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Stochastic Finance
An Introduction with Market Examples




ISBN 9781466594029
Published December 20, 2013 by Chapman and Hall/CRC
441 Pages - 104 B/W Illustrations

 
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Book 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.

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

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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

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