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

Size: px
Start display at page:

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

Transcription

1 Week 9 Binomial Trees : Hull, Ch

2 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2

3 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral Valuation. 4. Two-Step Binomial Model. 5. American Options. 6. Delta. 7. Dividends and the Binomial Tree. 8. Determining u and d. 9. Exotic Options. 10. Futures vs Option Formulas. 3

4 1. Introduction 4

5 Introduction to the Binomial Model What do we know about Option Pricing? Lower/Upper bounds: No assumptions, Arbitrage opportunity, Not very precise e.g. 3 c 18) Put-Call parity: No assumptions, Arbitrage opportunity, Relative pricing formula, not like F 0 =S 0 e rt Here we propose an option pricing model to find the theoretical price or fair price for a given option. To get this stronger result, we need to impose some structure: Assumption on the dynamics of S. Organization: (1) Simple Example, (2) Generalization, (3) Applications 5

6 2. One Step Binomial Model 6

7 A Simple Binomial Model A stock price is currently $20. In three months it will be either $22 or $18. Stock price = $20 Stock Price = $22 Stock Price = $18 7

8 A Call Option A 3-month call option on the stock has a strike price of $21. Stock price = $20 Option Price =? Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 8

9 ) ( ) ( 0 0 d d u u kt T kt c p c p e c c E e c But k, pu and pd are unknown. k = expected return on a risky project. k = r + risk premium. Call Option Price Today 9 Jorge Cruz Lopez - Bus 316: Derivative Securities

10 Setting Up a Riskless Portfolio Consider the Portfolio: long D shares short 1 call option Portfolio is riskless when 22D 1 = 18D or D = D 1 18D Remember: A riskless portfolio is a portfolio that has a fixed (and known) payoff in the future. 10

11 Valuing the Portfolio The riskless portfolio is: long 0.25 shares short 1 call option Assume that the risk-free rate is 12%. The value of the portfolio in 3 months is: = 4.50 = The value of the portfolio today is: 4.5e = Notice that we can discount at the riskless rate because this is a riskless portfolio!. 11

12 Valuing the Option Therefore, the portfolio that is: long 0.25 shares short 1 option is worth today The value of the share position today is: D 20 = = So now we can imply the value of the option today. The value of the option c today is: V 0 = D S 0 c = c c = Pretty COOL, eh? 12

13 Generalization An option maturing in T years written on a stock that is currently worth S. where S ƒ ƒ is the current option price u is a constant > 1 ƒ u is the option price in the upper state d is a constant < 1 ƒ d is the option price in the lower state S u ƒ u S d ƒ d 13 Jorge Cruz Lopez - Bus 316: Derivative Securities

14 Generalization Consider the portfolio that is long D shares and short one option. The payoff at time T is: S u D ƒ u S d D ƒ d The portfolio is riskless when S u D ƒ u = S d D ƒ d or D ƒu S u fd S d 14

15 Generalization Value of the portfolio at time T (maturity) is: S u D ƒ u or S d D ƒ d From the riskless portfolio, the value of the portfolio today is: (S u D ƒ u )e rt From the initial position, another expression for the portfolio value today is: S D f Hence the option price today is: f = S D (S u D ƒ u )e rt 15

16 Generalization Substituting for D we obtain: ƒ = [ p ƒ u + (1 p )ƒ d ]e rt where p e u rt d d 16 Jorge Cruz Lopez - Bus 316: Derivative Securities

17 3. Risk Neutral Valuation 17

18 Risk-Neutral Valuation ƒ = [ p ƒ u + (1 p )ƒ d ]e -rt The variables p and (1 p ) can be interpreted as the risk-neutral probabilities of up and down movements. Therefore, the value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate. S ƒ S u ƒ u S d ƒ d 18 Jorge Cruz Lopez - Bus 316: Derivative Securities

19 Irrelevance of Stock s Expected Return IMPORTANT: Notice that the stock growth rate and the probabilities of the stock moving up or down are irrelevant. That is, the expected return on the stock is irrelevant. WHY? This is because we re valuing the option in relative to the current stock price. This price contains all relevant information about the future prospects of the stock. 19

20 Original Example Revisited Proof that Risk Neutral valuation gives the same result as the no arbitrage argument: S = 20 ƒ S u = 22 ƒ u = 1 S d = 18 ƒ d = 0 p e u rt d d e Jorge Cruz Lopez - Bus 316: Derivative Securities

21 Valuing the Option S ƒ S u = 22 ƒ u = 1 S d = 18 ƒ d = 0 The value of the option today is: e [ ] = No Arbitrage Approach and Risk-Neutral Approach give the same result. 21

22 Pricing a Put No-Arbitrage Approach. Risk-Neutral Approach. Example: p (K = 40, T = 3/12) S0 = 40 S0 d = 35 and S0 u = 45 r = 8% 22

23 Pricing a Put: No Arbitrage Approach The portfolio is riskless when D S ƒ u u f d S d f = S D (S u D ƒ u )e rt = 40(-0.5) [45(-0.5) - 0] e 0.08*0.25 =

24 Pricing a Put: Risk Neutral Approach S u = 45 ƒ u = 0 S = 40 ƒ S d = 35 ƒ d = 5 p rt e d u d e The value of the option today is: e [ ( )5] = Jorge Cruz Lopez - Bus 316: Derivative Securities

25 4. Two-Step Binomial Trees 25

26 A Two-Step Example Same as the previous call example where p = Let each time step be 3 months. The tree is recombining (u and d constant). 26

27 Reminder: K=21 So = 20 Valuing a Call Option: Step by Step A Value at node B = e ( ) = Value at node A = e ( ) = Instead, we can proceed directly B C E D F 27 Jorge Cruz Lopez - Bus 316: Derivative Securities

28 Valuing a Call Option: The Direct Way S f u S f u d S u 2 S f uu u d S f ud dt f d d 2 S f dd f u = e -rdt [pf uu + (1-p)f ud ] f d = e -rdt [pf ud + (1-p)f dd ] f = e -rdt [p f u + (1-p) f d ] f = e -rdt [p {e -rdt [pf uu + (1-p)f ud ]} + (1-p) {e -rdt [pf ud + (1-p)f dd ]}] f = e -r2dt [p 2 f uu + 2p(1-p)f ud + (1-p) 2 f dd ] Check: sum prob = 1 28

29 A Put Option Example K=52; T=2; r=5% A B C E D F Try it yourself! 29

30 5. American Options 30

31 American Options Recall: Any time that the payoff from early exercise exceeds the price of the option, it is optimal to exercise early. C < (S 0 K) EE P < (K - S 0 ) EE 31

32 When the Put Option is American K=52; T=2; r=5% European American A B C E D F A B C E D F At this point the payoff from early exercise is greater than the price of the option. Therefore, we have early exercise. 32

33 6. Delta 33

34 Delta Delta (D) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock. In the binomial tree: D S ƒ u u f d S d The value of D varies from node to node. 34

35 7. Dividends and the Binomial Tree 35

36 Binomial Trees with Dividends With a percentage dividend (ds), the tree is still recombining. With a cash dividend ($D), the tree is not recombining anymore, so pricing becomes more complex. 36

37 8. Determining u and d 37

38 Determining u and d One way of matching the volatility is to set: u e s Dt d e s Dt 1 / u where s is the annual volatility and Dt is the length of the time step 38

39 9. Exotic Options 39

40 Applications: Exotic Options Pricing a Power Option European American Pricing a Chooser Option (Ch. 20) Pricing a Lookback Option (Ch. 20) 40

41 10. Futures vs Option Formulas 41

42 Difference Between Futures and Option Pricing Formulas? What should we do when on the Futures market we have: Fmarket S0 x exp(rt) What should we do when on the option market we have: cmarket cbinomial or pmarket pbinomial 42

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

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

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

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 V: Option Pricing Basics

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

More information

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

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.

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

Chapter 1: Financial Markets and Financial Derivatives

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

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

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

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

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

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 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

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

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

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

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

Properties of Stock Options. Chapter 10

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

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

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. Week 12 Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14. 1 Options on Stock Indices and Currencies Objective: To explain the basic asset pricing techniques used

More 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

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

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

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

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

Chapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

Chapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull Chapter 11 Properties of Stock Options 1 Notation c: European call option price p: European put option price S 0 : Stock price today K: Strike price T: Life of option σ: Volatility of stock price C: American

More information

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

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

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

Session IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics

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

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

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 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

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

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

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

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

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

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

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

More information

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

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)

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

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

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

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

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

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

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

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

Option Premium = Intrinsic. Speculative Value. Value

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

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

Week 13 Introduction to the Greeks and Portfolio Management:

Week 13 Introduction to the Greeks and Portfolio Management: Week 13 Introduction to the Greeks and Portfolio Management: Hull, Ch. 17; Poitras, Ch.9: I, IIA, IIB, III. 1 Introduction to the Greeks and Portfolio Management Objective: To explain how derivative portfolios

More 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

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

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

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

where N is the standard normal distribution function,

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

More information

Options/1. Prof. Ian Giddy

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

More information

Options: Valuation and (No) Arbitrage

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

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

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

Other variables as arguments besides S. Want those other variables to be observables.

Other variables as arguments besides S. Want those other variables to be observables. Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value

More information

9 Basics of options, including trading strategies

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

More information

Option Pricing with S+FinMetrics. PETER FULEKY Department of Economics University of Washington

Option Pricing with S+FinMetrics. PETER FULEKY Department of Economics University of Washington Option Pricing with S+FinMetrics PETER FULEKY Department of Economics University of Washington August 27, 2007 Contents 1 Introduction 3 1.1 Terminology.............................. 3 1.2 Option Positions...........................

More information

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

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

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

Martingale Pricing Applied to Options, Forwards and Futures

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

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

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

1.1 Some General Relations (for the no dividend case)

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,

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

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

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

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) 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.

More information

Black-Scholes. Ser-Huang Poon. September 29, 2008

Black-Scholes. Ser-Huang Poon. September 29, 2008 Black-Scholes Ser-Huang Poon September 29, 2008 A European style call (put) option is a right, but not an obligation, to purchase (sell) an asset at a strike price on option maturity date, T. An American

More information

CHAPTER 21: OPTION VALUATION

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

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

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

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

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

Stock. Call. Put. Bond. Option Fundamentals

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

More information

Other observable variables as arguments besides S.

Other observable variables as arguments besides S. Valuation of options before expiration Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value now of receiving p T at expiration?) Have candidate model

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

Options 1 OPTIONS. Introduction

Options 1 OPTIONS. Introduction Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or

More information

Valuation, Pricing of Options / Use of MATLAB

Valuation, Pricing of Options / Use of MATLAB CS-5 Computational Tools and Methods in Finance Tom Coleman Valuation, Pricing of Options / Use of MATLAB 1.0 Put-Call Parity (review) Given a European option with no dividends, let t current time T exercise

More information

Additional questions for chapter 4

Additional questions for chapter 4 Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two six-month periods it is expected to go up by 1% or go down by 1%. The risk-free interest rate is 8% per annum with

More information

Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5

Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5 Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options)

More information

Buy a number of shares,, and invest B in bonds. Outlay for portfolio today is S + B. Tree shows possible values one period later.

Buy a number of shares,, and invest B in bonds. Outlay for portfolio today is S + B. Tree shows possible values one period later. Replicating portfolios Buy a number of shares,, and invest B in bonds. Outlay for portfolio today is S + B. Tree shows possible values one period later. S + B p 1 p us + e r B ds + e r B Choose, B so that

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

Computational Finance Options

Computational Finance Options 1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to

More information

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 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

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

Bond Options, Caps and the Black Model

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

More information

4. Options Markets. 4.6. Applications

4. Options Markets. 4.6. Applications 4. Options Markets 4.6. Applications 4.6.1. Options on Stock Indices e.g.: Dow Jones Industrial (European) S&P 500 (European) S&P 100 (American) LEAPS S becomes to Se qt Basic Properties with Se qt c Se

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

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

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

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

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

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

More information

CHAPTER 7: PROPERTIES OF STOCK OPTION PRICES

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

More information

S 1 S 2. Options and Other Derivatives

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

More information

Trading Strategies Involving Options. Chapter 11

Trading Strategies Involving Options. Chapter 11 Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A risk-free bond and an option to create a principal-protected note A stock and an option Two or more options of the same type

More information

The Black-Scholes pricing formulas

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

More information