownian Motion8.6 Change of Numeraire in LIBOR Market Model8.7 Application in Credit Risk Modelling8.8 Exercises8.9 SummaryANNEXESA Annex A: Computations with Brownian MotionA.1 Moment Generating Function and Moments of Brownian MotionA.2 Probability of Brownian Motion PositionA.3 Brownian Motion Reflected at the OriginA.4 First Passage of a BarrierA.5 Alternative Brownian Motion SpecificationB Annex B: Ordinary IntegrationB.1 Riemann IntegralB.2 Riemann–Stieltjes IntegralB.3 Other Useful PropertiesB.4 ReferencesC Annex C: Brownian Motion VariabilityC.1 Quadratic VariationC.2 First VariationD Annex D: Norms D.1 Distance between Points D.2 Norm of a FunctionD.3 Norm of a Random VariableD.4 Norm of a Random ProcessD.5 ReferenceE Annex E: Convergence ConceptsE.1 Central Limit Theorem E.2 Mean-Square Convergence E.3 Almost Sure ConvergenceE.4 Convergence in ProbabilityE.5 SummaryAnswers to ExercisesReferencesIndex
- ISBN: 978-0-470-02170-5
- Editorial: John Wiley & Sons
- Encuadernacion: Rústica
- Páginas: 330
- Fecha Publicación: 01/04/2008
- Nº Volúmenes: 1
- Idioma: Inglés