Stochastic Finance: An Introduction with Market Examples, 1st Edition (Hardback) book cover

Stochastic Finance

An Introduction with Market Examples, 1st Edition

By Nicolas Privault

Chapman and Hall/CRC

441 pages | 104 B/W Illus.

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pub: 2013-12-20
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Description

Stochastic Finance: An Introduction with Market Examples presents an introduction to pricing and hedging in discrete and continuous time financial models without friction, emphasizing the complementarity of analytical and probabilistic methods. It demonstrates both the power and limitations of mathematical models in finance, covering the basics of finance and stochastic calculus, and builds up to special topics, such as options, derivatives, and credit default and jump processes. It details the techniques required to model the time evolution of risky assets.

The book discusses a wide range of classical topics including Black–Scholes pricing, exotic and American options, term structure modeling and change of numéraire, as well as models with jumps. The author takes the approach adopted by mainstream mathematical finance in which the computation of fair prices is based on the absence of arbitrage hypothesis, therefore excluding riskless profit based on arbitrage opportunities and basic (buying low/selling high) trading.

With 104 figures and simulations, along with about 20 examples based on actual market data, the book is targeted at the advanced undergraduate and graduate level, either as a course text or for self-study, in applied mathematics, financial engineering, and economics.

Reviews

"… a well-written book that manages to cover a wide range of topics in mathematical finance in a condensed way. … This is clearly something that lets the book stand out among other books on the subject. Another good feature of the book is that it provides a plethora of exercises at the end of each chapter. They are carefully chosen to practice the new concepts that were just introduced. … The book is aimed at advanced undergraduate or graduate students in applied mathematics, financial engineering, and economics. The level of the book is appropriate for the targeted audience. The concept of derivatives and their pricing is introduced well. … a useful book for getting a good overview about important concept in stochastic finance."

Journal of the American Statistical Association, March 2015

"This book gives an introduction to pricing and hedging in financial models for advanced undergraduate and graduate study in applied mathematics, financial engineering, and economics. It is written in a pedagogical tone, with an emphasis on complementarity between analytical and probabilistic methods. But the book is also a valuable reference for both academics and practitioners. … This book is written very well. … The textbook is successful in explaining difficult things plainly. … any reader of this book can acquire a more refined knowledge of option pricing and hedging strategies by computation of fair prices."

Mathematical Reviews, 2015

"… an elementary introduction to contemporary stochastic finance. … the theory is illustrated by numerical illustrations, including a large number of examples using actual market data."

Zentralblatt Math 1294

Table of Contents

Introduction

Assets, Portfolios and Arbitrage

Definitions and Formalism

Portfolio Allocation and Short-Selling

Arbitrage

Risk-Neutral Measures

Hedging of Contingent Claims

Market Completeness

Example

Exercises

Discrete-Time Model

Stochastic Processes

Portfolio Strategies

Arbitrage

Contingent Claims

Martingales and Conditional Expectation

Risk-Neutral Probability Measures

Market Completeness

Cox{Ross{Rubinstein (CRR) Market Model

Exercises

Pricing and Hedging in Discrete Time

Pricing of Contingent Claims

Hedging of Contingent Claims – Backward Induction

Pricing of Vanilla Options in the CRR Model

Hedging of Vanilla Options in the CRR model

Hedging of Exotic Options in the CRR Model

Convergence of the CRR Model

Exercises

Brownian Motion and Stochastic Calculus

Brownian Motion

Wiener Stochastic Integral

Itȏ Stochastic Integral

Deterministic Calculus

Stochastic Calculus

Geometric Brownian Motion

Stochastic Differential Equations

Exercises

The Black–Scholes PDE

Continuous-Time Market Model

Self-Financing Portfolio Strategies

Arbitrage and Risk-Neutral Measures

Market Completeness

Black–Scholes PDE

The Heat Equation

Solution of the Black–Scholes PDE

Exercises

Martingale Approach to Pricing and Hedging

Martingale Property of the Itȏ Integral

Risk-Neutral Measures

Girsanov Theorem and Change of Measure

Pricing by the Martingale Method

Hedging Strategies

Exercises

Estimation of Volatility

Historical Volatility

Implied Volatility

Black–Scholes Formula vs. Market Data

Local Volatility

Exotic Options

Generalities

Reexion Principle

Barrier Options

Lookback Options

Asian Options

Exercises

Contents vii

American Options

Filtrations and Information Flow

Martingales, Submartingales, and Supermartingales

Stopping Times

Perpetual American Options

Finite Expiration American Options

Exercises

Change of Numéraire and Forward Measures

Notion of Numéraire

Change of Numéraire

Foreign Exchange

Pricing of Exchange Options

Self-Financing Hedging by Change of Numéraire

Exercises

Forward Rate Modeling

Short-Term Models

Zero-Coupon Bonds

Forward Rates

HJM Model

Forward Vasicek Rates

Modeling Issues

BGM Model

Exercises

Pricing of Interest Rate Derivatives

Forward Measures and Tenor Structure

Bond Options

Caplet Pricing

Forward Swap Measures

Swaption Pricing on the LIBOR

Exercises

Default Risk in Bond Markets

Survival Probabilities and Failure Rate

Stochastic Default

Defaultable Bonds

Credit Default Swaps

Exercises

Stochastic Calculus for Jump Processes

Poisson Process

Compound Poisson Processes

Stochastic Integrals with Jumps

Itȏ Formula with Jumps

Stochastic Differential Equations with Jumps

Girsanov Theorem for Jump Processes

Exercises

Pricing and Hedging in Jump Models

Risk-Neutral Measures

Pricing in Jump Models

Black–Scholes PDE with Jumps

Exponential Models

Self-Financing Hedging with Jumps

Exercises

Basic Numerical Methods

The Heat Equation

Black–Scholes PDE

Euler Discretization

Milshtein Discretization

Appendix: Background on Probability Theory

Probability Spaces and Events

Probability Measures

Conditional Probabilities and Independence

Random Variables

Probability Distributions

Expectation of a Random Variable

Conditional Expectation

Moment Generating Functions

Exercises

Bibliography

Index

About the Series

Chapman and Hall/CRC Financial Mathematics Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
BUS027000
BUSINESS & ECONOMICS / Finance
MAT000000
MATHEMATICS / General
MAT029000
MATHEMATICS / Probability & Statistics / General