THE TIME-DISCRETE METHOD OF LINES FOR OPTIONS AND BONDS
|
|
- Eustace McKenzie
- 8 years ago
- Views:
Transcription
1 THE TIME-DISCRETE METHOD OF LINES FOR OPTIONS AND BONDS APDEApproach % " 24 BSV ViSfVs^i + pbi<rbs 1S 2V Sls a + + (T 9I)5'IVS 1 + (r q2)s2vs 3 rv Vt = 0 V(Si, %, 0) = max{0, Kmin(aiSi, 02%)} for «1,03 > 0 i 14 / // / ' Gunter H. Meyer Georgia Institute of Technology, USA World Scientific NEW JERSEY. LONDON S1NGAP0RE. BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
2 Contents Preface Acknowledgment v xv 1. Comments on the Pricing Equations in Finance Solutions and their properties 2 Example 1.1 Positivity of option prices and the Black Scholes formulas 5 Example 1.2 The early exercise boundary for piain American puts and calls 8 Example 1.3 Exercise boundaries for options with jump diffusion 10 Example 1.4 The early exercise premium for an American put 12 Example 1.5 The early exercise premium for an American call 14 Example 1.6 Strike price convexity 15 Example 1.7 Put-call parity 16 Example 1.8 Put-call symmetry for a CEV and Heston model 19 Example 1.9 Equations with an uncertain parameter Boundary conditions for the pricing equations The Fichera function for degenerate equations.. 29 Example 1.11 Boundary conditions for the heat equation. 33 Example 1.12 Boundary condition for the CEV Black Scholes equation at S = 0 34 xi
3 xii TAe T:me-ßwcre(e Met/iod o/.linea /or Optiona and Bonda ;4 f AppmacA Example 1.13 Boundary conditions for a discount bond at Example 1.14 Boundary conditions for the Black Scholes equation on two assets 37 Example 1.15 Boundary conditions for the Black Scholes equation with stochastic volatility v at S = 0 and v 0 38 Example 1.16 Boundary conditions for an Asian option The boundary condition at "infinity" 42 Example 1.17 CEV puts and calls 42 Example 1.18 Puts and calls with stochastic volatility.. 44 Example 1.19 The European max option 45 Example 1.20 An Asian average price call The Venttsel boundary conditions on "far but finite" boundaries 48 Example 1.21 A defaultable bond 50 Example 1.22 The Black Scholes equation with stochastic volatility Free boundaries The Method of Lines (MOL) for the Diffusion Equation The method of lines with continuous time (the vertical MOL) The method of lines with continuous x (the horizontal MOL) 61 Appendix 2.2 Stability of the time discrete three-level scheine for the heat equation The method of lines with continuous x for multidimensional problems 64 Appendix 2.3 Convergence of the line Gauss Seidel iteration for a model problem Free boundaries and the MOL in two dimensions The Riccati Transformation Method for Linear Two Point Boundary Value Problems The Riccati transformation on a fixed interval The Riccati transformation for a free boundary problem The numerical Solution of the sweep equations 81
4 Contents xiii Example 3.1 A real option for interest rate sensitive Investments 87 Appendix 3.3 Connection between the Riccati transformation, Gaussian elimination and the Brennan- Schwartz method European Options 93 Example 4.1 A piain European call 96 Example 4.2 A binary cash or nothing European call Example 4.3 A binary call with low volatility 107 Example 4.4 The Black Scholes Barenblatt equation for a CEV process American Puts and Calls 117 Example 5.1 An American put 117 Example 5.2 An American put with sub-optimal early exercise 123 Example 5.3 A put on an asset with a fixed dividend Example 5.4 An American lookback call 129 Example 5.5 An American strangle for power options Example 5.6 Jump diffusion with uncertain volatility Bonds and Options for One-Factor Interest Rate Models 153 Example 6.1 The Ho Lee model 158 Example 6.2 A one-factor CEV model 161 Example 6.3 An implied volatility for a call on a discount bond 165 Example 6.4 An American put on a discount bond Two-Dimensional Diffusion Problems in Finance Front tracking in Cartesian coordinates 185 Example 7.1 An American call on an asset with stochastic volatility. 185 Example 7.2 A European put on a combination of two assets 190 Example 7.3 A perpetual American put - MOL with overrelaxation 197 Example 7.4 An American call, its deltas and a vega
5 xiv The Time-Discrete Method of Lines for Options and Bonds A PDE Approach Example 7.5 American spread and exchange options Example 7.6 An American call option on the maximum of two assets American calls and puts in polar coordinates 220 Example 7.7 The basket call in polar coordinates 221 Example 7.8 A call on the minimum of two assets 223 Example 7.9 A put on the minimum of two assets 229 Example 7.10 A perpetual put on the minimum of two assets with uncertain correlation 237 Example 7.11 Implied correlation for a put on the sum of two assets A three-dimensional problem 245 Example 7.12 An American call with Heston volatility and a stochastic interest rate 246 Bibliography 261 fndez 265 About the Author 269
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 informationExotic Options Trading
Exotic Options Trading Frans de Weert John Wiley & Sons, Ltd Preface Acknowledgements 1 Introduction 2 Conventional Options, Forwards and Greeks 2.1 Call and Put Options and Forwards 2.2 Pricing Calls
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 informationA 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 informationMathematical Modeling and Methods of Option Pricing
Mathematical Modeling and Methods of Option Pricing This page is intentionally left blank Mathematical Modeling and Methods of Option Pricing Lishang Jiang Tongji University, China Translated by Canguo
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 Discrete-Time Stochastic
More informationOn 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 informationVehicle-Bridge Interaction Dynamics
Vehicle-Bridge Interaction Dynamics With Applications to High-Speed Railways Y. B. Yang National Taiwan University, Taiwan J. D. Yau Tamkang University, Taiwan Y. S. Wu Sinotech Engineering Consultants,
More informationThe Evaluation of Barrier Option Prices Under Stochastic Volatility. BFS 2010 Hilton, Toronto June 24, 2010
The Evaluation of Barrier Option Prices Under Stochastic Volatility Carl Chiarella, Boda Kang and Gunter H. Meyer School of Finance and Economics University of Technology, Sydney School of Mathematics
More informationOptions Markets: Introduction
Options Markets: Introduction Chapter 20 Option Contracts call option = contract that gives the holder the right to purchase an asset at a specified price, on or before a certain date put option = contract
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 informationHow To Model Money In An Excel 3D Program
EXCEL MODELING AND ESTIMATION IN CORPORATE FINANCE Third Edition CRAIG W. HOLDEN Max Barney Faculty Fellow and Associate Professor Kelley School of Business Indiana University PEARSON Pearson Education
More informationOther variables as arguments besides S. Want those other variables to be observables.
Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
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 informationENTERPRISE. - David L. Olson University of Nebraska, USA. Desheng Dash Wu University of Toronto, Canada. World Scientific
Financial Engineering and Risk Management Vol.1 ' ENTERPRISE T - David L. Olson University of Nebraska, USA Desheng Dash Wu University of Toronto, Canada World Scientific NEW JERSEY LONDON SINGAPORE BEIJING
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 informationOption Portfolio Modeling
Value of Option (Total=Intrinsic+Time Euro) Option Portfolio Modeling Harry van Breen www.besttheindex.com E-mail: h.j.vanbreen@besttheindex.com Introduction The goal of this white paper is to provide
More information1 The Black-Scholes Formula
1 The Black-Scholes Formula In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper,
More informationFIN-40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationFundamentals of Futures and Options (a summary)
Fundamentals of Futures and Options (a summary) Roger G. Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA Published 2013 by the Research Foundation of CFA Institute Summary prepared by Roger G.
More informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More informationConsider a European call option maturing at time T
Lecture 10: Multi-period Model Options Black-Scholes-Merton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More informationEXERCISES FROM HULL S BOOK
EXERCISES FROM HULL S BOOK 1. Three put options on a stock have the same expiration date, and strike prices of $55, $60, and $65. The market price are $3, $5, and $8, respectively. Explain how a butter
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 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 informationThe Complete Guide to Option Strategies
The Complete Guide to Option Strategies Advanced and Basic Strategies on Stocks, ETFs, Indexes, and Stock Indexes MICHAEL D. MULLANEY WILEY John Wiley & Sons, Inc. Contents Preface Acknowledgments xv xx
More informationSchonbucher Chapter 9: Firm Value and Share Priced-Based Models Updated 07-30-2007
Schonbucher Chapter 9: Firm alue and Share Priced-Based Models Updated 07-30-2007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spread-based models of default
More informationJorge 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 informationGeneral price bounds for discrete and continuous arithmetic Asian options
General price bounds for discrete and continuous arithmetic Asian options 1 Ioannis.Kyriakou@city.ac.uk in collaboration with Gianluca Fusai 1,2 Gianluca.Fusai.1@city.ac.uk 1 Cass Business School, City
More informationHow to use the Options/Warrants Calculator?
How to use the Options/Warrants Calculator? 1. Introduction Options/Warrants Calculator is a tool for users to estimate the theoretical prices of options/warrants in various market conditions by inputting
More informationIntroduction to Financial Models for Management and Planning
CHAPMAN &HALL/CRC FINANCE SERIES Introduction to Financial Models for Management and Planning James R. Morris University of Colorado, Denver U. S. A. John P. Daley University of Colorado, Denver U. S.
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 informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationChapter 21: Options and Corporate Finance
Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the
More informationAsian 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 informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call
More informationUnderlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)
INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the Put-Call-Forward
More informationA Day in the Life of a Trader
Siena, April 2014 Introduction 1 Examples of Market Payoffs 2 3 4 Sticky Smile e Floating Smile 5 Examples of Market Payoffs Understanding risk profiles of a payoff is conditio sine qua non for a mathematical
More informationThird Edition. Philippe Jorion GARP. WILEY John Wiley & Sons, Inc.
2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Third Edition Philippe Jorion GARP WILEY John Wiley & Sons, Inc.
More informationHow To Price Garch
2011 3rd International Conference on Information and Financial Engineering IPEDR vol.12 (2011) (2011) IACSIT Press, Singapore A Study on Heston-Nandi GARCH Option Pricing Model Suk Joon Byun KAIST Business
More informationPricing Interest-Rate- Derivative Securities
Pricing Interest-Rate- Derivative Securities John Hull Alan White University of Toronto This article shows that the one-state-variable interest-rate models of Vasicek (1977) and Cox, Ingersoll, and Ross
More informationLecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model Recall that the price of an option is equal to
More informationLEAN SUPPLY CHAIN MANAGEMENT TOPICS IN. Marc J. Schniederjans. Dara G. Schniederjans. Ashlyn M. Schniederjans. :.,.' If I.
TOPICS IN LEAN SUPPLY CHAIN MANAGEMENT Marc J. Schniederjans University of Nebraska-Lincoln, USA Dara G. Schniederjans Texas Tech University, USA Ashlyn M. Schniederjans Harvard University, USA :.,.' If
More informationOption Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25
Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1 Agenda Forward contracts Definition Determining forward
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 information1 Pricing options using the Black Scholes formula
Lecture 9 Pricing options using the Black Scholes formula Exercise. Consider month options with exercise prices of K = 45. The variance of the underlying security is σ 2 = 0.20. The risk free interest
More informationManagement of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I)
Management of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I) Pin Chung and Rachid Lassoued 5th September, 2014, Wicklow, Ireland 0 Agenda 1. Introduction 2. Review of
More informationValuation, Pricing of Options / Use of MATLAB
CS-5 Computational Tools and Methods in Finance Tom Coleman Valuation, Pricing of Options / Use of MATLAB 1.0 Put-Call Parity (review) Given a European option with no dividends, let t current time T exercise
More informationTutorial: Structural Models of the Firm
Tutorial: Structural Models of the Firm Peter Ritchken Case Western Reserve University February 16, 2015 Peter Ritchken, Case Western Reserve University Tutorial: Structural Models of the Firm 1/61 Tutorial:
More informationKnowledge Management
Series on Innovation and Knowledge - Vol. 11 Knowledge An Interdisciplinary Perspective Sajjad M Jasimuddin Aberystwyth University, UK World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG
More informationNumerical Methods for Engineers
Steven C. Chapra Berger Chair in Computing and Engineering Tufts University RaymondP. Canale Professor Emeritus of Civil Engineering University of Michigan Numerical Methods for Engineers With Software
More informationVolatility as an indicator of Supply and Demand for the Option. the price of a stock expressed as a decimal or percentage.
Option Greeks - Evaluating Option Price Sensitivity to: Price Changes to the Stock Time to Expiration Alterations in Interest Rates Volatility as an indicator of Supply and Demand for the Option Different
More informationOptions On Credit Default Index Swaps
Options On Credit Default Index Swaps Yunkang Liu and Peter Jäckel 20th May 2005 Abstract The value of an option on a credit default index swap consists of two parts. The first one is the protection value
More informationWeek 13 Introduction to the Greeks and Portfolio Management:
Week 13 Introduction to the Greeks and Portfolio Management: Hull, Ch. 17; Poitras, Ch.9: I, IIA, IIB, III. 1 Introduction to the Greeks and Portfolio Management Objective: To explain how derivative portfolios
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 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 Hans-Fredo List Swiss Reinsurance Company Mythenquai 50/60, CH-8022
More informationlife science data mining
life science data mining - '.)'-. < } ti» (>.:>,u» c ~'editors Stephen Wong Harvard Medical School, USA Chung-Sheng Li /BM Thomas J Watson Research Center World Scientific NEW JERSEY LONDON SINGAPORE.
More informationFinal Exam MØA 155 Financial Economics Fall 2009 Permitted Material: Calculator
University of Stavanger (UiS) Stavanger Masters Program Final Exam MØA 155 Financial Economics Fall 2009 Permitted Material: Calculator The number in brackets is the weight for each problem. The weights
More informationTABLE 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 informationEXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationTrading Strategies Involving Options. Chapter 11
Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A risk-free bond and an option to create a principal-protected note A stock and an option Two or more options of the same type
More informationSUPER COMPUTER CONSULTING INC.
SUPER COMPUTER CONSULTING INC. 1070 Westfield Way, Mundelein, IL 60060 USA Phone: (847) 837-0200 Fax: (847) 837-0228 e-mail: info@supercc.com http://www.supercc.com EXOTIC OPTIONS Including Second Generation
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationOpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem
OpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem Marc Henrard marc@opengamma.com OpenGamma Quantitative Research n. 1 November 2011 Abstract
More informationIntroduction Pricing Effects Greeks Summary. Vol Target Options. Rob Coles. February 7, 2014
February 7, 2014 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity The Basic Idea Basket split between risky asset and cash Chose weight of risky asset w to keep volatility
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 informationHedging Pension Liabilities
Hedging Pension Liabilities when there are incomplete markets and regulatory uncertainty Sampension, 2012 Outline Introduction 1 Introduction 2 3 4 Outline Introduction 1 Introduction 2 3 4 Sampension
More informationFinancial Options: Pricing and Hedging
Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial
More informationSession X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London
Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option
More informationJournal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997
Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 AN EMPIRICAL INVESTIGATION OF PUT OPTION PRICING: A SPECIFICATION TEST OF AT-THE-MONEY OPTION IMPLIED VOLATILITY Hongshik Kim,
More informationChapter 15 OPTIONS ON MONEY MARKET FUTURES
Page 218 The information in this chapter was last updated in 1993. Since the money market evolves very rapidly, recent developments may have superseded some of the content of this chapter. Chapter 15 OPTIONS
More informationCall Price as a Function of the Stock Price
Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived
More informationUnderlier Filters Category Data Field Description
Price//Capitalization Market Capitalization The market price of an entire company, calculated by multiplying the number of shares outstanding by the price per share. Market Capitalization is not applicable
More informationLecture 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 informationCHAPTER 11 INTRODUCTION TO SECURITY VALUATION TRUE/FALSE QUESTIONS
1 CHAPTER 11 INTRODUCTION TO SECURITY VALUATION TRUE/FALSE QUESTIONS (f) 1 The three step valuation process consists of 1) analysis of alternative economies and markets, 2) analysis of alternative industries
More informationn(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3
ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric
More informationMULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. EL-JAHEL
ISSN 1744-6783 MULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. EL-JAHEL Tanaka Business School Discussion Papers: TBS/DP04/12 London: Tanaka Business School, 2004 Multiple Defaults and Merton s Model
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a non-dividend paying stock whose price is initially S 0. Divide time into small
More informationb. June expiration: 95-23 = 95 + 23/32 % = 95.71875% or.9571875.9571875 X $100,000 = $95,718.75.
ANSWERS FOR FINANCIAL RISK MANAGEMENT A. 2-4 Value of T-bond Futures Contracts a. March expiration: The settle price is stated as a percentage of the face value of the bond with the final "27" being read
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
More informationHedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)
Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March
More information1.1 Some General Relations (for the no dividend case)
1 American Options Most traded stock options and futures options are of American-type while most index options are of European-type. The central issue is when to exercise? From the holder point of view,
More informationChapter 13 The Black-Scholes-Merton Model
Chapter 13 The Black-Scholes-Merton Model March 3, 009 13.1. The Black-Scholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
More informationBusiness. Control Administration. Alessandro Colantonio. Bay31 GmbH, Switzerland. Roberto Di Pietro. Universita di Roma Tre, Italy.
BEIJING SHANGHAI HONG TAIPEI Role Mining in Business Taming Role-Based Access Control Administration Alessandro Colantonio Bay31 GmbH, Switzerland Roberto Di Pietro Universita di Roma Tre, Italy Alberto
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (risk-neutral)
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationWrite clearly; the grade will also take into account the quality of the presentation and the clarity of the explanations
Name: Student-ID number: Write clearly; the grade will also take into account the quality of the presentation and the clarity of the explanations Question Points Score 1 29 2 17 3 19 4 2 5 2 6 1 Total:
More informationS 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 informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More informationFINANCIAL ENGINEERING CLUB TRADING 201
FINANCIAL ENGINEERING CLUB TRADING 201 STOCK PRICING It s all about volatility Volatility is the measure of how much a stock moves The implied volatility (IV) of a stock represents a 1 standard deviation
More informationWeek 12. Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14.
Week 12 Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14. 1 Options on Stock Indices and Currencies Objective: To explain the basic asset pricing techniques used
More informationFinancial Mathematics Exam
2014 Exam 2 Syllabus Financial Mathematics Exam The purpose of the syllabus for this examination is to develop knowledge of the fundamental concepts of financial mathematics and how those concepts are
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The risk-free interest rate r. 6. The
More informationHedging Exotic Options
Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Germany introduction 1-1 Models The Black Scholes model has some shortcomings: - volatility is not
More informationMathematical 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 informationModel-Free Boundaries of Option Time Value and Early Exercise Premium
Model-Free Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 33124-6552 Phone: 305-284-1885 Fax: 305-284-4800
More information