Ubbo F.Wiersema – Brownian Motion Calculus

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Description

Brownian Motion Calculus teaches the fundamentals of stochastic calculus with an emphasis on financial derivative valuation. It is meant to provide a simple introduction to technical literature. There is a clear divide between the mathematics that is useful for a first introduction and the more rigorous foundations that are better learned from the specified technical sources. The presence of completely worked-out activities makes the book appealing for self-study. The requirements are standard probability theory and ordinary calculus. The book website contains summary slides for revision and teaching.

ABOUT THE AUTHOR UBBO WIERSEMA earned degrees in Applied Mathematics from the University of Delft, Operations Research from the University of California, Berkeley, and Financial Economics and Financial Mathematics from the London School of Economics. In 1997, he joined The Business School for Financial Markets (the ICMA Centre) at the University of Reading, UK, to create and teach its Quantitative Finance curriculum. Prior to that, he worked as a derivatives mathematician in the City of London merchant firm Robert Fleming. Prior to that, he worked in Operations Research in the United States and Europe.
TABLE OF MATERIALS

Preface.

1 Brownian Movement

1.1 The Beginning.

Brownian Motion Specification 1.2

1.3 Application of Brownian Motion in Stock Price Dynamics

1.4 Brownian Motion Constructed from a Symmetric Random Walk

Brownian Motion Covariance 1.5

1.6 Brownian Motions with Correlated Motions

1.7 Increments in Brownian Motion.

1.8 Characteristics of a Brownian Motion Path

1.9 Exercising

1.10 Conclusion

Martingales 2

2.1 An Easy Example

Filtration (2.2).

2.3 Constrained Expectation

2.4 Martingale Formula

Steps for Martingale Analysis 2.5

2.6 Martingale Analysis Examples

2.7 Independent Increment Process

2.8 Exercising

2.9 Conclusion.

3 It is the Stochastic Integral.

3.1 How Does a Stochastic Integral Form?

3.2 Non-Random Step-Function Stochastic Integral

Stochastic Integral for Non-Anticipating Random Step Functions

3.4 General Random Integrands with Non-Anticipating Integrands

3.5 It Stochastic Integral Properties

3.6 Importance of Integrand Position

It is the integral of the Non-Random Integrand.

3.8 Area covered by a Brownian Motion Path

3.9 Exercising

3.10 Conclusion

3.11 Kiyosi It’s Tribute

Acknowledgment.

Calculus 4 It.

Stochastic Differential Notation (4.1).

Ordinary Calculus 4.2 Taylor Expansion

It is Formula as a System of Rules.

4.4 Exemplifications of Its Formula

4.5 Brownian Motion Lévy Characterization

4.6 Brownian Motion Combinations

Multiple Correlated Brownian Motions (4.7).

Revised version of 4.8 Area under a Brownian Motion Path.

Justification for Its Formula

4.10 Workouts

4.11 Conclusion

5 Differential Stochastic Equations

5.1 Stochastic Differential Equation Structure

SDE for Arithmetic Brownian Motion.

SDE for Geometric Brownian Motion.

SDE Ornstein-Uhlenbeck 5.4

5.5 SDE for Mean-Reversion.

Mean-Reversion SDE with Square-Root Diffusion

5.7 Square-Root Diffusion Process Expected Value

5.8 SDEs that are coupled

5.9 Validating an SDE Solution

5.10 Linear SDE General Solution Methods

Martingale Representation (5.11).

5.12 Workouts

5.13 Conclusion

6 Option Pricing

6.1 Method of Partial Differential Equation

Martingale Method in the One-Period Binomial Framework

Martingale Method in a Continuous-Time Framework

6.4 Risk-Neutral Method Overview

Some European Options Valued Using the Martingale Method.

6.6 Method Connections

Feynman-Ka, 6.6.1 PDE Method and Martingale Method are linked.

6.6.2 Multi-Period Binomial to Continuous Link

6.7 Workout.

6.8 Conclusion

7 Probability Change

7.1 Discrete Probability Mass Change

7.2 Normal Density Change

7.3 Brownian Motion Modification.

Girsanov Transformation 7.4

7.5 Revisited Use in Stock Price Dynamics

7.6 Change in General Drift

7.7 Application in Importance Sampling

7.8 Derivation of Conditional Expectations

7.9 Probability Change Concept

7.10 Workouts

7.11 Conclusion.

Numeraire 8

8.1 Numeraire Change

8.2 Price Dynamics in the Future

8.3 Option Valuation Using the Most Appropriate Numeraire

8.4 Relating Numeraire Change to Probability Change

8.5 Numeraire Modification for Geometric Brownian Motion

8.6 Numeraire Change in the LIBOR Market Model

8.7 Credit Risk Modeling Application

8.8 Exercising

8.9 Conclusion

ANNEXES.

Annex A: Brownian Motion Computations

A.1 Moment Generating Function and Brownian Motion Moments

A.2 Brownian Motion Position Probability

A.3 Brownian Motion As Seen at the Origin

A.4 First Barrier Passage

A.5 Brownian Motion Alternative Specification

Ordinary Integration (Annex B).

Riemann Integral (B.1)

B.2 Integral of Riemann-Stieltjes

B.3 Additional Useful Properties

B.4 Bibliography.

Brownian Motion Variability (Annex C).

Quadratic Variation (C.1)

C.2 The First Variation

Norms are listed in Annex D.

D.1 Point-to-Point Distance

D.2 A Function’s Norm

D.3 Random Variable Norm.

D.4 Random Process Norm.

D.5 Citation.

Convergence Concepts (Annex E).

E.1 Theorem of the Central Limit.

Mean-Square Convergence (E.2)

E.3 Almost Certain Convergence

E.4 Probability Convergence

E.5 Conclusion.

Exercise solutions

References.

Index.

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