Probability and Finance Theory

Kian Guan Lim

• Publisher: WSPC
• Year: 2015
• eBook ISBN: 9789814641951
• Edition: 2
• Topic: Economics
• Subtopic: Financial Engineering
• Index: Economics

This book is an introduction to the mathematical analysis of probability theory and provides some understanding of how probability is used to model random phenomena of uncertainty, specifically in the context of finance theory and applications. The integrated coverage of both basic probability theory and finance theory makes this book useful reading for advanced undergraduate students or for first-year postgraduate students in a quantitative finance course.

The book provides easy and quick access to the field of theoretical finance by linking the study of applied probability and its applications to finance theory all in one place. The coverage is carefully selected to include most of the key ideas in finance in the last 50 years.

The book will also serve as a handy guide for applied mathematicians and probabilists to easily access the important topics in finance theory and economics. In addition, it will also be a handy book for financial economists to learn some of the more mathematical and rigorous techniques so their understanding of theory is more rigorous. It is a must read for advanced undergraduate and graduate students who wish to work in the quantitative finance area.

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This book is an introduction to the mathematical analysis of probability theory and provides some understanding of how probability is used to model random phenomena of uncertainty, specifically in the context of finance theory and applications. The integrated coverage of both basic probability theory and finance theory makes this book useful reading for advanced undergraduate students or for first-year postgraduate students in a quantitative finance course.

The book provides easy and quick access to the field of theoretical finance by linking the study of applied probability and its applications to finance theory all in one place. The coverage is carefully selected to include most of the key ideas in finance in the last 50 years.

The book will also serve as a handy guide for applied mathematicians and probabilists to easily access the important topics in finance theory and economics. In addition, it will also be a handy book for financial economists to learn some of the more mathematical and rigorous techniques so their understanding of theory is more rigorous. It is a must read for advanced undergraduate and graduate students who wish to work in the quantitative finance area.

Request Inspection Copy

Readership: Advanced undergraduate students and 1st year post-graduate students in finance and economics, applied mathematicians, probabilists, financial economists.
Key Features:

• The book is a handy one for applied mathematicians and probabilists to easily access the important topics in finance theory and economics
• It is also a handy book for financial economists to learn some of the more mathematical and rigorous techniques so their understanding of theory is more rigorous
• It provides for very solid and useful learning for advanced undergraduate and graduate students who wish to work in the quantitative finance area

Chapter 1

PROBABILITY DISTRIBUTIONS

Probability is everywhere and is an important part of our lives. Most people make decisions based on the likelihood or chance, de facto probability, of one outcome versus others, that best enhances their well-being. For a more exact analytical modeling of financial problems and issues, a rigorous framework is required for studying this phenomenon of chance.

1.1 Basic Probability Concepts

A sample space Ω is the set of all possible simple outcomes of an experiment where each simple outcome or sample point ω_j is uniquely different from another, and each simple outcome is not a set containing other simple outcomes, i.e., not {ω₁, ω₂}, for example. An experiment could be anything happening with uncertainty in the outcomes, such as throwing of a dice in which case the sample space is Ω = {1, 2, 3, 4, 5, 6}, and ω_j = j for j = 1 or 2 or 3 or 4 or 5 or 6, or a more complicated case of investing in a portfolio of N stocks, in which case the sample space could be

where

denotes the return rate of the kth stock under the jth outcome. Another possible sample space could be

where

denotes the return rate of the equal-weighted portfolio P under the jth outcome.

Each simple outcome or sample point ω_j is also called an “atom” or “elementary event”. A more complicated outcome involving more than a sample point, such as {2, 4, 6} or “even numbers in a dice throw” which is a subset of Ω is called an event. Technically, a sample point {2} is also an event. Therefore, we shall use “events” as descriptions of outcomes, which may include the cases of sample points as events themselves.

As another example, in a simultaneous throw of two dices, the sample space consists of 36 sample points in the form (i, j) ∈ Ω where each i, j ∈ {1, 2, 3, 4, 5, 6}. An event could describe an outcome whereby the sum of the two numbers on the dices is larger than 8, in which case the event is said to happen if any of the following sample points or simple outcomes happen, (3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), and (6, 6). This event is described by the set {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)} ⊂ Ω. Another event could be an outcome whereby the sum of the two numbers on the dices is smaller than 4, in which case the event is described by {(1, 1), (1, 2), (2, 1)} ⊂ Ω. Another event could be “either the sum is larger than 10 or smaller than 4”, and represented by {(5, 6), (6, 5), (6, 6), (1, 1), (1, 2), (2, 1)} ⊂ Ω. Thus, an event is a subset of the sample space.

An event E is often a relevant and important description of what happens in an experiment, over and above sample points. In the November 2008 US presidential election, US citizens in each of the 50 states plus the District of Columbia voted by majority in each state for either the Democratic Party or the Republican Party. Each of the states plus DC is allocated a fixed number of electoral votes based on the population proportion. (Each electoral vote is rested on an elector, but electors in all but 2 states, Maine and Nebraska, cast their electoral college vote on a winner-takes-all basis from the result of the popular votes by the...

Table of contents

• Cover
• Halfitle
• Title
• Copyright
• Dedication
• Contents
• Preface to the Second Edition
• Preface to the First Edition
• From the First Edition
• About the Author
• Chapter 1: Probability Distributions
• Chapter 2: Conditional Probability
• Chapter 3: Laws of Probability
• Chapter 4: Theory of Risk and Utility
• Chapter 5: State Price and Risk-Neutral Probability
• Chapter 6: Single Period Asset Pricing Model
• Chapter 7: Stochastic Processes and Martingales
• Chapter 8: Dynamic Programming and Multi-period Asset Pricing
• Chapter 9: Continuous-Time Asset Pricing Model
• Chapter 10: Continuous-Time Option Pricing
• Chapter 11: Hedging and More Option Pricing
• Chapter 12: Brownian Motion and Technical Trading
• Chapter 13: Theory of Markov Chains and Credit Markets
• Chapter 14: Interest Rate Modeling and Derivatives
• Chapter 15: Risk Measures
• Appendix and Further Reading
• Index