OPTIONS. Options: Definitions. Definitions (Cont) Price of Call at Maturity and Payoff. Payoff from Holding Stock and Riskfree Bond
|
|
- Hilary Spencer
- 5 years ago
- Views:
Transcription
1 OPTIONS Professor Anant K. Sundaram THUNERBIR Spring 2003 Options: efinitions Contingent claim; derivative Right, not obligation when bought (but, not when sold) More general than might first appear Calls, puts American, European Can buy, sell (write)» Notation:» S, T, S T,»,» C, C T,» P, P T,» r, e rt efinitions (Cont) Price of Call at Maturity and Payoff Price of call at maturity: C T = Max{0, S T } Payoff S T Price of Put at Maturity and Payoff Payoff from Holding Stock and Riskfree Bond Price of put at maturity: P T = Max{0, S T } Payoff Payoff Payoff S T Buying a Stock S T Buying a Riskfree Bond S T 1
2 Put-Call Parity Basic idea in options theory. Establishes a formal link between the four fundamental types of securities, stocks, riskfree bonds, calls and puts. Says (in FV terms): S T + P T = C T + Put-Call Parity (Cont) Holding a portfolio that is long one share and one put option on that stock (with exercise price ) gives us a payoff that is identical to that of a portfolio long one call option (at the same exercise price and maturity as the put) and $ worth of riskfree bonds. In PV terms: S + P = C + PV() Buying Stock Buying Call Put-Call Parity: Graphical escription + + Buying Put Buying Riskfree Bond = Portfolio of Stock+Put = Portfolio of Call+Riskfree bond Put-Call parity: Example Say, = $105 on European calls and puts of the same maturity for the same underlying asset. S T P T C T S T + P T C T = = = = = = = = = Binomial Model Example = $50; T = 1 year; r = 5% S u = $55 S = $50 C =? S d = $45 C u =? C d =? Value of call if stock price goes up: C u = Max{0, S T } = Max{0, 5} = $5 Value of call if stock price goes down: C d = Max{0, S T } = Max{0, 5} = $0 2
3 We can create a synthetic portfolio of the underlying stock and a certain amount of riskfree bonds (whose values we already know), to mimic the payoff characteristics of the call option that we are trying to value. Once we create such a portfolio and figure out its value, we would have indirectly priced the option (by using the no-arbitrage logic). General principles in creating the synthetic portfolio:» Such a synthetic portfolio will consist of the underlying stock and the riskfree bond: long the stock, and short the bond.» There will be a specific ratio in which this portfolio value will be related to the option value, called the hedge ratio. Create the synthetic portfolio in this example:» Buy one share of stock ($50) and borrow $45/(1+r f ) = $45/(1.05) = $42.86.» You owe $45 to the lending institution one year from now.» If the stock goes up, this portfolio is worth:$55 $45 = $10» If the stock goes down, this portfolio is worth:$45 $45 = $0 Payoff comparison between synthetic portfolio and the option we are trying to value, given the same underlying states of nature, S u and S d : Portfolio Call Option S u $10 $5 S d $ 0 $0 The portfolio payoff is twice the payoff to the option fi The portfolio is worth twice as much as the option fi The option is worth half as much as the portfolio. Portfolio is worth: $50 $42.86 = $7.14 Therefore, the call option is worth $7.14/2 = $3.57 3
4 What is the put worth? Apply put-call parity: P = C + PV() S = $ $(50/1.05) $50 = $1.19 Multiperiod Binomial Option Valuation (Optional) Similar to single period valuation, except that we want to work backwards from the last period. For instance, if there are N-periods, start with Period N option values, then Period N 1, then N 2, and so forth. Note that in all periods but N, the value of the underlying security is the option value as derived from the subsequent period Equity in a levered firm is like a call option on the value of the firm s assets with an exercise price equal to the face value of debt... Value of equity + Value of debt Owning riskless debt is like owning risky debt and buying a put option with an exercise price equal to the face value of debt... Value of riskless debt = V V Value of risky debt Value of put option Value of equity plus debt = V = Firm value = Face value of debt + V Other corporate finance insights from options... An equity warrant is like a long-dated call option (after adjusting for dilution effects). Convertible debt is just a combination of straight debt plus a warrant. Buying insurance is like buying a put option. Other corporate finance insights from options... A loan guarantee is like a put option on the face value of the loan. A forward contract is the equivalent of buying a call and selling a put for an exercise price equal to the forward price, and for the maturity of the forward contract. 4
5 Assume that markets are perfect and that trading takes place continuously; Further assume that, over time, the movement in asset returns can be modeled as a diffusion process (this is a random walk in continuous time): ds/s = µdt + sdz This means the following: at any point in time, the expected instantaneous percentage change in asset prices is a function of two variables: (i) an expected rate of return µ in a small interval of time, dt, and (ii) an expected standard deviation in the rate of return, s, that gets moved around by a Weiner process, dz. B-S-M observed that we can create a hedge portfolio with the underlying asset and call options which will mimic the payoff from having risk-free debt. Since we know the rate of return on risk free debt, we can calculate its price; we can also observe the price of the underlying asset; thus we can derive the value of the only unknown, the call option. Once we know the value of the call, we can derive the value of the put using put-call parity. The mathematics produces the following formulae for pricing (European) call and put options: C = SN(d 1 ) e rt N(d 2 ) P = e rt N( d 2 ) SN( d 1 ), N = Cumulative normal probability that s variable takes on value d 1 ln(s/) + [r+s 2 /2]T d 1 = s T d 2 = d 1 s T Thus, need to know five variables: S,, T, r, and s, to value European options. N(d 1 ) is the option delta i.e., the amount of shares to buy in order to create the replicating portfolio for pricing one option. The amount to borrow in order to create this portfolio is e rt N(d 2 ). 5
6 B-S-M for a ividend-paying When the underlying asset pays dividends, one additional variable is introduced into the option pricing model: the (expected) annualized dividend yield. Call this annualized dividend yield y. B-S-M for a ividend-paying ividends lower the value of the underlying asset (since the ex-dividend price is lower by the extent of the pershare dividend) and therefore lower call option values (and increase put option values). B-S-M for a ividend-paying Early exercise may be optimal in the case of dividend-paying stocks if the dividends are high enough. The reason is that, by exercising prior to the dividend and obtaining the underlying asset, the option holder gets the dividends in addition to the underlying asset. B-S-M for a ividend-paying Likelihood of early exercise will also be affected by the time to expiration. B-S-M for a ividend-paying Stock (Cont) With dividends, the B-S-M model gets modified slightly to the following: C = Se yt N(d 1 ) e rt N(d 2 ) P = e rt N( d 2 ) Se yt N( d 1 ), where y is the annualized dividend yield B-S-M for a ividend-paying and, ln(s/) + [r y+s 2 /2]T d 1 = s T d 2 = d 1 s T Thus, we need to know six variables: S,, T, r, y, and s, to value American options on dividend-paying stocks. 6
7 Calculating Annualized Volatility Can be calculated using daily stock price data (But: How far back to go? How many data points?) Step 1: Calculate time series of returns using the formula ln(p t+1 /P t ) where ln is the natural logarithm, P t+1 is the asset price on day t+1 and P t the price on day t. Step 2: Calculate the s of the time series of returns. Calculating Annualized Volatility (Cont) Step 3: Multiply this daily s by the square root of 250 (If using weekly data, multiply by the square root of 50), to get the annualized s. Note: We can also calculate implied volatilities based on observed option price data. Volatility Calculation: Example Suppose the following are daily stock prices: ay 1 $25 ay 2 $26 ay 3 $23 ay 4 $24 ay 5 $25 : : : : ay 59 $25 ay 60 $22 Volatility Calculation: Example Step 1: Calculate daily returns: ay 1 $25 ln(26/25) = 3.92% ay 2 $26 ln(23/26) = 12.26% ay 3 $23 ln(24/23) = 4.26% ay 4 $24 ln(25/24) = 4.08% ay 5 $25 : : : : : : : ay 59 $25 : ay 60 $22 ln(22/25) = 12.78% aily returns Volatility Calculation: Example Step 2: Calculate the standard deviation of the daily returns ( STEV in Excel); suppose this worked out to 2.3%. Step 3: Since we are using daily data, multiply this number by the square root of 250: (0.023)*( 250) =.3637 The volatility of this stock (annualized) for B-S-M input purposes is 36.37% B-S-M Example: American Option Consider the following data for IBM on October 9, 2002:» Stock price, S = $57.05» Exercise price, = $60.00» Time to maturity, T = 1 month (26 days)» Riskfree rate (3 month T-bill yield), r F = 1.60%» ividend yield = 1.10%» Volatility, s = 42.75% 7
8 B-S-M Example: American Option Using the NUMA options calculator:» Price of the call = $1.45» Price of the put = $4.38 The actual price of the call and put were $1.45 and $4.30 respectively. Pretty decent estimates! How Changes in Underlying Variables Affect Call & Put Values Variable Impact on Call Impact on Put Explanation Increase in S Call: Further in-the-money Put: Further out-of-the-money Increase in Increase in s Increase in T Increase in r f Call and Put: Reduces the PV Increase in y of the underlying asset Call: Less to gain on exercise Put: More to gain on exercise Call: on t care about downside Put: on t care about upside Call and Put: Increased prob. of finishing in-the-money Call and Put: Reduces the PV of exercise price 8
B. Combinations. 1. Synthetic Call (Put-Call Parity). 2. Writing a Covered Call. 3. Straddle, Strangle. 4. Spreads (Bull, Bear, Butterfly).
1 EG, Ch. 22; Options I. Overview. A. Definitions. 1. Option - contract in entitling holder to buy/sell a certain asset at or before a certain time at a specified price. Gives holder the right, but not
More informationOptions Markets: Introduction
17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value
More informationAppendix: Basics of Options and Option Pricing Option Payoffs
Appendix: Basics of Options and Option Pricing An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise
More informationCHAPTER 10 OPTION PRICING - II. Derivatives and Risk Management By Rajiv Srivastava. Copyright Oxford University Press
CHAPTER 10 OPTION PRICING - II Options Pricing II Intrinsic Value and Time Value Boundary Conditions for Option Pricing Arbitrage Based Relationship for Option Pricing Put Call Parity 2 Binomial Option
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More informationIn general, the value of any asset is the present value of the expected cash flows on
ch05_p087_110.qxp 11/30/11 2:00 PM Page 87 CHAPTER 5 Option Pricing Theory and Models In general, the value of any asset is the present value of the expected cash flows on that asset. This section will
More informationRisk Management Using Derivatives Securities
Risk Management Using Derivatives Securities 1 Definition of Derivatives A derivative is a financial instrument whose value is derived from the price of a more basic asset called the underlying asset.
More informationCHAPTER 27: OPTION PRICING THEORY
CHAPTER 27: OPTION PRICING THEORY 27-1 a. False. The reverse is true. b. True. Higher variance increases option value. c. True. Otherwise, arbitrage will be possible. d. False. Put-call parity can cut
More informationOption Pricing. Simple Arbitrage Relations. Payoffs to Call and Put Options. Black-Scholes Model. Put-Call Parity. Implied Volatility
Simple Arbitrage Relations Payoffs to Call and Put Options Black-Scholes Model Put-Call Parity Implied Volatility Option Pricing Options: Definitions A call option gives the buyer the right, but not the
More informationImportant Concepts LECTURE 3.2: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL. Applications of Logarithms and Exponentials in Finance
Important Concepts The Black Scholes Merton (BSM) option pricing model LECTURE 3.2: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL Black Scholes Merton Model as the Limit of the Binomial Model Origins
More informationAppendix A Financial Calculations
Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY
More informationRisk Neutral Valuation, the Black-
Risk Neutral Valuation, the Black- Scholes Model and Monte Carlo Stephen M Schaefer London Business School Credit Risk Elective Summer 01 C = SN( d )-PV( X ) N( ) N he Black-Scholes formula 1 d (.) : cumulative
More informationCHAPTER 17 OPTIONS AND CORPORATE FINANCE
CHAPTER 17 OPTIONS AND CORPORATE FINANCE Answers to Concept Questions 1. A call option confers the right, without the obligation, to buy an asset at a given price on or before a given date. A put option
More informationEconomic Risk and Decision Analysis for Oil and Gas Industry CE School of Engineering and Technology Asian Institute of Technology
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More information2 The binomial pricing model
2 The binomial pricing model 2. Options and other derivatives A derivative security is a financial contract whose value depends on some underlying asset like stock, commodity (gold, oil) or currency. The
More information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
More informationFinancial Markets & Risk
Financial Markets & Risk Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA259 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com Session 3 Derivatives Binomial
More informationFINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other
More informationOptions, Futures and Structured Products
Options, Futures and Structured Products Jos van Bommel Aalto Period 5 2017 Options Options calls and puts are key tools of financial engineers. A call option gives the holder the right (but not the obligation)
More informationCash Flows on Options strike or exercise price
1 APPENDIX 4 OPTION PRICING In general, the value of any asset is the present value of the expected cash flows on that asset. In this section, we will consider an exception to that rule when we will look
More informationLECTURE 12. Volatility is the question on the B/S which assumes constant SD throughout the exercise period - The time series of implied volatility
LECTURE 12 Review Options C = S e -δt N (d1) X e it N (d2) P = X e it (1- N (d2)) S e -δt (1 - N (d1)) Volatility is the question on the B/S which assumes constant SD throughout the exercise period - The
More informationAny asset that derives its value from another underlying asset is called a derivative asset. The underlying asset could be any asset - for example, a
Options Week 7 What is a derivative asset? Any asset that derives its value from another underlying asset is called a derivative asset. The underlying asset could be any asset - for example, a stock, bond,
More informationValuing Put Options with Put-Call Parity S + P C = [X/(1+r f ) t ] + [D P /(1+r f ) t ] CFA Examination DERIVATIVES OPTIONS Page 1 of 6
DERIVATIVES OPTIONS A. INTRODUCTION There are 2 Types of Options Calls: give the holder the RIGHT, at his discretion, to BUY a Specified number of a Specified Asset at a Specified Price on, or until, a
More informationFINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other
More informationOptions (2) Class 20 Financial Management,
Options (2) Class 20 Financial Management, 15.414 Today Options Option pricing Applications: Currency risk and convertible bonds Reading Brealey and Myers, Chapter 20, 21 2 Options Gives the holder the
More informationDynamic Hedging and PDE Valuation
Dynamic Hedging and PDE Valuation Dynamic Hedging and PDE Valuation 1/ 36 Introduction Asset prices are modeled as following di usion processes, permitting the possibility of continuous trading. This environment
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationReview of Derivatives I. Matti Suominen, Aalto
Review of Derivatives I Matti Suominen, Aalto 25 SOME STATISTICS: World Financial Markets (trillion USD) 2 15 1 5 Securitized loans Corporate bonds Financial institutions' bonds Public debt Equity market
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationForwards, Futures, Options and Swaps
Forwards, Futures, Options and Swaps A derivative asset is any asset whose payoff, price or value depends on the payoff, price or value of another asset. The underlying or primitive asset may be almost
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationSOA Exam MFE Solutions: May 2007
Exam MFE May 007 SOA Exam MFE Solutions: May 007 Solution 1 B Chapter 1, Put-Call Parity Let each dividend amount be D. The first dividend occurs at the end of months, and the second dividend occurs at
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationAttempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS MTHE6026A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are
More informationBlack-Scholes-Merton Model
Black-Scholes-Merton Model Weerachart Kilenthong University of the Thai Chamber of Commerce c Kilenthong 2017 Weerachart Kilenthong University of the Thai Chamber Black-Scholes-Merton of Commerce Model
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More information******************************* The multi-period binomial model generalizes the single-period binomial model we considered in Section 2.
Derivative Securities Multiperiod Binomial Trees. We turn to the valuation of derivative securities in a time-dependent setting. We focus for now on multi-period binomial models, i.e. binomial trees. This
More information( 0) ,...,S N ,S 2 ( 0)... S N S 2. N and a portfolio is created that way, the value of the portfolio at time 0 is: (0) N S N ( 1, ) +...
No-Arbitrage Pricing Theory Single-Period odel There are N securities denoted ( S,S,...,S N ), they can be stocks, bonds, or any securities, we assume they are all traded, and have prices available. Ω
More informationHedging and Pricing in the Binomial Model
Hedging and Pricing in the Binomial Model Peter Carr Bloomberg LP and Courant Institute, NYU Continuous Time Finance Lecture 2 Wednesday, January 26th, 2005 One Period Model Initial Setup: 0 risk-free
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 19 & 6 th, 1 he Black-Scholes-Merton Model for Options plus Applications Where we are Previously: Modeling the Stochastic Process for Derivative
More informationAttempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS MTHE6026A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are
More informationFixed-Income Options
Fixed-Income Options Consider a two-year 99 European call on the three-year, 5% Treasury. Assume the Treasury pays annual interest. From p. 852 the three-year Treasury s price minus the $5 interest could
More informationThéorie Financière. Financial Options
Théorie Financière Financial Options Professeur André éfarber Options Objectives for this session: 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationFIN 451 Exam Answers, November 8, 2007
FIN 45 Exam Answers, November 8, 007 Phil Dybvig This is a closed-book examination. Answer all questions as directed. Mark your answers directly on the examination. On the valuation question, make sure
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More informationECO OPTIONS AND FUTURES SPRING Options
ECO-30004 OPTIONS AND FUTURES SPRING 2008 Options These notes describe the payoffs to European and American put and call options the so-called plain vanilla options. We consider the payoffs to these options
More information#10. Problems 1 through 10, part a). Fi8000 Practice Set #1 Check Solutions 1. Payoff. Payoff #8 Payoff S
Problems 1 through 1, part a). #1 #2 #3-1 -1-1 #4 #5 #6-1 -1-1 #7 #8 #9-1 -1-1 #1-1 Fi8 Practice et #1 Check olutions 1 Problem b) Profitable Range c) Maximum Profit c) Maximum Loss 1 < $22.8 $12.8 Unlimited
More informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationI. Reading. A. BKM, Chapter 20, Section B. BKM, Chapter 21, ignore Section 21.3 and skim Section 21.5.
Lectures 23-24: Options: Valuation. I. Reading. A. BKM, Chapter 20, Section 20.4. B. BKM, Chapter 21, ignore Section 21.3 and skim Section 21.5. II. Preliminaries. A. Up until now, we have been concerned
More informationValuing Stock Options: The Black-Scholes-Merton Model. Chapter 13
Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 1 The Black-Scholes-Merton Random Walk Assumption l Consider a stock whose price is S l In a short period of time of length t the return
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationChapter 17. Options and Corporate Finance. Key Concepts and Skills
Chapter 17 Options and Corporate Finance Prof. Durham Key Concepts and Skills Understand option terminology Be able to determine option payoffs and profits Understand the major determinants of option prices
More informationValuation of Options: Theory
Valuation of Options: Theory Valuation of Options:Theory Slide 1 of 49 Outline Payoffs from options Influences on value of options Value and volatility of asset ; time available Basic issues in valuation:
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 3. Uncertainty and Risk Uncertainty and risk lie at the core of everything we do in finance. In order to make intelligent investment and hedging decisions, we need
More informationDerivatives Options on Bonds and Interest Rates. Professor André Farber Solvay Business School Université Libre de Bruxelles
Derivatives Options on Bonds and Interest Rates Professor André Farber Solvay Business School Université Libre de Bruxelles Caps Floors Swaption Options on IR futures Options on Government bond futures
More informationRisk and Return and Portfolio Theory
Risk and Return and Portfolio Theory Intro: Last week we learned how to calculate cash flows, now we want to learn how to discount these cash flows. This will take the next several weeks. We know discount
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives November 5, 212 Option Analysis and Modeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis and Modeling:
More informationPractical Hedging: From Theory to Practice. OSU Financial Mathematics Seminar May 5, 2008
Practical Hedging: From Theory to Practice OSU Financial Mathematics Seminar May 5, 008 Background Dynamic replication is a risk management technique used to mitigate market risk We hope to spend a certain
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationChapter 22: Real Options
Chapter 22: Real Options-1 Chapter 22: Real Options I. Introduction to Real Options A. Basic Idea => firms often have the ability to wait to make a capital budgeting decision => may have better information
More informationChapter 5. Risk Handling Techniques: Diversification and Hedging. Risk Bearing Institutions. Additional Benefits. Chapter 5 Page 1
Chapter 5 Risk Handling Techniques: Diversification and Hedging Risk Bearing Institutions Bearing risk collectively Diversification Examples: Pension Plans Mutual Funds Insurance Companies Additional Benefits
More informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
More informationFixed Income and Risk Management
Fixed Income and Risk Management Fall 2003, Term 2 Michael W. Brandt, 2003 All rights reserved without exception Agenda and key issues Pricing with binomial trees Replication Risk-neutral pricing Interest
More informationLecture 1 Definitions from finance
Lecture 1 s from finance Financial market instruments can be divided into two types. There are the underlying stocks shares, bonds, commodities, foreign currencies; and their derivatives, claims that promise
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationCHAPTER 20 Spotting and Valuing Options
CHAPTER 20 Spotting and Valuing Options Answers to Practice Questions The six-month call option is more valuable than the six month put option since the upside potential over time is greater than the limited
More information3.1 Simple Interest. Definition: I = Prt I = interest earned P = principal ( amount invested) r = interest rate (as a decimal) t = time
3.1 Simple Interest Definition: I = Prt I = interest earned P = principal ( amount invested) r = interest rate (as a decimal) t = time An example: Find the interest on a boat loan of $5,000 at 16% for
More informationActuarialBrew.com. Exam MFE / 3F. Actuarial Models Financial Economics Segment. Solutions 2014, 2nd edition
ActuarialBrew.com Exam MFE / 3F Actuarial Models Financial Economics Segment Solutions 04, nd edition www.actuarialbrew.com Brewing Better Actuarial Exam Preparation Materials ActuarialBrew.com 04 Please
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 18 & 5 th, 13 he Black-Scholes-Merton Model for Options plus Applications 11.1 Where we are Last Week: Modeling the Stochastic Process for
More informationThis chapter discusses the valuation of European currency options. A European
Options on Foreign Exchange, Third Edition David F. DeRosa Copyright 2011 David F. DeRosa CHAPTER 3 Valuation of European Currency Options This chapter discusses the valuation of European currency options.
More informationEvaluating the Black-Scholes option pricing model using hedging simulations
Bachelor Informatica Informatica Universiteit van Amsterdam Evaluating the Black-Scholes option pricing model using hedging simulations Wendy Günther CKN : 6052088 Wendy.Gunther@student.uva.nl June 24,
More informationDepartment of Mathematics. Mathematics of Financial Derivatives
Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, 2009 2pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5 1. (a) Suppose 0 < E 1 < E 3 and E 2
More informationLecture 8. Spring Semester, Rutgers University. Lecture 8. Options Markets and Pricing. Prof. Paczkowski
Rutgers University Spring Semester, 2009 (Rutgers University) Spring Semester, 2009 1 / 31 Part I Assignment (Rutgers University) Spring Semester, 2009 2 / 31 Assignment (Rutgers University) Spring Semester,
More informationStochastic Modelling in Finance
in Finance Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH April 2010 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationLecture 16: Delta Hedging
Lecture 16: Delta Hedging We are now going to look at the construction of binomial trees as a first technique for pricing options in an approximative way. These techniques were first proposed in: J.C.
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationFIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More informationOPTION PRICING: BASICS
1 OPTION PRICING: BASICS The ingredients that make an op?on 2 An op?on provides the holder with the right to buy or sell a specified quan?ty of an underlying asset at a fixed price (called a strike price
More informationS u =$55. S u =S (1+u) S=$50. S d =$48.5. S d =S (1+d) C u = $5 = Max{55-50,0} $1.06. C u = Max{Su-X,0} (1+r) (1+r) $1.06. C d = $0 = Max{48.
Fi8000 Valuation of Financial Assets Spring Semester 00 Dr. Isabel katch Assistant rofessor of Finance Valuation of Options Arbitrage Restrictions on the Values of Options Quantitative ricing Models Binomial
More informationCHAPTER 7 INVESTMENT III: OPTION PRICING AND ITS APPLICATIONS IN INVESTMENT VALUATION
CHAPTER 7 INVESTMENT III: OPTION PRICING AND ITS APPLICATIONS IN INVESTMENT VALUATION Chapter content Upon completion of this chapter you will be able to: explain the principles of option pricing theory
More informationChapter 5. Financial Forwards and Futures. Copyright 2009 Pearson Prentice Hall. All rights reserved.
Chapter 5 Financial Forwards and Futures Introduction Financial futures and forwards On stocks and indexes On currencies On interest rates How are they used? How are they priced? How are they hedged? 5-2
More informationMS-E2114 Investment Science Exercise 10/2016, Solutions
A simple and versatile model of asset dynamics is the binomial lattice. In this model, the asset price is multiplied by either factor u (up) or d (down) in each period, according to probabilities p and
More informationOption pricing models
Option pricing models Objective Learn to estimate the market value of option contracts. Outline The Binomial Model The Black-Scholes pricing model The Binomial Model A very simple to use and understand
More informationOptions in Corporate Finance
FIN 614 Corporate Applications of Option Theory Professor Robert B.H. Hauswald Kogod School of Business, AU Options in Corporate Finance The value of financial and managerial flexibility: everybody values
More informationFinal Exam. Please answer all four questions. Each question carries 25% of the total grade.
Econ 174 Financial Insurance Fall 2000 Allan Timmermann UCSD Final Exam Please answer all four questions. Each question carries 25% of the total grade. 1. Explain the reasons why you agree or disagree
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationTEACHING NOTE 98-04: EXCHANGE OPTION PRICING
TEACHING NOTE 98-04: EXCHANGE OPTION PRICING Version date: June 3, 017 C:\CLASSES\TEACHING NOTES\TN98-04.WPD The exchange option, first developed by Margrabe (1978), has proven to be an extremely powerful
More informationMS-E2114 Investment Science Lecture 10: Options pricing in binomial lattices
MS-E2114 Investment Science Lecture 10: Options pricing in binomial lattices A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationGlobal Financial Management. Option Contracts
Global Financial Management Option Contracts Copyright 1997 by Alon Brav, Campbell R. Harvey, Ernst Maug and Stephen Gray. All rights reserved. No part of this lecture may be reproduced without the permission
More informationChapter 22: Real Options
Chapter 22: Real Options-1 Chapter 22: Real Options I. Introduction to Real Options A. Basic Idea B. Valuing Real Options Basic idea: can use any of the option valuation techniques developed for financial
More informationK = 1 = -1. = 0 C P = 0 0 K Asset Price (S) 0 K Asset Price (S) Out of $ In the $ - In the $ Out of the $
Page 1 of 20 OPTIONS 1. Valuation of Contracts a. Introduction The Value of an Option can be broken down into 2 Parts 1. INTRINSIC Value, which depends only upon the price of the asset underlying the option
More informationDiscrete Probability Distribution
1 Discrete Probability Distribution Key Definitions Discrete Random Variable: Has a countable number of values. This means that each data point is distinct and separate. Continuous Random Variable: Has
More informationArbitrage, Martingales, and Pricing Kernels
Arbitrage, Martingales, and Pricing Kernels Arbitrage, Martingales, and Pricing Kernels 1/ 36 Introduction A contingent claim s price process can be transformed into a martingale process by 1 Adjusting
More informationCourse MFE/3F Practice Exam 2 Solutions
Course MFE/3F Practice Exam Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 A Chapter 16, Black-Scholes Equation The expressions for the value
More informationSOLUTIONS. Solution. The liabilities are deterministic and their value in one year will be $ = $3.542 billion dollars.
Illinois State University, Mathematics 483, Fall 2014 Test No. 1, Tuesday, September 23, 2014 SOLUTIONS 1. You are the investment actuary for a life insurance company. Your company s assets are invested
More informationChapter 21: Option Valuation
Chapter 21: Option Valuation-1 Chapter 21: Option Valuation I. The Binomial Option Pricing Moel Intro: 1. Goal: to be able to value options 2. Basic approach: 3. Law of One Price: 4. How it will help:
More information