Mathematical Modeling and Methods of Option Pricing

Size: px
Start display at page:

Download "Mathematical Modeling and Methods of Option Pricing"

Transcription

1 Mathematical Modeling and Methods of Option Pricing

2 This page is intentionally left blank

3 Mathematical Modeling and Methods of Option Pricing Lishang Jiang Tongji University, China Translated by Canguo Li \Hp World Scientific

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. First published in Chinese in 2003 by Higher Education Press. MATHEMATICAL MODELING AND METHODS OF OPTION PRICING Copyright 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN Printed in Singapore by World Scientific Printers (S) Pte Ltd

5 Preface This text is based on the lecture notes of a graduate-level course "Mathematical Theory of Financial Derivatives", which I gave first at Department of Mathematics, University of Iowa, U.S.A., and later at Department of Applied Mathematics, Tongji University, Shanghai, China. In this course, I intended to present a systematic and in-depth introduction to the Black- Scholes-Merton's option pricing theory from the perspective of partial differential equation theory. It is the author's hope that this text may contribute to filling a gap in the existing literature. Option is a financial derivative. Therefore its price depends on the underlying asset's price movement. In the case of the continuous time model, the movement of the underlying asset's price can be described by a stochastic differential equation. Consequently, according to the idea of Black and Scholes, the option price can be modeled as a terminal-boundary problem for a partial differential equation (PDE). Therefore it is reasonable to adopt the existing theory and methods of PDE as a fundamental approach to the study of the option pricing theory. This includes establishing the PDE models for various types of options, deriving the pricing formulas as solutions of the corresponding PDE problems, making qualitative and in-depth analysis of the structure of the option price, and designing efficient algorithms for solving option pricing problems from the viewpoint of numerical solutions of PDE problems. As a textbook for graduate students in applied mathematics, the depth and scope of this book must be appropriate. In order to limit the prerequisites, we tried our best to make this text self-contained when topics of modern mathematics are involved. In fact, we only assume a basic knowledge of calculus, linear algebra, elementary probability theory, and mathematical physics equations. When topics of stochastic analysis, numerical V

6 vi Mathematical Modeling and Methods of Option Pricing methods of PDE and free boundary problems are encountered in the text, only a brief presentation of the basic concepts and results is given. That is, the conclusion is stated, the basic idea of the proof is explained, but the details are not presented, and references are provided to guide the reader for further study. Furthermore, we restrict our discussion to those financial topics whose option pricing can be formulated as a PDE problem via the A-hedging technique, to illustrate the basic idea of the PDE approach. The book is organized as follows: Fundamental concepts of financial derivatives are introduced in Chapter 1, and basics of stochastic analysis are covered in Chapter 4. Chapters 2, 3, and Chapter 5 form the core of this book. In these three chapters, in addition to presenting the mathematical models, algorithms and formulas of option pricing, we expound the basic ideas behind the Black-Scholes-Merton option pricing theory from several perspectives and levels: starting from the arbitrage-free assumption, via the A-hedging technique, put the investors in a risk-neutral world where all risky assets have the same expected return the risk-free interest rate, then option as a contingent claim is given a fair market price independent of the risk preference of each individual investor. In the case of the continuous time model, the pricing formula for European vanilla option is the well-known Black-Scholes formula. In Chapter 6 and 7.7, we study American option pricing problems. Since American option offers early exercise, the holder needs to select the optimal exercise strategy to get optimal returns. Mathematically, this is modeled as a free boundary problem, where the free boundary is the optimal exercise boundary of an American option. Since it is a nonlinear problem, explicit closed form solution does not exist in general, hence qualitative analysis and quantitative numerical solution play an important role. Naturally, American option pricing as free boundary problem becomes the central topic and apex of the book, where the power of the theory and methods of PDE are fully demonstrated. In Chapters 7-9, we study the models and solution methods for multi-asset options and various types of path-dependent options. New multi-dimensional PDE pricing models are introduced in those chapters, which include not only the multi-dimensional Black-Scholes equation, but also various types of terminal-boundary problems for hyperparabolic equations. In addition to studying various methods of numerical solution, we are particularly interested in the possibility of reducing a multi-dimensional problem to a one-dimensional problem. Finally, in Chapter 10, we study the inverse problem of option pricing, that is, how to recover the volatility of the underlying asset from the information of its option market. It is called the

7 Preface vii implied volatility problem. We first derive the Dupire's method from the PDE viewpoint, and then proceed to work in the optimal control framework, thus obtain a system of partial differential equations and propose a well-posed algorithm for recovering the implied volatility. I would like to thank my colleagues and students at Financial Mathematics Group in Tongji University, who have read earlier versions of the manuscript and made helpful suggestions. My special thanks go to Mrs. Xiaoping Zhang, my editor of the original Chinese edition of this text at the Higher Education Press(Beijing), for her expertise and enthusiastic work, and to Dr. Canguo Li for his elegant and painstaking translation work. The publication of the English edition of this text would not be possible without their efforts. Lishang Jiang Tongji University, 2004

8 This page is intentionally left blank

9 Contents Preface 1. Risk Management and Financial Derivatives Risk and Risk Management Forward Contracts and Futures Options Option Pricing Types of Traders 6 2. Arbitrage-Free Principle Financial Market and Arbitrage-Free Principle European Option Pricing and Call-Put Parity American Option Pricing and Early Exercise Dependence of Option Pricing on the Strike Price Binomial Tree Methods Discrete Models of Option Pricing An Example One-Period and Two-State Model Binomial Tree Method of European Option Pricing (I) Non-Dividend-Paying Binomial Tree Method of European Options (II) Dividend-Paying Binomial Tree Method of American Option Pricing Call-Put Symmetry 48 v 4. Brownian Motion and I to Formula 55 ix

10 x Mathematical Modeling and Methods of Option Pricing 4.1 Random Walk and Brownian Motion Continuous Models of Asset Price Movement Quadratic Variation Theorem Ito Integral Ito Formula European Option Pricing Black-Scholes Formula History Black-Scholes Equation Black-Scholes Formula Generalized Black-Scholes Model (I) Dividend-Paying Options Generalized Black-Scholes Model (II) Binary Options and Compound Options Numerical Methods (I) Finite Difference Method Numerical Methods (II) Binomial Tree Method and Finite Difference Method Properties of European Option Price Risk Management American Option Pricing and Optimal Exercise Strategy Perpetual American Option Models of American Options Decomposition of American Options Properties of American Option Price Optimal Exercise Boundary Numerical Method (I) Finite Difference Method Numerical Methods(II) Line Method Other Types of American Options Multi-Asset Option Pricing Stochastic Models of Multi-Assets Pricing Black-Scholes Equation Black-Scholes Formula Rainbow Options Basket Options Quanto Options American Multi-Asset Options 222

11 Contents xi 8. Path-Dependent Options (I) Weakly Path-Dependent Options Barrier Options Time-Dependent Barrier Options Reset Options Modified Barrier Options Path-Dependent Options (II) Strongly Path-Dependent Options Asian Options Model and Simplification Valuation Formula for European-Style Geometric Average Asian Option Call-Put Parities for Asian Options Lookback Option Numerical Methods Implied Volatility Preliminaries Dupire Method Optimal Control Method Numerical Method 320 Bibliography 323 Index 327

Financial Derivatives Futures, Forwards, Swaps, Options, Corporate Securities, and Credit Default Swaps

Financial Derivatives Futures, Forwards, Swaps, Options, Corporate Securities, and Credit Default Swaps Financial Derivatives Futures, Forwards, Swaps, Options, Corporate Securities, and Credit Default Swaps World Scientific Lecture Notes in Economics ISSN: 2382-6118 Series Editor: Dirk Bergemann (Yale University,

More information

TEACHING UNDERGRADUATE MATHEMATICS

TEACHING UNDERGRADUATE MATHEMATICS TEACHING UNDERGRADUATE MATHEMATICS This page is intentionally left blank TEACHING UNDERGRADUATE Editors Bob Burn MATHEMATICS Reader in Mathematics Education, Agder College, Norway John Appleby Senior Lecturer

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

More information

PRICING DERIVATIVE SECURITIES SECOND EDITION. Pricing Derivative Securities Downloaded from

PRICING DERIVATIVE SECURITIES SECOND EDITION. Pricing Derivative Securities Downloaded from PRICING DERIVATIVE SECURITIES SECOND EDITION This page intentionally left blank T W E P P S University of Virginia. USA PRICING DERIVATIVE SECURITIES SECOND EDITION World Scientific NEWJERSEY * LONDON

More information

MATHEMATICAL LOGIC FOR COMPUTER SCIENCE

MATHEMATICAL LOGIC FOR COMPUTER SCIENCE MATHEMATICAL LOGIC FOR COMPUTER SCIENCE Second Edition WORLD SCIENTIFIC SERIES IN COMPUTER SCIENCE 25: Computer Epistemology A Treatise on the Feasibility of the Unfeasible or Old Ideas Brewed New (T Vamos)

More information

LECTURES ON CHERN-WE11 THEORY

LECTURES ON CHERN-WE11 THEORY LECTURES ON CHERN-WE11 THEORY AND WITTEN DEFORMATIONS This page is intentionally left blank Nankai Tracts in Mathematics - Vol. 4 LECTURES ON CHERN-WEN THEORY AND WITTEN DEFORMATIONS Weiping Zhang Nankai

More information

Bariatric Surgery. Obesity. Care and. Obesity Care and Bariatric Surgery Downloaded from www.worldscientific.com

Bariatric Surgery. Obesity. Care and. Obesity Care and Bariatric Surgery Downloaded from www.worldscientific.com Obesity Care and Bariatric Surgery This page intentionally left blank Obesity Care and Bariatric Surgery Editors Kenric M Murayama University of Hawaii, USA Shanu N Kothari Gundersen Lutheran Health System,

More information

Numerical Methods in Finance with C++

Numerical Methods in Finance with C++ Numerical Methods in Finance with C++ Driven by concrete computational problems in quantitative finance, this book provides aspiring quant developers with the numerical techniques and programming skills

More information

E-Commerce Operations Management Downloaded from www.worldscientific.com -COMMERCE. by 37.44.207.139 on 06/15/16. For personal use only.

E-Commerce Operations Management Downloaded from www.worldscientific.com -COMMERCE. by 37.44.207.139 on 06/15/16. For personal use only. -COMMERCE O p e r a t i o n s M a n a g e m e n t 2nd Edition This page intentionally left blank -COMMERCE O p e r a t i o n s M a n a g e m e n t 2nd Edition Marc J. Schniederjans University of Nebraska-Lincoln,

More information

Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time Arbitrage Theory in Continuous Time THIRD EDITION TOMAS BJORK Stockholm School of Economics OXTORD UNIVERSITY PRESS 1 Introduction 1 1.1 Problem Formulation i 1 v. 2 The Binomial Model 5 2.1 The One Period

More information

Social Services Administration In Hong Kong

Social Services Administration In Hong Kong Social Services Administration In Hong Kong Tneoretical Issues ana Case Studies This page is intentionally left blank Social Services Administration In Hong Kong Theoretical Issues ana Case Studies Editors

More information

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of Discrete-Time Stochastic

More information

Basic Black-Scholes: Option Pricing and Trading

Basic Black-Scholes: Option Pricing and Trading Basic Black-Scholes: Option Pricing and Trading BSc (HONS 1 st Timothy Falcon Crack Class), PGDipCom, MCom, PhD (MIT), IMC Contents Preface Tables Figures ix xi xiii 1 Introduction to Options 1 1.1 Hedging,

More information

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

More information

More Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options

More Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

More information

BINOMIAL OPTION PRICING

BINOMIAL OPTION PRICING Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing

More information

NANOCOMPUTING. Computational Physics for Nanoscience and Nanotechnology

NANOCOMPUTING. Computational Physics for Nanoscience and Nanotechnology NANOCOMPUTING Computational Physics for Nanoscience and Nanotechnology NANOCOMPUTING Computational Physics for Nanoscience and Nanotechnology James J Y Hsu National Cheng Kung University, Taiwan National

More information

A Simulation-Based lntroduction Using Excel

A Simulation-Based lntroduction Using Excel Quantitative Finance A Simulation-Based lntroduction Using Excel Matt Davison University of Western Ontario London, Canada CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint

More information

Master of Mathematical Finance: Course Descriptions

Master of Mathematical Finance: Course Descriptions Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support

More information

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ.

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. 金融隨機計算 : 第一章 Black-Scholes-Merton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. Derivative Contracts Derivatives, also called contingent claims, are

More information

The Black-Scholes pricing formulas

The Black-Scholes pricing formulas The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

A Core Curriculum. by Russian Academy of Sciences Moscow on 01/03/17. For personal use only. HAEMATOLOGY

A Core Curriculum. by Russian Academy of Sciences Moscow on 01/03/17. For personal use only. HAEMATOLOGY HAEMATOLOGY A Core Curriculum This page intentionally left blank HAEMATOLOGY A Core Curriculum Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific

More information

Derivatives: Principles and Practice

Derivatives: Principles and Practice Derivatives: Principles and Practice Rangarajan K. Sundaram Stern School of Business New York University New York, NY 10012 Sanjiv R. Das Leavey School of Business Santa Clara University Santa Clara, CA

More information

OPTIONS, FUTURES, & OTHER DERIVATI

OPTIONS, FUTURES, & OTHER DERIVATI Fifth Edition OPTIONS, FUTURES, & OTHER DERIVATI John C. Hull Maple Financial Group Professor of Derivatives and Risk Manage, Director, Bonham Center for Finance Joseph L. Rotinan School of Management

More information

The Black-Scholes Model

The Black-Scholes Model The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

More information

THE TIME-DISCRETE METHOD OF LINES FOR OPTIONS AND BONDS

THE TIME-DISCRETE METHOD OF LINES FOR OPTIONS AND BONDS THE TIME-DISCRETE METHOD OF LINES FOR OPTIONS AND BONDS APDEApproach % " 24 BSV ViSfVs^i + pbi

More information

Market Risk Analysis. Quantitative Methods in Finance. Volume I. The Wiley Finance Series

Market Risk Analysis. Quantitative Methods in Finance. Volume I. The Wiley Finance Series Brochure More information from http://www.researchandmarkets.com/reports/2220051/ Market Risk Analysis. Quantitative Methods in Finance. Volume I. The Wiley Finance Series Description: Written by leading

More information

PIONEERS of MICROBIOLOGY and the NOBEL PRIZE

PIONEERS of MICROBIOLOGY and the NOBEL PRIZE PIONEERS of MICROBIOLOGY and the NOBEL PRIZE This page is intentionally left blank PIONEERS of MICROBIOLOGY and the NOBEL PRIZE Ulf Lagerkvist Goteborg University, Sweden Vfe World Scientific wl New Jersey

More information

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

More information

Pricing of a worst of option using a Copula method M AXIME MALGRAT

Pricing of a worst of option using a Copula method M AXIME MALGRAT Pricing of a worst of option using a Copula method M AXIME MALGRAT Master of Science Thesis Stockholm, Sweden 2013 Pricing of a worst of option using a Copula method MAXIME MALGRAT Degree Project in Mathematical

More information

Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D

Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS

More information

UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:

UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date: UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,

More information

Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation

Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation Ying Peng, Bin Gong, Hui Liu, and Yanxin Zhang School of Computer Science and Technology, Shandong University,

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

European Call Option Pricing using the Adomian Decomposition Method

European Call Option Pricing using the Adomian Decomposition Method Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 1, pp. 75 85 (2014) http://campus.mst.edu/adsa European Call Option Pricing using the Adomian Decomposition Method Martin

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1 Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. Black-Scholes-Merton formula 2. Binomial trees

More information

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial

More information

A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model Applied Mathematical Sciences, vol 8, 14, no 143, 715-7135 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting

More information

Analysis of Financial Time Series

Analysis of Financial Time Series Analysis of Financial Time Series Analysis of Financial Time Series Financial Econometrics RUEY S. TSAY University of Chicago A Wiley-Interscience Publication JOHN WILEY & SONS, INC. This book is printed

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13. Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

More information

INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION

INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION 14 April 2016 (pm) Subject ST6 Finance and Investment Specialist Technical B Time allowed: Three hours 1. Enter all the candidate and examination details

More information

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

More information

Pricing Barrier Options under Local Volatility

Pricing Barrier Options under Local Volatility Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly

More information

BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract

BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple

More information

Algebra and Discrete Mathematics Vol. 3

Algebra and Discrete Mathematics Vol. 3 Algebra and Discrete Mathematics Vol. 3 Algebra and Discrete Mathematics ISSN: 1793-5873 Managing Editor: Rüdiger Göbel (University Duisburg-Essen, Germany) Editorial Board: Elisabeth Bouscaren, Manfred

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More information

NEW WORLDS IN C J 1-3. New Worlds in Astroparticle Physics Downloaded from www.worldscientific.com

NEW WORLDS IN C J 1-3. New Worlds in Astroparticle Physics Downloaded from www.worldscientific.com NEW WORLDS IN Proceedings of the Fourth International Workshop n C J 1-3 n This page is intentionally left blank NEW WORLDS IN Proceedings of the Fourth International Workshop n ) I m editors Alexander

More information

SURGICAL CARE MALFORMATIONS

SURGICAL CARE MALFORMATIONS SURGICAL CARE OF MAJOR Newborn MALFORMATIONS This page intentionally left blank SURGICAL CARE OF MAJOR Newborn MALFORMATIONS editors Stephen E Dolgin Schneider Children s Hospital NS-LIJ Health System,

More information

Options on Foreign Exchange. 3rd Edition. Wiley Finance

Options on Foreign Exchange. 3rd Edition. Wiley Finance Brochure More information from http://www.researchandmarkets.com/reports/2242927/ Options on Foreign Exchange. 3rd Edition. Wiley Finance Description: Praise for Options on Foreign Exchange, 3rd Edition

More information

AN INTRODUCTION TO OPTIONS TRADING. Frans de Weert

AN INTRODUCTION TO OPTIONS TRADING. Frans de Weert AN INTRODUCTION TO OPTIONS TRADING Frans de Weert AN INTRODUCTION TO OPTIONS TRADING The Securities & Investment Institute Mission Statement: To set standards of professional excellence and integrity

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Syllabus for the TEMPUS SEE PhD Course (Podgorica, April 4 29, 2011) Franz Kappel 1 Institute for Mathematics and Scientific Computing University of Graz Žaneta Popeska 2 Faculty

More information

Quark Confinement and the Hadron Spectrum III

Quark Confinement and the Hadron Spectrum III Quark Confinement and the Hadron Spectrum III Newport News, Virginia, USA 7-12 June 1998 Editor Nathan Isgur Jefferson Laboratory, USA 1lhWorld Scientific.,., Singapore - New Jersey- London -Hong Kong

More information

ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL

ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinant-wise effect of option prices when

More information

FX Options and Smile Risk_. Antonio Castagna. )WILEY A John Wiley and Sons, Ltd., Publication

FX Options and Smile Risk_. Antonio Castagna. )WILEY A John Wiley and Sons, Ltd., Publication FX Options and Smile Risk_ Antonio Castagna )WILEY A John Wiley and Sons, Ltd., Publication Preface Notation and Acronyms IX xiii 1 The FX Market 1.1 FX rates and spot contracts 1.2 Outright and FX swap

More information

Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

More information

SOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD. KENNEDY HAYFORD, (B.Sc. Mathematics)

SOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD. KENNEDY HAYFORD, (B.Sc. Mathematics) SOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD BY KENNEDY HAYFORD, (B.Sc. Mathematics) A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University

More information

Properties of the SABR model

Properties of the SABR model U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT

More information

AN ACCESSIBLE TREATMENT OF MONTE CARLO METHODS, TECHNIQUES, AND APPLICATIONS IN THE FIELD OF FINANCE AND ECONOMICS

AN ACCESSIBLE TREATMENT OF MONTE CARLO METHODS, TECHNIQUES, AND APPLICATIONS IN THE FIELD OF FINANCE AND ECONOMICS Brochure More information from http://www.researchandmarkets.com/reports/2638617/ Handbook in Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics. Wiley Handbooks

More information

Numerical Methods for Pricing Exotic Options

Numerical Methods for Pricing Exotic Options Numerical Methods for Pricing Exotic Options Dimitra Bampou Supervisor: Dr. Daniel Kuhn Second Marker: Professor Berç Rustem 18 June 2008 2 Numerical Methods for Pricing Exotic Options 0BAbstract 3 Abstract

More information

Option pricing. Vinod Kothari

Option pricing. Vinod Kothari Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

More information

Student Solutions Manual for Mathematical Methods for Physics and Engineering, third edition

Student Solutions Manual for Mathematical Methods for Physics and Engineering, third edition Student Solutions Manual for Mathematical Methods for Physics and Engineering, third edition Mathematical Methods for Physics and Engineering, third edition, is a highly acclaimed undergraduate textbook

More information

Lecture 3: Put Options and Distribution-Free Results

Lecture 3: Put Options and Distribution-Free Results OPTIONS and FUTURES Lecture 3: Put Options and Distribution-Free Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distribution-free results? option

More information

Valuation of American Options

Valuation of American Options Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political

More information

Applied Mathematics. Dr. Carlos Marques, Chair Mathematics Dept School of Arts & Sciences

Applied Mathematics. Dr. Carlos Marques, Chair Mathematics Dept School of Arts & Sciences Applied Mathematics Dr. Carlos Marques, Chair Mathematics Dept. carlos.marques@farmingdale.edu 631-420-2182 School of Arts & Sciences Bachelor of Science Degree The Applied Mathematics Bachelor of Science

More information

Numerical methods for American options

Numerical methods for American options Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

More information

Asian Option Pricing Formula for Uncertain Financial Market

Asian Option Pricing Formula for Uncertain Financial Market Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei

More information

Handbook in. Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics

Handbook in. Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics Handbook in Monte Carlo Simulation Applications in Financial Engineering, Risk Management, and Economics PAOLO BRANDIMARTE Department of Mathematical Sciences Politecnico di Torino Torino, Italy WlLEY

More information

Path-dependent options

Path-dependent options Chapter 5 Path-dependent options The contracts we have seen so far are the most basic and important derivative products. In this chapter, we shall discuss some complex contracts, including barrier options,

More information

Lecture 6 Black-Scholes PDE

Lecture 6 Black-Scholes PDE Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent

More information

Table of Contents. Montessori Algebra for the Adolescent Michael J. Waski"

Table of Contents. Montessori Algebra for the Adolescent Michael J. Waski Table of Contents I. Introduction II. Chapter of Signed Numbers B. Introduction and Zero Sum Game C. Adding Signed Numbers D. Subtracting Signed Numbers 1. Subtracting Signed Numbers 2. Rewriting as Addition

More information

COURSE OUTLINE. MATHEMATICS 101 Intermediate Algebra

COURSE OUTLINE. MATHEMATICS 101 Intermediate Algebra Degree Applicable I. Catalog Statement COURSE OUTLINE MATHEMATICS 101 Intermediate Algebra Glendale Community College October 2013 Mathematics 101 is an accelerated course of Intermediate Algebra. Topics

More information

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

More information

VALUATION IN DERIVATIVES MARKETS

VALUATION IN DERIVATIVES MARKETS VALUATION IN DERIVATIVES MARKETS September 2005 Rawle Parris ABN AMRO Property Derivatives What is a Derivative? A contract that specifies the rights and obligations between two parties to receive or deliver

More information

Understanding Options and Their Role in Hedging via the Greeks

Understanding Options and Their Role in Hedging via the Greeks Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200 Options are priced assuming that

More information

Monte Carlo Methods and Models in Finance and Insurance

Monte Carlo Methods and Models in Finance and Insurance Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Monte Carlo Methods and Models in Finance and Insurance Ralf Korn Elke Korn Gerald Kroisandt f r oc) CRC Press \ V^ J Taylor & Francis Croup ^^"^ Boca Raton

More information

Notes on Black-Scholes Option Pricing Formula

Notes on Black-Scholes Option Pricing Formula . Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading

More information

Applied Computational Economics and Finance

Applied Computational Economics and Finance Applied Computational Economics and Finance Mario J. Miranda and Paul L. Fackler The MIT Press Cambridge, Massachusetts London, England Preface xv 1 Introduction 1 1.1 Some Apparently Simple Questions

More information

One-state Variable Binomial Models for European-/American-Style Geometric Asian Options

One-state Variable Binomial Models for European-/American-Style Geometric Asian Options One-state Variable Binomial Models for European-/American-Style Geometric Asian Options Min Dai Laboratory of Mathematics and Applied Mathematics, and Dept. of Financial Mathematics, Peking University,

More information

American Capped Call Options on Dividend-Paying Assets

American Capped Call Options on Dividend-Paying Assets American Capped Call Options on Dividend-Paying Assets Mark Broadie Columbia University Jerome Detemple McGill University and CIRANO This article addresses the problem of valuing American call options

More information

Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

More information

Algebra I Credit Recovery

Algebra I Credit Recovery Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

More information

Monte Carlo Methods for American Option Pricing

Monte Carlo Methods for American Option Pricing COPENHAGEN BUSINESS SCHOOL MSc in Advanced Economics and Finance Master s Thesis Monte Carlo Methods for American Option Pricing Alberto Barola February 2013 Academic Supervisor Jesper Lund Number of characters

More information

The Valuation of Currency Options

The Valuation of Currency Options The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction

More information

Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management

Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Vincenzo Bochicchio, Niklaus Bühlmann, Stephane Junod and Hans-Fredo List Swiss Reinsurance Company Mythenquai 50/60, CH-8022

More information

American and European. Put Option

American and European. Put Option American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example

More information

Ind AS 102 Share-based Payments

Ind AS 102 Share-based Payments Ind AS 102 Share-based Payments Mayur Ankolekar Consulting Actuary Current Issues in Pension Seminar at Mumbai The Institute of Actuaries of India August 21, 2015 Page 1 Session Objectives 1. To appreciate

More information

Statistics for Experimenters

Statistics for Experimenters Statistics for Experimenters Design, Innovation, and Discovery Second Edition GEORGE E. P. BOX J. STUART HUNTER WILLIAM G. HUNTER WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION FACHGEBIETSBGCHEREI

More information

Black-Scholes-Merton approach merits and shortcomings

Black-Scholes-Merton approach merits and shortcomings Black-Scholes-Merton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The Black-Scholes and Merton method of modelling derivatives prices was first introduced

More information

P-1. Preface. Thank you for choosing ACTEX.

P-1. Preface. Thank you for choosing ACTEX. Preface P- Preface Thank you for choosing ACTEX. Since Exam MFE was introduced in May 007, there have been quite a few changes to its syllabus and its learning objectives. To cope with these changes, ACTEX

More information

How to Manage the Maximum Relative Drawdown

How to Manage the Maximum Relative Drawdown How to Manage the Maximum Relative Drawdown Jan Vecer, Petr Novotny, Libor Pospisil, Columbia University, Department of Statistics, New York, NY 27, USA April 9, 26 Abstract Maximum Relative Drawdown measures

More information

S 1 S 2. Options and Other Derivatives

S 1 S 2. Options and Other Derivatives Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating

More information

MATH3075/3975 Financial Mathematics

MATH3075/3975 Financial Mathematics MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the Black-Scholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per

More information

4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

More information

CAPM Option Pricing. Sven Husmann a, Neda Todorova b

CAPM Option Pricing. Sven Husmann a, Neda Todorova b CAPM Option Pricing Sven Husmann a, Neda Todorova b a Department of Business Administration, European University Viadrina, Große Scharrnstraße 59, D-15230 Frankfurt (Oder, Germany, Email: husmann@europa-uni.de,

More information

Two-State Option Pricing

Two-State Option Pricing Rendleman and Bartter [1] present a simple two-state model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.

More information

THE BLACK-SCHOLES MODEL AND EXTENSIONS

THE BLACK-SCHOLES MODEL AND EXTENSIONS THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that

More information