# Options on an Asset that Yields Continuous Dividends

Size: px
Start display at page:

Transcription

1 Finance 400 A. Penati - G. Pennacchi Options on an Asset that Yields Continuous Dividends I. Risk-Neutral Price Appreciation in the Presence of Dividends Options are often written on what can be interpreted as an asset that pays a continuous dividend that is proportional to the asset s price. Let us consider how to value such options for a general case, then we will look at specific examples. Suppose there is a European call option that is written on an asset that pays a continuous dividend yield equal to q. Usingthe risk-neutral valuation technique: c = e rτ Ê [max (0,S T X)] (1) where the risk-neutral expectation, Ê [ ], requires that the expected rate of return on the underlying asset equals the risk-free rate, r. Thisasset stotal rate of return is composed of both dividend payments and price appreciation (capital gains). If no dividends are paid, then Ê [S T ]=Se rτ. However, with dividends, the expected asset price appreciates at rate r q, rather than r. This is because with dividends paid out at rate q, price appreciation must be at rate r q to keep the total expected rate of return = q + r q = r. Thus, the risk-neutral expectation of S T is Ê [S T ] = Se (r q)τ () = Se qτ e rτ = S e rτ wherewedefine S Se qτ. The above shows that the value of an option on a dividend-paying asset with current price S equals the value of an option on a non-dividend paying asset having current price S = Se qτ. Hence, we can value a European call or put option on a dividend-paying asset using the Black-Scholes formula, but replacing S with S = Se qτ. c = Se qτ N (d 1 ) Xe rτ N (d ) (3) 1

2 and ³ where d 1 = ln ³ Se qτ + X r+ σ II. Stock Index Options p = Xe rτ N ( d ) Se qτ N ( d 1 ) (4) τ σ τ,andd = d 1 σ τ. Several exchanges trade options on U.S. and foreign stock market indices. They are cash settled: upon exercise, the holder receives the difference between the index and the strike price. Index options can be American or European. A stock index equals the value of a stock portfolio. If this portfolio is assumed to follow a diffusion process with a constant rate of return variance, we can apply the previous formula. q is set equal to the average dividend yield on the stocks during the life of the option. Example: Suppose a stock index currently equals Consider the value of a put option with exercise price of 450 and time until maturity of 15 weeks. Assuming a (historically estimated) q =.7%, σ = 17%, r =6%,andτ =15/5, we have p = Xe rτ N ( d ) Se qτ N ( d 1 )=\$3.94 III. Foreign Currency Options Foreign currency options are sold over-the-counter and on exchanges. Buying a foreign currency allows us to invest in a foreign bond paying the foreign interest rate r f.thus,aunit of foreign currency can be considered an asset having a dividend yield equal to r f.define the current U.S. \$ value of a unit of foreign currency as S, which is the spot exchange rate. S can then be viewed as the current U.S. \$ value of an investment paying a dividend of r f.thusthe risk-neutral expectation of this asset s value at date T is Ê [S T ] = Se (r q)τ (5) = Se (r r f)τ If the exchange rate is assumed to have a constant rate of return volatility equal to σ, the

3 value of European puts and calls on a unit of foreign currency are c = Se r f τ N (d 1 ) Xe rτ N (d ) (6) ³ Se r f τ ³ + r+ σ p = Xe rτ N ( d ) Se r f τ N ( d 1 ) (7) where d 1 = ln X σ,andd τ = d 1 σ τ. Recall that the forward exchange rate, F,isgivenby 1 τ F = Se (r r f)τ (8) Substituting Fe rτ = Se r f τ into the above currency option formulas, they can be re-written in terms of the forward rate. c = e rτ [FN (d 1 ) XN (d )] (9) where d 1 = ln( F X )+ σ σ τ τ IV. Futures Options p = e rτ [XN ( d ) FN ( d 1 )] (10),andd = d 1 σ τ. Upon exercising a call option on a futures contract, the writer must deliver a long position in an underlying futures contract plus cash equal to the difference between the current futures price and the option strike price, F T X. Upon exercising a put option on a futures contract, the writer must deliver a short position in an underlying futures contract plus cash equal to the difference between the strike price and the current futures price, X F T. Since the holder can immediately close out the futures position at no cost, a futures option is essentially one that is written on a futures price. Options are written on futures of commodities, equities, bonds, and currencies. Futures options can be attractive since it may be easier to deliver an underlying futures contract rather than a physical commodity or asset. 1 This is the same formula derived earlier, but with interest rates quoted on a continuously-compounded basis. 3

4 The relationship between spot and futures prices is often written in terms of the cost of carry formula: F = Se (c y)τ Se ατ (11) where α c y is the cost of carry less any convenience yield. For example, the cost of carry for a dividend-yielding security is c = r q. Commodities are often considered to have a convenience yield, in which case y is non-zero. Assuming α is non-random, if S has a volatility of σ, sodoesf. Since futures options are written on F T,notS T,weneedtoconsiderthe risk-neutral expectation of F T. The undiscounted profit from a long position in a futures contract which was entered into at the initial futures price of F andhelduntilthematurityofthecontractiss T F = F T F. Since it costs zero to enter into a futures contract, what would be the expected profit ina risk-neutral world? It must be zero. Therefore, or Ê [F T F ]=0 (1) F = Ê [F T ] (13) Thus, while a non-dividend paying asset would be expected to grow at rate r, futures prices would be expected to grow at rate 0. Hence, futures are like assets with a dividend yield q = r. From this we can derive the value of futures options as: c = e rτ [FN (d 1 ) XN (d )] (14) p = e rτ [XN ( d ) FN ( d 1 )] (15) Technically, this is the profit on a forward contract. Futures contracts are marked-to-market daily, so that this would be the profit on a futures contract over a single day. However, in a risk-neutral world, the relationship in (1) holds for either forward or futures contracts. 4

5 where d 1 = ln( F X )+ σ τ σ,andd τ = d 1 σ τ. Example: Let the current futures price written on a stock index and having a time until maturity of 15 weeks be Consider the value of a futures put option on this stock index having a strike price of 475 and also having a maturity of 15 weeks. Assuming σ =15%,r =6%, τ =15/5, and setting F = and X = 475, we have p = e rτ [XN ( d ) FN ( d 1 )]=\$8.95 (16) One can derive a put-call parity relationship for European futures options: At date t: Portfolio A: one futures call plus a bond having an initial value equal to Xe rτ. Portfolio B: one futures put plus a bond having an initial value equal to Fe rτ plus a long position in a futures contract. At date T: Portfolio A: max (0,F T X)+X =max(f T,X) Portfolio B: max(0,x F T )+F + F T F = max (F T,X) Since Portfolios A and B have the same date T value, their values at date t must be the same, implying c + Xe rτ = p + Fe rτ (17) 5

### Finance 400 A. Penati - G. Pennacchi. Option Pricing

Finance 400 A. Penati - G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary

### Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

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

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

### Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.

Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards

### Lecture 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

### Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies

Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies No-arbitrage bounds on option prices Binomial option pricing Black-Scholes-Merton

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

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

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

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

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### Chapter 1: Financial Markets and Financial Derivatives

Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange

### Options. Moty Katzman. September 19, 2014

Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to

### Figure S9.1 Profit from long position in Problem 9.9

Problem 9.9 Suppose that a European call option to buy a share for \$100.00 costs \$5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances

### Session 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

### Option 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

### Introduction to Options. Derivatives

Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

### Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

### Option Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)

Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.

### A SNOWBALL CURRENCY OPTION

J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA E-mail address: gshim@ajou.ac.kr ABSTRACT. I introduce

### CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given

### Chapter 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)

### American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

### Options: Valuation and (No) Arbitrage

Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

### Jung-Soon Hyun and Young-Hee Kim

J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest

### Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

### CHAPTER 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

### FIN-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

### Part V: Option Pricing Basics

erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, put-call parity introduction

### Digital Options. and d 1 = d 2 + σ τ, P int = e rτ[ KN( d 2) FN( d 1) ], with d 2 = ln(f/k) σ2 τ/2

Digital Options The manager of a proprietary hedge fund studied the German yield curve and noticed that it used to be quite steep. At the time of the study, the overnight rate was approximately 3%. The

### Introduction to Binomial Trees

11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths

### Chapter 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

### Lecture 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

### 1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

### Options/1. Prof. Ian Giddy

Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

### CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the put-call parity theorem as follows: P = C S + PV(X) + PV(Dividends)

### Chapter 21 Valuing Options

Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher

### Lecture 5: Forwards, Futures, and Futures Options

OPTIONS and FUTURES Lecture 5: Forwards, Futures, and Futures Options Philip H. Dybvig Washington University in Saint Louis Spot (cash) market Forward contract Futures contract Options on futures Copyright

### 1 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,

### Bond Options, Caps and the Black Model

Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2

### Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US\$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price

Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption

### Week 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

### DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options

DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis review of pricing formulas assets versus futures practical issues call options

### CHAPTER 7: PROPERTIES OF STOCK OPTION PRICES

CHAPER 7: PROPERIES OF SOCK OPION PRICES 7.1 Factors Affecting Option Prices able 7.1 Summary of the Effect on the Price of a Stock Option of Increasing One Variable While Keeping All Other Fixed Variable

### CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support

### ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005

ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005 Options: General [1] Define the following terms associated with options: a. Option An option is a contract which gives the holder

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008. Options

FIN-40008 FINANCIAL INSTRUMENTS 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

### Option pricing. Vinod Kothari

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

### Lecture 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

### Convenient 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

### Martingale Pricing Applied to Options, Forwards and Futures

IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the

### 2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

### Options, Futures, and Other Derivatives 7 th Edition, Copyright John C. Hull 2008 2

Mechanics of Options Markets Chapter 8 Options, Futures, and Other Derivatives, 7th Edition, Copyright John C. Hull 2008 1 Review of Option Types A call is an option to buy A put is an option to sell A

### Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9

Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying

### Factors 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

### Option Premium = Intrinsic. Speculative Value. Value

Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An Option-Pricing Formula Investment in

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### Mid-Term Spring 2003

Mid-Term Spring 2003 1. (1 point) You want to purchase XYZ stock at \$60 from your broker using as little of your own money as possible. If initial margin is 50% and you have \$3000 to invest, how many shares

### The Black-Scholes pricing formulas

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

### Chapter 20 Understanding Options

Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interest-rate options (III) Commodity options D) I, II, and III

### Use the option quote information shown below to answer the following questions. The underlying stock is currently selling for \$83.

Problems on the Basics of Options used in Finance 2. Understanding Option Quotes Use the option quote information shown below to answer the following questions. The underlying stock is currently selling

### Test 4 Created: 3:05:28 PM CDT 1. The buyer of a call option has the choice to exercise, but the writer of the call option has: A.

Test 4 Created: 3:05:28 PM CDT 1. The buyer of a call option has the choice to exercise, but the writer of the call option has: A. The choice to offset with a put option B. The obligation to deliver the

### EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals

EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline Contents 1 Call options and put options 1 2 Payoffs on options

### The Black-Scholes Model

Chapter 4 The Black-Scholes Model 4. Introduction Easily the best known model of option pricing, the Black-Scholes model is also one of the most widely used models in practice. It forms the benchmark model

### Chapter 3: Commodity Forwards and Futures

Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique

### Forwards and Futures

Prof. Alex Shapiro Lecture Notes 16 Forwards and Futures I. Readings and Suggested Practice Problems II. Forward Contracts III. Futures Contracts IV. Forward-Spot Parity V. Stock Index Forward-Spot Parity

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

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

### 9 Basics of options, including trading strategies

ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European

### t = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3

MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate

### 1.1 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,

### or enters into a Futures contract (either on the IPE or the NYMEX) with delivery date September and pay every day up to maturity the margin

Cash-Futures arbitrage processes Cash futures arbitrage consisting in taking position between the cash and the futures markets to make an arbitrage. An arbitrage is a trade that gives in the future some

### Chapter 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

### Mechanics of Options Markets

Mechanics of Options Markets Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 8) Liuren Wu (Baruch) Options Markets Mechanics Options Markets 1 / 22 Outline 1 Definition

### Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined

### Lecture 12. Options Strategies

Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same

### S 1 S 2. Options and Other Derivatives

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

### TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder

### Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.

Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero

### Determination of Forward and Futures Prices. Chapter 5

Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets

### The Valuation of Currency Options

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

### A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

1 Exam FM Questions Practice Exam 1 1. Consider the following yield curve: Year Spot Rate 1 5.5% 2 5.0% 3 5.0% 4 4.5% 5 4.0% Find the four year forward rate. A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 2.

### 2. How is a fund manager motivated to behave with this type of renumeration package?

MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff

### Lecture 5: Put - Call Parity

Lecture 5: Put - Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible

### Fundamentals 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.

### Final 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

### The 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

### Options (1) Class 19 Financial Management, 15.414

Options (1) Class 19 Financial Management, 15.414 Today Options Risk management: Why, how, and what? Option payoffs Reading Brealey and Myers, Chapter 2, 21 Sally Jameson 2 Types of questions Your company,

### Introduction, Forwards and Futures

Introduction, Forwards and Futures Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapters: 1,2,3,5) Liuren Wu Introduction, Forwards & Futures Option Pricing, Fall, 2007 1 / 35

### Accounting for Derivatives. Rajan Chari Senior Manager rchari@deloitte.com

Accounting for Derivatives Rajan Chari Senior Manager rchari@deloitte.com October 3, 2005 Derivative Instruments to be Discussed Futures Forwards Options Swaps Copyright 2004 Deloitte Development LLC.

### Options 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

### Assumptions: No transaction cost, same rate for borrowing/lending, no default/counterparty risk

Derivatives Why? Allow easier methods to short sell a stock without a broker lending it. Facilitates hedging easily Allows the ability to take long/short position on less available commodities (Rice, Cotton,

### CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus \$250 1329.78 = \$332,445. The closing futures price for the March contract was 1364.00,

### Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

### CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options

CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options Paul Obour Supervisor: Dr. Antony Ware University of Calgary PRMIA Luncheon - Bankers Hall, Calgary May 8, 2012 Outline 1 Introductory

### Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).

Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.

### Options. + Concepts and Buzzwords. Readings. Put-Call Parity Volatility Effects

+ Options + Concepts and Buzzwords Put-Call Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options

### a. What is the portfolio of the stock and the bond that replicates the option?

Practice problems for Lecture 2. Answers. 1. A Simple Option Pricing Problem in One Period Riskless bond (interest rate is 5%): 1 15 Stock: 5 125 5 Derivative security (call option with a strike of 8):?

### The Black-Scholes Formula

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

### CHAPTER 22 Options and Corporate Finance

CHAPTER 22 Options and Corporate Finance Multiple Choice Questions: I. DEFINITIONS OPTIONS a 1. A financial contract that gives its owner the right, but not the obligation, to buy or sell a specified asset

### 11 Option. Payoffs and Option Strategies. Answers to Questions and Problems

11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of \$80 and a cost of \$5. Graph the profits and losses at expiration for various