1. Assume that a (European) call option exists on this stock having on exercise price of $155.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1. Assume that a (European) call option exists on this stock having on exercise price of $155."

Transcription

1 MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on risk-free investments is 6% per period. 1. Assume that a (European) call option exists on this stock having on exercise price of $155. (a) How could you form a portfolio based on the stock and the call so as to achieve a risk-free hedge? (b) Compute the price of the call. 2. Answer the above two questions if the exercise price was $18. Exercise 2. Calls, hedge [6] A stock s current price is $1. There are two possible prices at the end of the year: $15 or % 75. A call option to buy one share at $1 at the end of the year sells at $2. Suppose that you are told that a) writing 3 calls, b) buying 2 stocks and c) borrowing $14 today is a perfect hedge portfolio. 1. What is the risk free rate of interest? Exercise 3. Call (RWJ 21.16) [5] You bought a 1-share call contranct three weeks ago. The expirations date of the call is five weeks from today. On that date, the price of the underlying stock will be either $12 or $95. The two states are equally likely to occur. Currently, the stock sells for $96; its strike price is $112. You are able to purchase 32 shares of the stock. You are able to borrow money at 1% per annum, annually compounded. 1. What is the value of your call contract? Exercise 4. Stock A is expected next period to have either a price of 1 or 3. The current stock price for A is 2 and the per period risk free interest rate is 2%. What is the price of a one period call option on A with exercise price of 15? (a). (b) 7.65 (c) 8.72 (d) 15. (e) I choose not to answer. Exercise 5. A stock will next period either be priced at 12% or 8% of todays price. The stock is currently priced at 5. The risk free interest rate is 6%. Consider a call option written on this stock with exercise price equal to 6. What is todays price on this option? (a). (b) 2.3 1

2 (c) 5. (d) 1. (e) I choose not to answer. Exercise 6. HAL [6] You are interested in the computer company HAL computers. Its stock is currently priced at 9. The stock price is expected to either go up by 25% or down by 2% each six months. The annual risk free interest rate is 2%. Your broker now calls you with an interesting offer. You pay C now for the following opportunity: In month 6 you can choose whether or not to buy a call option on HAL computers with 6 months maturity (i.e. expiry is 12 months from now). This option has an exercise price of $9, and costs $1,5. (You have an option on an option.) 1. If C is the fair price for this compound option, find C. 2. If you do not have any choice after 6 months, you have to buy the option, what is then the value of the contract? 2

3 Empirical Solutions MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] 1. We have the following payoff next period: S u = S d = 15 Call option with exercise price 155. C u = max(, ) = 2 C C d = max(, ) = 2. How to get a risk free hedge by buying m call options? Need payoff in each period to be equal S d + mc d = S u + mc u m = S d S u = = 1.25 C u C d 2 Need to sell 1.25 options to create a risk free hedge. 3. To price the option, we find the parameters of the binomial option pricing model. Start by finding u and d: S u = us = u16 = 175 Solve for u and d: Then find the risk-neutral probabilities: q = (1 + r f ) d u d S d = ds = d16 = 15 u = = 1.1 d = =.973 = = =.685 3

4 Finding terminal payoffs C u = max(s u K, ) = max( , ) = 2 Then find price C d = max(s d K, ) = max(15 155, ) = C = qc u + (1 q)c d = = r f With an exercise price of 18, the call will never be exercised. It is worthless. Exercise 2. Calls, hedge [6] 1. A perfect hedge implies that total payoffs are zero in each state. Hence, Hence, Exercise 3. Call (RWJ 21.16) [5] Payoffs t = t = 1 S = 75 S = 15 3C 6 15 Buy stock Borrow (1+r) 14(1+r) 14(1 + r) = 15( for it to be perfect hedge ) r = = 1 14 = 71 7 % 1. Ignoring all the extraneous information, let us find the price of one Call. S = 96 S u = 12 S d = 95 Find u and d: Value.5974 Exercise 4. The stock has price movements: u = = 1.25 d = = S u = 3 S = 2 S d = 1 4

5 The Call has payoffs: C =? C u = (3 15) = 15 C d = u = 3/2 = 1.5, d = 1/2 =.5, q = =.52, C = (.52 15) = (b) is correct Exercise 5. The stock price can never exceed the exercise price, current value equals zero. is correct. Exercise 6. HAL [6] Stock price movement, u = 1.25, d =.8. S uu = S = 9 S u = 1125 S d = 72 S ud = 9 S dd = 576 Start by calculating the value of an option at time 1, with exercise price 9. q = er t d u d = e = q =.322 C u = e r t (qc uu + (1 q)c ud ) = e.2.5 ( ) = 3, C d = You are now offered to pay C for the ability to choose to pay $1,5 in month 6 for this option. This can be analyzed on a tree. Clearly, you will not pay 1,5 in the down state, but will pay it in the up-state. 5

6 3, , 5 = C =? Calculate the value the usual way C = e r t (q (1 q) ) = e.2.5 ( ) = This is the current value of the option on an option. For the second part of the question, if you do not have any choice, this is the picture 3, , 5 = C =? 15 C = e r t (q (1 q) ( 15)) = e.2.5 ( ( 15)) =

One Period Binomial Model

One Period Binomial Model FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

More information

Lecture 4: Derivatives

Lecture 4: Derivatives Lecture 4: Derivatives School of Mathematics Introduction to Financial Mathematics, 2015 Lecture 4 1 Financial Derivatives 2 uropean Call and Put Options 3 Payoff Diagrams, Short Selling and Profit Derivatives

More information

Lecture 11: Risk-Neutral Valuation Steven Skiena. skiena

Lecture 11: Risk-Neutral Valuation Steven Skiena.  skiena Lecture 11: Risk-Neutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Risk-Neutral Probabilities We can

More information

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

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:

More information

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

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

More information

Lecture 5: Put - Call Parity

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

More information

Lecture 9. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 8

Lecture 9. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 8 Lecture 9 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 8 Lecture 9 1 Risk-Neutral Valuation 2 Risk-Neutral World 3 Two-Steps Binomial

More information

Week 1: Futures, Forwards and Options derivative three Hedge: Speculation: Futures Contract: buy or sell

Week 1: Futures, Forwards and Options derivative three Hedge: Speculation: Futures Contract: buy or sell Week 1: Futures, Forwards and Options - A derivative is a financial instrument which has a value which is determined by the price of something else (or an underlying instrument) E.g. energy like coal/electricity

More information

Lecture 3: Put Options and Distribution-Free Results

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

More information

Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values

Option 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 information

Lecture 21 Options Pricing

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

More information

UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Final Exam. December Date:

UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Final Exam. December Date: UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2013 MBA Final Exam December 2013 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open

More information

Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?

Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions

More information

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

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

More information

Chapter 2 Introduction to Option Management

Chapter 2 Introduction to Option Management Chapter 2 Introduction to Option Management The prize must be worth the toil when one stakes one s life on fortune s dice. Dolon to Hector, Euripides (Rhesus, 182) In this chapter we discuss basic concepts

More information

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

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

More information

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

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

More information

BUS-495 Fall 2011 Final Exam: December 14 Answer Key

BUS-495 Fall 2011 Final Exam: December 14 Answer Key Name: Score: BUS-495 Fall 011 Final Exam: December 14 Answer Key 1. (6 points; 1 point each) If you are neutral on a particular stock for the short term and looking to generate some income using options,

More information

Caput Derivatives: October 30, 2003

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

More information

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. Put-Call Parity II. One-Period Binomial Option Pricing III. Adding Periods to the Binomial Model IV. Black-Scholes

More information

7: The CRR Market Model

7: The CRR Market Model Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein

More information

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

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

More information

Option pricing. Vinod Kothari

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

More information

Lecture 11. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7

Lecture 11. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 11 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating

More information

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs) Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A four-year dollar-denominated European put option

More information

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

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

More information

Option Valuation. Chapter 21

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

More information

Part A: The put call parity relation is: call + present value of exercise price = put + stock price.

Part A: The put call parity relation is: call + present value of exercise price = put + stock price. Corporate Finance Mod 20: Options, put call parity relation, Practice Problem s ** Exercise 20.1: Put Call Parity Relation! One year European put and call options trade on a stock with strike prices of

More information

Finance 2 for IBA (30J201) F.Feriozzi Resit exam June 14 th, 2011. Part One: Multiple-Choice Questions (45 points)

Finance 2 for IBA (30J201) F.Feriozzi Resit exam June 14 th, 2011. Part One: Multiple-Choice Questions (45 points) Question 1 Finance 2 for IBA (30J201) F.Feriozzi Resit exam June 14 th, 2011 Part One: Multiple-Choice Questions (45 points) Assume that financial markets are perfect and that the market value of a levered

More information

Two-State Option Pricing

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

More information

Pricing Options: Pricing Options: The Binomial Way FINC 456. The important slide. Pricing options really boils down to three key concepts

Pricing Options: Pricing Options: The Binomial Way FINC 456. The important slide. Pricing options really boils down to three key concepts Pricing Options: The Binomial Way FINC 456 Pricing Options: The important slide Pricing options really boils down to three key concepts Two portfolios that have the same payoff cost the same. Why? A perfectly

More information

Chapter 21 Valuing Options

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

More information

Factors Affecting Option Prices

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

More information

Binomial lattice model for stock prices

Binomial lattice model for stock prices Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

More information

FINANCIAL OPTION ANALYSIS HANDOUTS

FINANCIAL 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 information

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

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

More information

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

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

More information

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

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

More information

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

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

More information

Numerical Methods for Option Pricing

Numerical Methods for Option Pricing Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

More information

Consider a European call option maturing at time T

Consider 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 information

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

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):?

More information

BINOMIAL OPTION PRICING

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

More information

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

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

More information

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

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

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12. Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral

More information

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. 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

More information

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

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

More information

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

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

More information

The Binomial Option Pricing Model André Farber

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

More information

BUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING

BUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING BUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING 3. Suppose there are only two possible future states of the world. In state 1 the stock price rises by 50%. In state 2, the stock price drops by 25%. The

More information

10 Binomial Trees. 10.1 One-step model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1

10 Binomial Trees. 10.1 One-step model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 10 Binomial Trees 10.1 One-step model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t

More information

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

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

More information

Chapter 21: Options and Corporate Finance

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

More information

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).

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.

More information

Lecture 17/18/19 Options II

Lecture 17/18/19 Options II 1 Lecture 17/18/19 Options II Alexander K. Koch Department of Economics, Royal Holloway, University of London February 25, February 29, and March 10 2008 In addition to learning the material covered in

More information

American and European. Put Option

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

More information

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

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

More information

Two-State Model of Option Pricing

Two-State Model of Option Pricing Rendleman and Bartter [1] put forward a simple two-state model of option pricing. As in the Black-Scholes model, to buy the stock and to sell the call in the hedge ratio obtains a risk-free portfolio.

More information

Options Markets: Introduction

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

More information

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

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

More information

Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan

Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction

More information

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

Session 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 information

Introduction to Binomial Trees

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

More information

Lecture 12. Options Strategies

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

More information

Binomial trees and risk neutral valuation

Binomial trees and risk neutral valuation Binomial trees and risk neutral valuation Moty Katzman September 19, 2014 Derivatives in a simple world A derivative is an asset whose value depends on the value of another asset. Call/Put European/American

More information

DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options

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

More information

Put-Call Parity. chris bemis

Put-Call Parity. chris bemis Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain

More information

Figure S9.1 Profit from long position in Problem 9.9

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

More information

Protective Put Strategy Profits

Protective Put Strategy Profits Chapter Part Options and Corporate Finance: Basic Concepts Combinations of Options Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options

More information

Lecture 4: Properties of stock options

Lecture 4: Properties of stock options Lecture 4: Properties of stock options Reading: J.C.Hull, Chapter 9 An European call option is an agreement between two parties giving the holder the right to buy a certain asset (e.g. one stock unit)

More information

Lecture 10. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7

Lecture 10. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 10 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 10 1 Binomial Model for Stock Price 2 Option Pricing on Binomial

More information

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. 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

More information

Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

More information

1 The Black-Scholes model: extensions and hedging

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

More information

Financial Options: Pricing and Hedging

Financial 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 information

The Black-Scholes Formula

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

More information

CHAPTER 20 Understanding Options

CHAPTER 20 Understanding Options CHAPTER 20 Understanding Options Answers to Practice Questions 1. a. The put places a floor on value of investment, i.e., less risky than buying stock. The risk reduction comes at the cost of the option

More information

The n-period Binomial Model

The n-period Binomial Model FIN-0008 FINANCIAL INSTRMENTS SPRING 2008 The n-period Binomial Model Introduction Once we have understood the one period binomial model it is very easy to extend the model to two or more periods so that

More information

Mathematical Finance: Stochastic Modeling of Asset Prices

Mathematical Finance: Stochastic Modeling of Asset Prices Mathematical Finance: Stochastic Modeling of Asset Prices José E. Figueroa-López 1 1 Department of Statistics Purdue University Worcester Polytechnic Institute Department of Mathematical Sciences December

More information

Covered Call Option Strategy

Covered Call Option Strategy The covered call option strategy, also known as a buy write strategy, is implemented by writing (selling) a call option contract while owning an equivalent number of shares of the underlying stock. This

More information

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

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

More information

Introduction to Mathematical Finance

Introduction to Mathematical Finance Introduction to Mathematical Finance R. J. Williams Mathematics Department, University of California, San Diego, La Jolla, CA 92093-0112 USA Email: williams@math.ucsd.edu DO NOT REPRODUCE WITHOUT PERMISSION

More information

Dynamic Trading Strategies

Dynamic Trading Strategies Dynamic Trading Strategies Concepts and Buzzwords Multi-Period Bond Model Replication and Pricing Using Dynamic Trading Strategies Pricing Using Risk- eutral Probabilities One-factor model, no-arbitrage

More information

Hedging with Foreign Currency Options. Kris Kuthethur Murthy Vanapalli

Hedging with Foreign Currency Options. Kris Kuthethur Murthy Vanapalli Hedging with Foreign Currency Options Kris Kuthethur Murthy Vanapalli Foreign Currency Option Financial instrument that gives the holder the right, but not the obligation, to sell or buy currencies at

More information

Lecture 8. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 1

Lecture 8. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 8 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 8 1 One-Step Binomial Model for Option Price 2 Risk-Neutral Valuation

More information

Contents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM

Contents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM Contents 1 Put-Call Parity 1 1.1 Review of derivative instruments................................................ 1 1.1.1 Forwards............................................................ 1 1.1.2 Call

More information

Option Basics. c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153

Option Basics. c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153 Option Basics c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153 The shift toward options as the center of gravity of finance [... ] Merton H. Miller (1923 2000) c 2012 Prof. Yuh-Dauh Lyuu,

More information

CHAPTER 20: OPTIONS MARKETS: INTRODUCTION

CHAPTER 20: OPTIONS MARKETS: INTRODUCTION CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X

More information

Solutions to Lectures on Corporate Finance, Second Edition. Peter Bossaerts and Bernt Arne Ødegaard

Solutions to Lectures on Corporate Finance, Second Edition. Peter Bossaerts and Bernt Arne Ødegaard Solutions to Lectures on Corporate Finance, Second Edition Peter Bossaerts and Bernt Arne Ødegaard 2006 Contents 1 Finance 1 2 Axioms of modern corporate finance 2 3 On Value Additivity 3 4 On the Efficient

More information

CHAPTER 21: OPTION VALUATION

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)

More information

Questions and Answers

Questions and Answers MA3245 Financial Mathematics I Suggested Solutions of Tutorial 1 (Semester 2/03-04) Questions and Answers 1. What is the difference between entering into a long forward contract when the forward price

More information

Rate of Return. Reading: Veronesi, Chapter 7. Investment over a Holding Period

Rate of Return. Reading: Veronesi, Chapter 7. Investment over a Holding Period Rate of Return Reading: Veronesi, Chapter 7 Investment over a Holding Period Consider an investment in any asset over a holding period from time 0 to time T. Suppose the amount invested at time 0 is P

More information

Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS

Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS HW 6 Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS 4. Cost Payoff Profit Call option, X = 85 3.82 5.00 +1.18 Put option, X = 85 0.15 0.00-0.15 Call option, X = 90 0.40 0.00-0.40

More information

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008. Options

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

More information

Introduction to Options. Derivatives

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

More information

Valuing Options / Volatility

Valuing Options / Volatility Chapter 5 Valuing Options / Volatility Measures Now that the foundation regarding the basics of futures and options contracts has been set, we now move to discuss the role of volatility in futures and

More information

European Options Pricing Using Monte Carlo Simulation

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

More information

Expected payoff = 1 2 0 + 1 20 = 10.

Expected payoff = 1 2 0 + 1 20 = 10. Chapter 2 Options 1 European Call Options To consolidate our concept on European call options, let us consider how one can calculate the price of an option under very simple assumptions. Recall that the

More information

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

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

More information