Introduction to Stochastic Calculus with Applications, Second Edition
Containing many solved examples and exercises, this book gives a simple but rigorous treatment of stochastic calculus and its applications, including a range of advanced topics.
TABLE OF CONTENTS Introduction to Stochastic Calculus with Applications, Second Edition
Chapter 11: Applications in Finance Stock and FX Options
In this chapter the fundamentals of Mathematics of Option Pricing are given. The concept of arbitrage is introduced, and a martingale characterization of models that don't admit arbitrage is given, the First Fundamental Theorem of asset pricing. The theory of pricing by no-arbitrage is presented first in the Finite Market model, and then in a general Semimartingale Model, where the martingale representation property is used. Change of measure and its application as the change of numeraire are given as a corollary to Girsanov's theorem and general Bayes formula for expectations. They represent the main techniques used for pricing foreign exchange options, exotic options (asian, lookback, barrier options) and interest rates options.
11.1 Financial Derivatives and Arbitrage
A financial derivative or a contingent claim on an asset is a contract that allows purchase or sale of this asset in the future on terms that are specified in the contract. An option on stock is a basic example.
Definition 11.1
A call option on stock is a contract that gives its holder the right to buy this stock in the future at the price K written in the contract, called the exercise price or the strike.
A European call option allows the holder to exercise the contract (that is, to buy this stock at K) at a particular date T, called the maturity or the expiration date. An American option allows the holder to exercise the contract at any time before or at T.
A contract that gives its...
Copyright Imperial College Press 2005 under license agreement with Books24x7
