There are not many calculus books that are very accessible to students without a strong mathematical background and the large majority of financial derivatives students do not have a strong quantitative background. This book provides a short introduction to the subject with examples of its use in mathematical finance e.g pricing of derivatives. Wiersma assumes only a basic knowledge of calculus and probability and guides the student through the book with examples and exercises (complemented by the website/disk). Wiersma has been teaching the subject for many years and the book will be based on his tried and tested course notes.
Stochastic (or Brownian motion) calculus provides the formal framework and quantitative tools necessary for modeling stochastic processes (random functions, such as market fluctuations) and has found wide application in financial mathematics for modeling the evolution in time of stock and bond prices. Brownian Motion Calculus provides a short, clear introduction to this subject for readers without a strong mathematical background. Readers with only a basic knowledge of calculus and probability will be able to follow the book's examples and exercises (enhanced with a Web site and disk).
Preface.1 Brownian Motion.
1.1 Origins.
1.2 Brownian Motion Specification.
1.3 Use of Brownian Motion in Stock Price Dynamics.
1.4 Construction of Brownian Motion from a Symmetric Random Walk.
1.5 Covariance of Brownian Motion.
1.6 Correlated Brownian Motions.
1.7 Successive Brownian Motion Increments.
1.8 Features of a Brownian Motion Path.
1.9 Exercises.
1.10 Summary.
2 Martingales.
2.1 Simple Example.
2.2 Filtration.
2.3 Conditional Expectation.
2.4 Martingale Description.
2.5 Martingale Analysis Steps.
2.6 Examples of Martingale Analysis.
2.7 Process of Independent Increments.
2.8 Exercises.
2.9 Summary.
3 Itō Stochastic Integral.
3.1 How a Stochastic Integral Arises.
3.2 Stochastic Integral for Non-Random Step-Functions.
3.3 Stochastic Integral for Non-Anticipating Random Step-Functions.
3.4 Extension to Non-Anticipating General Random Integrands.
3.5 Properties of an Itō Stochastic Integral.
3.6 Significance of Integrand Position.
3.7 Itō integral of Non-Random Integrand.
3.8 Area under a Brownian Motion Path.
3.9 Exercises.
3.10 Summary.
3.11 A Tribute to Kiyosi Itō.
Acknowledgment.
4 Itō Calculus.
4.1 Stochastic Differential Notation.
4.2 Taylor Expansion in Ordinary Calculus.
4.3 Itō’s Formula as a Set of Rules.
4.4 Illustrations of Itō’s Formula.
4.5 Lévy Characterization of Brownian Motion.
4.6 Combinations of Brownian Motions.
4.7 Multiple Correlated Brownian Motions.
4.8 Area under a Brownian Motion Path – Revisited.
4.9 Justification of Itō’s Formula.
4.10 Exercises.
4.11 Summary.
5 Stochastic Differential Equations.
5.1 Structure of a Stochastic Differential Equation.
5.2 Arithmetic Brownian Motion SDE.
5.3 Geometric Brownian Motion SDE.
5.4 Ornstein–Uhlenbeck SDE.
5.5 Mean-Reversion SDE.
5.6 Mean-Reversion with Square-Root Diffusion SDE.
5.7 Expected Value of Square-Root Diffusion Process.
5.8 Coupled SDEs.
5.9 Checking the Solution of a SDE.
5.10 General Solution Methods for Linear SDEs.
5.11 Martingale Representation.
5.12 Exercises.
5.13 Summary.
6 Option Valuation.
6.1 Partial Differential Equation Method.
6.2 Martingale Method in One-Period Binomial Framework.
6.3 Martingale Method in Continuous-Time Framework.
6.4 Overview of Risk-Neutral Method.
6.5 Martingale Method Valuation of Some European Options.
6.6 Links between Methods.
6.6.1 Feynman-Kač Link between PDE Method and Martingale Method.
6.6.2 Multi-Period Binomial Link to Continuous.
6.7 Exercise.
6.8 Summary.
7 Change of Probability.
7.1 Change of Discrete Probability Mass.
7.2 Change of Normal Density.
7.3 Change of Brownian Motion.
7.4 Girsanov Transformation.
7.5 Use in Stock Price Dynamics – Revisited.
7.6 General Drift Change.
7.7 Use in Importance Sampling.
7.8 Use in Deriving Conditional Expectations.
7.9 Concept of Change of Probability.
7.10 Exercises.
7.11 Summary.
8 Numeraire.
8.1 Change of Numeraire.
8.2 Forward Price Dynamics.
8.3 Option Valuation under most Suitable Numeraire.
8.4 Relating Change of Numeraire to Change of Probability.
8.5 Change of Numeraire for Geometric Brownian Motion.
8.6 Change of Numeraire in LIBOR Market Model.
8.7 Application in Credit Risk Modelling.
8.8 Exercises.
8.9 Summary.
ANNEXES.
A Annex A: Computations with Brownian Motion.
A.1 Moment Generating Function and Moments of Brownian Motion.
A.2 Probability of Brownian Motion Position.
A.3 Brownian Motion Reflected at the Origin.
A.4 First Passage of a Barrier.
A.5 Alternative Brownian Motion Specification.
B Annex B: Ordinary Integration.
B.1 Riemann Integral.
B.2 Riemann–Stieltjes Integral.
B.3 Other Useful Properties.
B.4 References.
C Annex C: Brownian Motion Variability.
C.1 Quadratic Variation.
C.2 First Variation.
D Annex D: Norms.
D.1 Distance between Points.
D.2 Norm of a Function.
D.3 Norm of a Random Variable.
D.4 Norm of a Random Process.
D.5 Reference.
E Annex E: Convergence Concepts.
E.1 Central Limit Theorem.
E.2 Mean-Square Convergence.
E.3 Almost Sure Convergence.
E.4 Convergence in Probability.
E.5 Summary.
Answers to Exercises.
References.
Index.