# Lecture 11. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 7

Size: px
Start display at page:

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

Transcription

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

2 Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating Portfolio Sergei Fedotov (University of Manchester) / 7

3 American Option An American Option is one that may be exercised at any time prior to expire (t = T). Sergei Fedotov (University of Manchester) / 7

4 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! Sergei Fedotov (University of Manchester) / 7

5 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. Sergei Fedotov (University of Manchester) / 7

6 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. We can buy stock for S and option for P and immediately exercise the option by selling stock for E. Sergei Fedotov (University of Manchester) / 7

7 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. We can buy stock for S and option for P and immediately exercise the option by selling stock for E. E (P + S) > 0 Sergei Fedotov (University of Manchester) / 7

8 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. Sergei Fedotov (University of Manchester) / 7

9 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. Sergei Fedotov (University of Manchester) / 7

10 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. American Put Option: P m n = max { max(e S m n,0),e r t ( pp m+1 n+1 ) } + (1 p)pm+1 n, where S m n is the n-th possible value of stock price at time-step m t. Sergei Fedotov (University of Manchester) / 7

11 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. American Put Option: P m n = max { max(e S m n,0),e r t ( pp m+1 n+1 ) } + (1 p)pm+1 n, where S m n is the n-th possible value of stock price at time-step m t. Final condition: P N n = max ( E S N n,0 ), where n = 0,1,2,...,N, E is the strike price. Sergei Fedotov (University of Manchester) / 7

12 Example: Evaluation of American Put Option on Two-Step Tree We assume that over each of the next two years the stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%. Find the value of a 2-year American put with a strike price of \$52 on a stock whose current price is \$50. Sergei Fedotov (University of Manchester) / 7

13 Example: Evaluation of American Put Option on Two-Step Tree We assume that over each of the next two years the stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%. Find the value of a 2-year American put with a strike price of \$52 on a stock whose current price is \$50. In this case u = 1.2, d = 0.8, r = 0.05, E = 52. Risk-neutral probability: p = e = Sergei Fedotov (University of Manchester) / 7

14 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = Sergei Fedotov (University of Manchester) / 7

15 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Sergei Fedotov (University of Manchester) / 7

16 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Payoff: E S = = 12 > Early exercise is optimal! P d = 12 Sergei Fedotov (University of Manchester) / 7

17 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Payoff: E S = = 12 > Early exercise is optimal! P d = 12 P 0 = e ( ) = Sergei Fedotov (University of Manchester) / 7

18 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Payoff: E S = = 12 > Early exercise is optimal! P d = 12 P 0 = e ( ) = Payoff: E S = = 2 < Early exercise is not optimal at the initial node Sergei Fedotov (University of Manchester) / 7

19 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) Sergei Fedotov (University of Manchester) / 7

20 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Sergei Fedotov (University of Manchester) / 7

21 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Consider replicating portfolio of shares held long and N bonds held short. Sergei Fedotov (University of Manchester) / 7

22 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Consider replicating portfolio of shares held long and N bonds held short. The value of portfolio: Π = S NB. A pair (,N) is called a trading strategy. How to find (,N) such that Π T = C T and Π 0 = C 0? Sergei Fedotov (University of Manchester) / 7

23 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. Sergei Fedotov (University of Manchester) / 7

24 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. Sergei Fedotov (University of Manchester) / 7

25 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Sergei Fedotov (University of Manchester) / 7

26 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Current value: C 0 = S 0 NB 0. Sergei Fedotov (University of Manchester) / 7

27 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Current value: C 0 = S 0 NB 0. Prove: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. (Exercise sheet 5) Sergei Fedotov (University of Manchester) / 7

### 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

### 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

### Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6

Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static

### 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

### 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 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

### 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):?

### 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

### 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

### 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

### 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

### 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

### 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

### 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

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

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

### Lecture 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

### 1 Introduction to Option Pricing

ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 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)

### 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

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall

### 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

### 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

### 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)

### 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

### 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

### 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

### 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

### How To Value Options In Black-Scholes Model

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

### 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

### 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

### 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

### 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

### 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

### 1 Further Pricing Relationships on Options

1 Further Pricing Relationships on Options 1.1 The Put Option with a Higher Strike must be more expensive Consider two put options on the same stock with the same expiration date. Put option #1 has a strike

### 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

### 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

### 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

### 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.

### 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

### 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,

### 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

### K 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.

Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.

### 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

### 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

### CHAPTER 11: ARBITRAGE PRICING THEORY

CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times

### 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

### 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:

### 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

### 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

### Payoff (Riskless bond) Payoff(Call) Combined

Short-Answer 1. Is the payoff to stockholders most similar to the payoff on a long put, a long call, a short put, a short call or some combination of these options? Long call 2. ebay s current stock price

### EXERCISES FROM HULL S BOOK

EXERCISES FROM HULL S BOOK 1. Three put options on a stock have the same expiration date, and strike prices of \$55, \$60, and \$65. The market price are \$3, \$5, and \$8, respectively. Explain how a butter

### 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

### 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

### 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.

### 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 }

### 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:

### Call Price as a Function of the Stock Price

Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived

### ASimpleMarketModel. 2.1 Model Assumptions. Assumption 2.1 (Two trading dates)

2 ASimpleMarketModel In the simplest possible market model there are two assets (one stock and one bond), one time step and just two possible future scenarios. Many of the basic ideas of mathematical finance

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Name: 1 (5) a b c d e TRUE/FALSE 1 (2) TRUE FALSE. 2 (5) a b c d e. 3 (5) a b c d e 2 (2) TRUE FALSE. 4 (5) a b c d e.

Name: Thursday, February 28 th M375T=M396C Introduction to Actuarial Financial Mathematics Spring 2013, The University of Texas at Austin In-Term Exam I Instructor: Milica Čudina Notes: This is a closed

### Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging

Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in

### 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

### 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.

### Fundamentals of Futures and Options (a summary)

Fundamentals of Futures and Options (a summary) Roger G. Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA Published 2013 by the Research Foundation of CFA Institute Summary prepared by Roger G.

### 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

### Stochastic Processes and Advanced Mathematical Finance. Multiperiod Binomial Tree Models

Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Stochastic Processes and Advanced

### ONE PERIOD MODELS t = TIME = 0 or 1

BASIC INSTRUMENTS: * S t : STOCK * B t : RISKLESS BOND ONE PERIOD MODELS t = TIME = 0 or 1 B 0 = 1 B 1 = 1 + r or e r * FORWARD CONTRACT: AGREEMENT TO SWAP \$\$ FOR STOCK - AGREEMENT TIME: t = 0 - AGREEMENT

### An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing

An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted

### Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive 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

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

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

### LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

### 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)

### Lecture 4: Futures and Options Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 4: Futures and Options Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Foreign Currency Futures Assume that

### How To Price A Call Option

Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min

### 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

### 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