Mathematical Modeling and Methods of Option Pricing


 Bethanie Gregory
 1 years ago
 Views:
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 CataloguinginPublication 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 graduatelevel 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 indepth introduction to the Black ScholesMerton'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 terminalboundary 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 indepth 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 selfcontained 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 Ahedging 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 BlackScholesMerton option pricing theory from several perspectives and levels: starting from the arbitragefree assumption, via the Ahedging technique, put the investors in a riskneutral world where all risky assets have the same expected return the riskfree 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 wellknown BlackScholes 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 79, we study the models and solution methods for multiasset options and various types of pathdependent options. New multidimensional PDE pricing models are introduced in those chapters, which include not only the multidimensional BlackScholes equation, but also various types of terminalboundary problems for hyperparabolic equations. In addition to studying various methods of numerical solution, we are particularly interested in the possibility of reducing a multidimensional problem to a onedimensional 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 wellposed 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. ArbitrageFree Principle Financial Market and ArbitrageFree Principle European Option Pricing and CallPut 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 OnePeriod and TwoState Model Binomial Tree Method of European Option Pricing (I) NonDividendPaying Binomial Tree Method of European Options (II) DividendPaying Binomial Tree Method of American Option Pricing CallPut 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 BlackScholes Formula History BlackScholes Equation BlackScholes Formula Generalized BlackScholes Model (I) DividendPaying Options Generalized BlackScholes 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 MultiAsset Option Pricing Stochastic Models of MultiAssets Pricing BlackScholes Equation BlackScholes Formula Rainbow Options Basket Options Quanto Options American MultiAsset Options 222
11 Contents xi 8. PathDependent Options (I) Weakly PathDependent Options Barrier Options TimeDependent Barrier Options Reset Options Modified Barrier Options PathDependent Options (II) Strongly PathDependent Options Asian Options Model and Simplification Valuation Formula for EuropeanStyle Geometric Average Asian Option CallPut 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 World Scientific Lecture Notes in Economics ISSN: 23826118 Series Editor: Dirk Bergemann (Yale University,
More informationTEACHING 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 informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationPRICING 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 informationMATHEMATICAL 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 informationLECTURES ON CHERNWE11 THEORY
LECTURES ON CHERNWE11 THEORY AND WITTEN DEFORMATIONS This page is intentionally left blank Nankai Tracts in Mathematics  Vol. 4 LECTURES ON CHERNWEN THEORY AND WITTEN DEFORMATIONS Weiping Zhang Nankai
More informationBariatric 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 informationNumerical 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 informationECommerce 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 NebraskaLincoln,
More informationArbitrage 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 informationSocial 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 informationStephane 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 DiscreteTime Stochastic
More informationBasic BlackScholes: Option Pricing and Trading
Basic BlackScholes: 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 informationEuropean 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 informationMore 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 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS 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 informationBINOMIAL 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 optionpricing
More informationNANOCOMPUTING. 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 informationA SimulationBased lntroduction Using Excel
Quantitative Finance A SimulationBased 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 informationMaster 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. BlackScholesMerton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. TsingHua Univ.
金融隨機計算 : 第一章 BlackScholesMerton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. TsingHua Univ. Derivative Contracts Derivatives, also called contingent claims, are
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationCS 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 informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationA 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 informationDerivatives: 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 informationOPTIONS, 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 informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationTHE TIMEDISCRETE METHOD OF LINES FOR OPTIONS AND BONDS
THE TIMEDISCRETE METHOD OF LINES FOR OPTIONS AND BONDS APDEApproach % " 24 BSV ViSfVs^i + pbi
More informationMarket 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 informationPIONEERS 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 informationTABLE OF CONTENTS. A. PutCall 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. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationPricing 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 informationExam 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 informationUCLA 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 informationParallel 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 informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationEuropean Call Option Pricing using the Adomian Decomposition Method
Advances in Dynamical Systems and Applications ISSN 09735321, Volume 9, Number 1, pp. 75 85 (2014) http://campus.mst.edu/adsa European Call Option Pricing using the Adomian Decomposition Method Martin
More informationMoreover, 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 informationFinancial 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. BlackScholesMerton formula 2. Binomial trees
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes 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 informationA Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model
Applied Mathematical Sciences, vol 8, 14, no 143, 7157135 HIKARI Ltd, wwwmhikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting
More informationAnalysis of Financial Time Series
Analysis of Financial Time Series Analysis of Financial Time Series Financial Econometrics RUEY S. TSAY University of Chicago A WileyInterscience Publication JOHN WILEY & SONS, INC. This book is printed
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 11. The BlackScholes Model: Hull, Ch. 13.
Week 11 The BlackScholes Model: Hull, Ch. 13. 1 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2 The BlackScholes Model 1.
More informationINSTITUTE 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 informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationPricing 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 informationBINOMIAL 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 informationAlgebra and Discrete Mathematics Vol. 3
Algebra and Discrete Mathematics Vol. 3 Algebra and Discrete Mathematics ISSN: 17935873 Managing Editor: Rüdiger Göbel (University DuisburgEssen, Germany) Editorial Board: Elisabeth Bouscaren, Manfred
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE 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 informationNEW WORLDS IN C J 13. New Worlds in Astroparticle Physics Downloaded from www.worldscientific.com
NEW WORLDS IN Proceedings of the Fourth International Workshop n C J 13 n This page is intentionally left blank NEW WORLDS IN Proceedings of the Fourth International Workshop n ) I m editors Alexander
More informationSURGICAL 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 NSLIJ Health System,
More informationOptions 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 informationAN 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 informationProbability 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 informationQuark Confinement and the Hadron Spectrum III
Quark Confinement and the Hadron Spectrum III Newport News, Virginia, USA 712 June 1998 Editor Nathan Isgur Jefferson Laboratory, USA 1lhWorld Scientific.,., Singapore  New Jersey London Hong Kong
More informationON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL
ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinantwise effect of option prices when
More informationFX 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 informationCaput 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 informationSOLVING 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 informationProperties 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 informationAN 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 informationNumerical 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 informationOption 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 informationStudent 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 informationLecture 3: Put Options and DistributionFree Results
OPTIONS and FUTURES Lecture 3: Put Options and DistributionFree Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distributionfree results? option
More informationValuation 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 informationApplied Mathematics. Dr. Carlos Marques, Chair Mathematics Dept School of Arts & Sciences
Applied Mathematics Dr. Carlos Marques, Chair Mathematics Dept. carlos.marques@farmingdale.edu 6314202182 School of Arts & Sciences Bachelor of Science Degree The Applied Mathematics Bachelor of Science
More informationNumerical 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 informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationHandbook 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 informationPathdependent options
Chapter 5 Pathdependent 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 informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More informationTable 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 informationCOURSE 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 informationHedging 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 informationVALUATION 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 informationUnderstanding 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 379961200 Options are priced assuming that
More informationMonte 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 informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationApplied 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 informationOnestate Variable Binomial Models for European/AmericanStyle Geometric Asian Options
Onestate Variable Binomial Models for European/AmericanStyle Geometric Asian Options Min Dai Laboratory of Mathematics and Applied Mathematics, and Dept. of Financial Mathematics, Peking University,
More informationAmerican Capped Call Options on DividendPaying Assets
American Capped Call Options on DividendPaying Assets Mark Broadie Columbia University Jerome Detemple McGill University and CIRANO This article addresses the problem of valuing American call options
More informationFinite 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 BlackScholes
More informationAlgebra 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 informationMonte 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 informationThe 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 informationRisk/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 HansFredo List Swiss Reinsurance Company Mythenquai 50/60, CH8022
More informationAmerican 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 informationInd AS 102 Sharebased Payments
Ind AS 102 Sharebased 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 informationStatistics 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 informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
More informationP1. 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 informationHow 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 informationS 1 S 2. Options and Other Derivatives
Options and Other Derivatives The OnePeriod Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating
More informationMATH3075/3975 Financial Mathematics
MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the BlackScholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per
More information4. Factor polynomials over complex numbers, describe geometrically, and apply to realworld 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 informationCAPM 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, D15230 Frankfurt (Oder, Germany, Email: husmann@europauni.de,
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate 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 informationTHE BLACKSCHOLES MODEL AND EXTENSIONS
THE BLACSCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the BlackScholes pricing model of a European option by calculating the expected value of the option. We will assume that
More information