Introduction to Option Pricing Theory

by Kallianpur, Gopinath; Karandikar, Rajeeva L.

• ISBN13: 9780817641085
• ISBN10: 0817641084
• Format: Hardcover
• Copyright: 2000-02-01
• Publisher: Birkhauser

Summary

Since the appearance of seminal works by R. Merton, and F. Blackand M. Scholes, stochastic processes have assumed an increasinglyimportant role in the development of the mathematical theory offinance. This work examines, in some detail, that part of stochasticfinance pertaining to option pricing theory. Thus the exposition isconfined to areas of stochastic finance that are relevant to thetheory, omitting such topics as futures and term-structure.This self-contained work begins with five introductory chapterson stochastic analysis, making it accessible to readers with little orno prior knowledge of stochastic processes or stochastic analysis.These chapters cover the essentials of Ito's theory of stochasticintegration, integration with respect to semimartingales, Girsanov'sTheorem, and a brief introduction to stochastic differentialequations.Subsequent chapters treat more specialized topics, includingoption pricing in discrete time, continuous time trading, arbitrage,complete markets, European options (Black and Scholes Theory),American options, Russian options, discrete approximations, and assetpricing with stochastic volatility. In several chapters, new resultsare presented. A unique feature of the book is its emphasis onarbitrage, in particular, the relationship between arbitrage andequivalent martingale measures (EMM), and the derivation of necessaryand sufficient conditions for no arbitrage (NA).{\it Introduction to Option Pricing Theory} is intended forstudents and researchers in statistics, applied mathematics, business,or economics, who have a background in measure theory and havecompleted probability theory at the intermediate level. The worklends itself to self-study, as well as to a one-semester course at thegraduate level.

Table of Contents

Preface	ix
Stochastic Integration	1	(46)
Notation and definitions	1	(4)
The predictable σ field	5	(1)
The Ito integral	6	(13)
Quadratic variation of a continuous martingale	19	(7)
The stochastic integral w.r.t. continuous local martingales	26	(6)
Stochastic integral w.r.t. continuous semimartingales	32	(6)
Integration w.r.t. semimartingales	38	(9)
Ito's Formula and its Applications	47	(24)
Preliminaries	47	(4)
Ito's formula for continuous semimartingales	51	(3)
Ito's formula for r.c.l.l. semimartingales	54	(2)
Applications	56	(6)
Application to geometric Brownian motion	62	(1)
Local time and the Tanaka formula	62	(1)
Brownian motion and the heat equation	63	(8)
Representation of Square Integrable Martingales	71	(8)
The Ito representation	71	(3)
The Kunita--Watanabe representation	74	(5)
Stochastic Differential Equations	79	(16)
Preliminaries	79	(2)
Existence and uniqueness of solutions	81	(10)
The Feynman--Kac formula	91	(2)
The Ornstein--Uhlenbeck process (O.U.P)	93	(2)
Girsanov's Theorem	95	(8)
Auxiliary results	95	(3)
Girsanov's Theorem	98	(5)
Option Pricing in Discrete Time	103	(20)
Arbitrage opportunities	103	(6)
Option pricing: an example	109	(2)
European call option	111	(3)
Complete markets	114	(3)
The American option	117	(6)
Introduction to Continuous Time Trading	123	(14)
Introduction	123	(2)
A general model	125	(1)
Trading strategies and arbitrage opportunities	126	(7)
Examples	133	(1)
Contingent claims and complete markets	134	(3)
Arbitrage and Equivalent Martingale Measures	137	(32)
Introduction	137	(2)
Necessary and sufficient conditions for NA	139	(8)
A general model of stock prices	147	(4)
The separation theorem	151	(4)
Orlicz spaces	155	(2)
No arbitrage with controlled risk	157	(4)
Fractional Brownian motion (1/2 < H < 1) and existence of arbitrage opportunities	161	(4)
Extension to geometric Gladyshev processes	165	(4)
Complete Markets	169	(22)
Definition	169	(3)
Representation of martingales	172	(5)
Examples of complete markets	177	(3)
Geometric Brownian motion (GBM)	177	(2)
Diffusion model for stock prices	179	(1)
Equivalent martingale measures	180	(3)
Incomplete markets	183	(4)
Completeness and underlying filtration	187	(4)
Black and Scholes Theory	191	(14)
Preliminaries	191	(3)
The Black--Scholes PDE	194	(1)
Explicit solution of the Black-Scholes PDE	195	(4)
The Black-Scholes formula	199	(2)
Diffusion model	201	(4)
Discrete Approximations	205	(10)
The binomial model	205	(2)
A binomial Feynman--Kac formula	207	(1)
Approximation of the Black--Scholes PDE	208	(3)
Approximation to the Black--Scholes formula	211	(4)
The American Options	215	(10)
Model	215	(2)
Upper and lower bounds	217	(3)
American claims in complete markets	220	(5)
Asset Pricing with Stochastic Volatility	225	(16)
Introduction	225	(1)
Incompleteness of the market	226	(6)
Asymptotic analysis for models with two scales	232	(3)
Filtering of the stochastic volatility	235	(4)
PDE when S is observed	239	(2)
The Russian Options	241	(24)
Introduction and background	241	(2)
The Russian put option	243	(1)
A free boundary problem for the put option	243	(7)
Proofs of the lemmas	250	(5)
The Russian call option (or the option for selling short)	255	(1)
The F.B.P. for the call option	255	(10)
References	265	(4)
Index	269