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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 put options options on futures Copyright c Philip H. Dybvig 2004

2 Riskless bond: 1 r Binomial asset pricing: review Stock: 1 u d V Derivative security:??? u V d where Value = p u V u +p d V d = r 1 (π uv u +π dv d ) p u = r 1r d u d p d = r 1u r u d and πu = r d u d π d = u r u d. Interpretation: all investments are fair gambles, subject to discounting to account for impatience and a probability adjustment to account for risk pricing. 2

3 Expected excess returns Some orders of magnitude Common stock indices: 4 5% per year or 4 5%/ or 2 basis points daily Individual common stocks: 50% 150% of the index expected excess return. Standard deviation Common stock indices: 15 20% per year or 15 20%/ % per day Individual common stocks: 35% 40% per year Theoretical observation: for the usual case, standard deviations over short periods are almost exactly the same in actual probabilities as in risk-neutral probabilities. 3

4 Digression: review of interest rates and compounding Simple interest: the amount paid for use of money is Prt, where P is principal, r is the interest rate, and t is the time interval (most commonly measured in years). For example, the interest on \$200 for 2 months at a 12% annual rate is \$200 12% 2/12 = \$4. Compound interest: interest is recorded periodically and added to the principal; consequently there is interest on the interest. For example, if 12% interest is compounded semi annually and we start with \$1000, then after three years we will accumulate \$1000 (1+12%/2) 2 3 \$ Continuous compounding: the value in the limit as interest is recorded more and more frequently is given by the exponential function. For example, if 12% interest is compounded continuously and we start with \$1000, then after three years we will have \$1000 exp(12% 3) = \$1000e 0.36 \$ Over short horizons, all three are almost exactly the same! 4

5 Binomial parameters in practice Most texts seem to have unreasonably complicated expressions for u, d, and r in binomial models. From the theory, we know that a good choice is r = 1+r f t u = 1+r f t+σ t d = 1+r f t σ t with π u = π d = 1/2 and t the time increment. This has the two essential features: it equates risk-neutral expected stock and bond returns, and it has the right standard deviation. In addition, it has a continuous stock price (like Black-Scholes) as a limit. One alternative is to choose the positive solution of ud = 1 and u d = 2σ t (good for option price as a function of time) or a recombining trinomial (good for including some dependence of variance on the stock price). 5

6 European call option The purchaser obtains the right to buy one share of stock from the issuer at the maturity date for the pre-specified strike or exercise price. Ifthestockpriceexceedsthestrikepriceatmaturity, theoptionisin the money. In this case, it is optimal for the owner (the purchaser) to exercise the option. The option is worth the difference between the stock price and the strike price, which is the amount the owner pockets by exercising the option. If the strike price exceeds the stock price at maturity, the option is out of the money. In this case, it is optimal for the owner to let the option expire without exercising it, because exercise would cost the owner the difference between the stock price and the strike price. The price of the option is zero. Summary: At maturity (T), the value is max(s T X,0), or the maximum of the stock price less the exercise price and zero. An American call option is just like the European option, but the option of exercising is available anytime (once only) at or before the maturity date. 6

7 Call Price call option price T=0.00 T=0.25 T=0.50 T= stock price 7

8 More on call options The call option pays off when the underlying stock goes up but does not obligate the owner when the underlying stock goes down. For this privilege, the purchaser pays a price (also called a premium) up front. The market price of the option depends on the exercise price, the stock price, the time to maturity, the volatility of the underlying stock, the riskless interest rate and the anticipated size of dividends before maturity. The most important influence on an option s value day-to-day is the underlying stock s price. The next most important influence is changing volatility of the underlying stock price. In fact, one interesting use of option prices is to compute implied volatilities, that is the level of volatility ( vol ) implicit in the option price. The other influences either do not change on a daily basis (the exercise price and the anticipated remaining dividends) or have movements with relatively minor effect on the option price at normal maturities (time to maturity and the riskless interest rate). 8

9 In-class exercise: call option valuation Consider the binomial model with u = 2, d = 1/2, and r = 1. What are the risk-neutral probabilities? Assuming the stock price is initially \$100, what is the price of a call option with a \$90 strike price maturing in two periods? stock call option reminder: π u = r d u d π d = u r u d. 9

10 European put options The purchaser obtains the right to sell one share of stock to the issuer at the maturity date for the pre-specified strike (or exercise) price. If the strike price S exceeds the stock price S T at maturity, the option is in the money. In this case, it is optimal for the owner to exercise the option. The in-the-money option is worth the excess, if any, of the stock price and the strike price, which is the amount the owner pockets by exercising the option. This assumes no frictions. If the stock price exceeds the strike price at maturity, the option is out of the money In this case, it is optimal for the owner to let the option expire without exercising it, because the exercise would cost the owner the difference between the stock price and the strike price. The price of the option is zero. Summary: At maturity (T), the European put is worth max(0,x S T ), which is the larger of zero and the net value of exercising. An American put option is just like the European option, but the option of exercising is available anytime (once only) at or before maturity. 10

11 Put Price put option price T=0.00 T=0.25 T=0.50 T= stock price Put price as a function of stock price and maturity 11

12 More on put options The put option pays off when the underlying stock goes down but does not obligate the owner when the underlying stock goes up. For this privelege, the purchaser pays a price (premium) up front. The market price of the option depends on the exercise price, the stock price, the time to maturity, the volatility of the underlying stock, the riskless interest rate, and the anticipated size of dividends before maturity. Same as for call options, the most important influence on the put s value dayto-day is changes in the underlying stock s price, and the next most important influence is changes in the volatility of the underlying stock. Put and call prices both increase when volatility increases, while they move in opposite directions (call up, put down) if the underlying stock price increases. 12

13 Sensitivities of options market price When this call price put price Stock price Volatility Strike price Time to maturity Interest rate Dividend Since the largest changes on a day-to-day basis are the stock price and the volatility, we should think of buying a call as buying the stock and buying vol, and a call as selling the stock and buying vol. 13

14 In-class exercise: European put option Consider the binomial model with u = 2, d = 1/2, and r = 5/4 (interest rate is 25%). What are the risk-neutral probabilities? Assuming the stock price is initially \$100, what is the price of a European put option with a \$125 strike price maturing in two periods? stock European put reminder: π u = r d u d π d = u r u d. 14

15 Binomial model strategy: American versus European stock options Binomial valuation follows the steps: Compute the tree from the underlying, starting at the beginning Evaluate the option at the end based on the contract terms Step back through the tree to compute the value one node at a time For a European value is the value of holding into next period For an American option, the value is the larger of holding it into the next period or exercising now. 15

16 In-class exercise: American put option Consider the binomial model with u = 2, d = 1/2, and r = 5/4 (interest rate is 25%). What are the risk-neutral probabilities? Assuming the stock price is initially \$100, what is the price of an American put option with a \$125 strike price maturing in two periods? stock European put reminder: π u = r d u d π d = u r u d. 16

17 Futures: 0 Futures are not assets! F u F F d F 0 = π u(f u F)+π d(f d F) So E [ F] = 0 or F t = E t[f t+1 ] Because we do not put money into a futures contract (and our margin account adjusted daily to leave no value in the futures), the price in risk-neutral probabilities has expected growth 0, not expected growth at the interest rate like assets. 17

18 In-class exercise: Futures option valuation Suppose the initial stock index is \$50 and that the stock price doubles or falls by half each period. Assume further that the riskfree rate is 5/4. Assuming there are no dividends, compute the value to day of a call option maturing one period from now with strike \$50 on the index futures contract maturing two periods from now. stock index index futures futures option reminder: π u = r d u d π d = u r u d. 18

19 Binomial pricing of options! Wrap-up single-period: use risk-neutral probabilities and discounted expected value multiple-period: use single period valuation again and again strategy: generate a valuation tree for the underlying value the option at the terminal date based on contract terms use single-period valuation to step through the tree call option example 19

### OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options

OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (risk-neutral)

### 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

### Lecture 4: The Black-Scholes model

OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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):?

### 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

### 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

### Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

### 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

### 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

### 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

### 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

### 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

### 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

### Goals. Options. Derivatives: Definition. Goals. Definitions Options. Spring 2007 Lecture Notes 4.6.1 Readings:Mayo 28.

Goals Options Spring 27 Lecture Notes 4.6.1 Readings:Mayo 28 Definitions Options Call option Put option Option strategies Derivatives: Definition Derivative: Any security whose payoff depends on any other

### 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

### 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

### 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

### CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

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

### 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

### 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

### 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

### Futures Price d,f \$ 0.65 = (1.05) (1.04)

24 e. Currency Futures In a currency futures contract, you enter into a contract to buy a foreign currency at a price fixed today. To see how spot and futures currency prices are related, note that holding

### 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

### Option Pricing Basics

Option Pricing Basics Aswath Damodaran Aswath Damodaran 1 What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called

### 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

### 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)

### 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)

### 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

### 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:

### where N is the standard normal distribution function,

The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at

### Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week

Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics

### 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

### 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

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

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

### EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0

EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS. Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002

DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS 1. Background Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002 It is now becoming increasingly accepted that companies

### 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

### 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

### Online Appendix: Payoff Diagrams for Futures and Options

Online Appendix: Diagrams for Futures and Options As we have seen, derivatives provide a set of future payoffs based on the price of the underlying asset. We discussed how derivatives can be mixed and

### Stock. Call. Put. Bond. Option Fundamentals

Option Fundamentals Payoff Diagrams hese are the basic building blocks of financial engineering. hey represent the payoffs or terminal values of various investment choices. We shall assume that the maturity

### 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 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

### 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

### Properties of Stock Options. Chapter 10

Properties of Stock Options Chapter 10 1 Notation c : European call option price C : American Call option price p : European put option price P : American Put option price S 0 : Stock price today K : Strike

### Option Payoffs. Problems 11 through 16: Describe (as I have in 1-10) the strategy depicted by each payoff diagram. #11 #12 #13 #14 #15 #16

Option s Problems 1 through 1: Assume that the stock is currently trading at \$2 per share and options and bonds have the prices given in the table below. Depending on the strike price (X) of the option

### 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

### CHAPTER 5 OPTION PRICING THEORY AND MODELS

1 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. In this section, we will consider an exception to that rule

### Employee Options, Restricted Stock and Value. Aswath Damodaran 1

Employee Options, Restricted Stock and Value 1 Basic Proposition on Options Any options issued by a firm, whether to management or employees or to investors (convertibles and warrants) create claims on

### 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

### 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

### 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

### 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

### BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract

BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple

### 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

### 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

### 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

### call option put option strike price/exercise price expiration date/maturity

OPTIONS Objective This chapter introduces the readers to the concept of options which include calls and puts. All basic concepts like option buyer and seller, European and American options, and payoff

### 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

### Quantitative Strategies Research Notes

Quantitative Strategies Research Notes December 1992 Valuing Options On Periodically-Settled Stocks Emanuel Derman Iraj Kani Alex Bergier SUMMARY In some countries, for example France, stocks bought or

### 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 }

### 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 on an Asset that Yields Continuous Dividends

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

### Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008

: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of

### Research on Option Trading Strategies

Research on Option Trading Strategies An Interactive Qualifying Project Report: Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree

### Options and Derivative Pricing. U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University.

Options and Derivative Pricing U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University. e-mail: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,

### 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

### 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

### 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