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


 Angela Parker
 2 years ago
 Views:
Transcription
1 Lecture 9 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 8
2 Lecture 9 1 RiskNeutral Valuation 2 RiskNeutral World 3 TwoSteps Binomial Tree Sergei Fedotov (University of Manchester) / 8
3 RiskNeutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. NoArbitrage Principle: d < ert < u. Sergei Fedotov (University of Manchester) / 8
4 RiskNeutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. NoArbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. Sergei Fedotov (University of Manchester) / 8
5 RiskNeutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. NoArbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. We interpret the variable 0 p 1 as the probability of an up movement in the stock price. This formula is known as a riskneutral valuation. Sergei Fedotov (University of Manchester) / 8
6 RiskNeutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. NoArbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. We interpret the variable 0 p 1 as the probability of an up movement in the stock price. This formula is known as a riskneutral valuation. The probability of up q or down movement 1 q in the stock price plays no role whatsoever! Why??? Sergei Fedotov (University of Manchester) / 8
7 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] Sergei Fedotov (University of Manchester) / 8
8 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d Sergei Fedotov (University of Manchester) / 8
9 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d Sergei Fedotov (University of Manchester) / 8
10 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. Sergei Fedotov (University of Manchester) / 8
11 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the riskfree interest rate r. Since the expected return is r, this is a riskneutral world. Sergei Fedotov (University of Manchester) / 8
12 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the riskfree interest rate r. Since the expected return is r, this is a riskneutral world. In Real World: E [S T ] = S 0 e µt. In RiskNeutral World: E p [S T ] = S 0 e rt Sergei Fedotov (University of Manchester) / 8
13 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the riskfree interest rate r. Since the expected return is r, this is a riskneutral world. In Real World: E [S T ] = S 0 e µt. In RiskNeutral World: E p [S T ] = S 0 e rt RiskNeutral Valuation: C 0 = e rt E p [C T ] The option price is the expected payoff in a riskneutral world, discounted at riskfree rate r. Sergei Fedotov (University of Manchester) / 8
14 TwoStep Binomial Tree Now the stock price changes twice, each time by either a factor of u > 1 or d < 1. We assume that the length of the time step is t such that T = 2 t. After two time steps the stock price will be S 0 u 2, S 0 ud or S 0 d 2. Sergei Fedotov (University of Manchester) / 8
15 TwoStep Binomial Tree Now the stock price changes twice, each time by either a factor of u > 1 or d < 1. We assume that the length of the time step is t such that T = 2 t. After two time steps the stock price will be S 0 u 2, S 0 ud or S 0 d 2. The call option expires after two time steps producing payoffs C uu, C ud and C dd respectively. Sergei Fedotov (University of Manchester) / 8
16 Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. Sergei Fedotov (University of Manchester) / 8
17 Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. We apply the riskneutral valuation backward in time: C u = e r t (pc uu + (1 p)c ud ), C d = e r t (pc ud + (1 p)c dd ). Sergei Fedotov (University of Manchester) / 8
18 Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. We apply the riskneutral valuation backward in time: C u = e r t (pc uu + (1 p)c ud ), C d = e r t (pc ud + (1 p)c dd ). Current option price: C 0 = e r t (pc u + (1 p)c d ). Sergei Fedotov (University of Manchester) / 8
19 Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a riskneutral world that the upper, middle, and lower final nodes are reached. Sergei Fedotov (University of Manchester) / 8
20 Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a riskneutral world that the upper, middle, and lower final nodes are reached. Finally, the current call option price is C 0 = e rt E p [C T ], T = 2 t. Sergei Fedotov (University of Manchester) / 8
21 Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a riskneutral world that the upper, middle, and lower final nodes are reached. Finally, the current call option price is C 0 = e rt E p [C T ], T = 2 t. The current put option price can be found in the same way: P 0 = e 2r t ( p 2 P uu + 2p(1 p)p ud + (1 p) 2 P dd ) or P 0 = e rt E p [P T ]. Sergei Fedotov (University of Manchester) / 8
22 TwoStep Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Riskfree interest rate is 10%. Find the current option price. Sergei Fedotov (University of Manchester) / 8
23 TwoStep Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Riskfree interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Sergei Fedotov (University of Manchester) / 8
24 TwoStep Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Riskfree interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Riskneutral probability p = er t d u d = e = Sergei Fedotov (University of Manchester) / 8
25 TwoStep Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Riskfree interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Riskneutral probability p = er t d u d = e = The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, = 38.4, and 40 (0.8) 2 = Sergei Fedotov (University of Manchester) / 8
26 TwoStep Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Riskfree interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Riskneutral probability p = er t d u d = e = The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, = 38.4, and 40 (0.8) 2 = We obtain P uu = 0, P ud = 0 and P dd = = 6.4. Sergei Fedotov (University of Manchester) / 8
27 TwoStep Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Riskfree interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Riskneutral probability p = er t d u d = e = The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, = 38.4, and 40 (0.8) 2 = We obtain P uu = 0, P ud = 0 and P dd = = 6.4. Thus the value of put( option is ) P 0 = e ( ) = Sergei Fedotov (University of Manchester) / 8
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 OneStep Binomial Model for Option Price 2 RiskNeutral Valuation
More informationLecture 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 informationLecture 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 informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationBinomial 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 informationLectures. 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 information10 Binomial Trees. 10.1 Onestep 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 Onestep model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t
More informationIntroduction 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 informationLecture 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 Putcall
More informationLecture 11: RiskNeutral Valuation Steven Skiena. skiena
Lecture 11: RiskNeutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena RiskNeutral Probabilities We can
More informationACTS 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 fouryear dollardenominated European put option
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationChapter 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 riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More information1. 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 informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationConsider a European call option maturing at time T
Lecture 10: Multiperiod Model Options BlackScholesMerton 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 informationOption 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 informationa. 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 informationLecture 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 BlackScholes Equation and Replicating Portfolio 2 Static
More informationBINOMIAL 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 optionpricing
More informationBuy 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 informationDERIVATIVE 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 information7: 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 CoxRossRubinstein
More informationAmerican 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 informationTABLE OF CONTENTS. A. PutCall 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. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationOPTIONS 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 (riskneutral)
More informationPart 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, putcall parity introduction
More informationOptions. + Concepts and Buzzwords. Readings. PutCall Parity Volatility Effects
+ Options + Concepts and Buzzwords PutCall Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options
More informationFinance 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 informationAmerican 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 informationNumerical 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 informationOne Period Binomial Model
FIN40008 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 informationChapter 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 informationLecture 4: Derivatives
Lecture 4: Derivatives School of Mathematics Introduction to Financial Mathematics, 2015 Lecture 4 1 Financial Derivatives 2 uropean Call and Put Options 3 Payoff Diagrams, Short Selling and Profit Derivatives
More informationChapter 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 informationOption 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 informationUCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Final Exam. December Date:
UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2013 MBA Final Exam December 2013 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open
More informationStock. 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 informationDynamic Trading Strategies
Dynamic Trading Strategies Concepts and Buzzwords MultiPeriod Bond Model Replication and Pricing Using Dynamic Trading Strategies Pricing Using Risk eutral Probabilities Onefactor model, noarbitrage
More informationFinancial 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 equitylinked securities requires an understanding of financial
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationChapter 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 informationCall 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
More informationHull, 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. OnePerio Binomial Moel Creating synthetic options (replicating options)
More informationUCLA 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 informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationCurrency Options (2): Hedging and Valuation
Overview Chapter 9 (2): Hedging and Overview Overview The Replication Approach The Hedging Approach The Riskadjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic
More informationLecture 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 information1 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
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.
More informationOptions 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 informationPricing 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 informationOption Basics. c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153
Option Basics c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153 The shift toward options as the center of gravity of finance [... ] Merton H. Miller (1923 2000) c 2012 Prof. YuhDauh Lyuu,
More informationLecture 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 informationLecture 3: Put Options and DistributionFree Results
OPTIONS and FUTURES Lecture 3: Put Options and DistributionFree Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distributionfree results? option
More informationAdditional questions for chapter 4
Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two sixmonth periods it is expected to go up by 1% or go down by 1%. The riskfree interest rate is 8% per annum with
More informationOverview. 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 Noarbitrage bounds on option prices Binomial option pricing BlackScholesMerton
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationFinancial 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. BlackScholesMerton formula 2. Binomial trees
More informationThe 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 nondividend paying stock whose price is initially S 0. Divide time into small
More informationBINOMIAL 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
More informationFINANCIAL 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 informationIntroduction 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 informationOptions/1. Prof. Ian Giddy
Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls PutCall Parity Combinations and Trading Strategies Valuation Hedging Options2
More informationOptions 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 PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationLecture 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 informationCall 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 informationBlackScholes. SerHuang Poon. September 29, 2008
BlackScholes SerHuang 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 informationSession 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第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS 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
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime 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 informationFIN 673 Pricing Real Options
FIN 673 Pricing Real Options Professor Robert B.H. Hauswald Kogod School of Business, AU From Financial to Real Options Option pricing: a reminder messy and intuitive: lattices (trees) elegant and mysterious:
More informationFinance 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 informationLecture 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 informationCHAPTER 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 putcall parity theorem as follows: P = C S + PV(X) + PV(Dividends)
More informationEC3070 FINANCIAL DERIVATIVES
BINOMIAL OPTION PRICING MODEL A OneStep Binomial Model The Binomial Option Pricing Model is a simple device that is used for determining the price c τ 0 that should be attributed initially to a call option
More informationA Comparison of Option Pricing Models
A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More information1.1 Some General Relations (for the no dividend case)
1 American Options Most traded stock options and futures options are of Americantype while most index options are of Europeantype. The central issue is when to exercise? From the holder point of view,
More informationCaput 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 informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 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 informationOptions pricing in discrete systems
UNIVERZA V LJUBLJANI, FAKULTETA ZA MATEMATIKO IN FIZIKO Options pricing in discrete systems Seminar II Mentor: prof. Dr. Mihael Perman Author: Gorazd Gotovac //2008 Abstract This paper is a basic introduction
More informationName: 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 InTerm Exam I Instructor: Milica Čudina Notes: This is a closed
More informationBUS 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 informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationThe Discrete Binomial Model for Option Pricing
The Discrete Binomial Model for Option Pricing Rebecca Stockbridge Program in Applied Mathematics University of Arizona May 4, 2008 Abstract This paper introduces the notion of option pricing in the context
More informationTwoState Model of Option Pricing
Rendleman and Bartter [1] put forward a simple twostate model of option pricing. As in the BlackScholes model, to buy the stock and to sell the call in the hedge ratio obtains a riskfree portfolio.
More informationwhere N is the standard normal distribution function,
The BlackScholesMerton 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 informationLecture 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 informationOption 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 OptionPricing Formula Investment in
More informationOption 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
More informationS 1 S 2. Options and Other Derivatives
Options and Other Derivatives The OnePeriod Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating
More informationUse the option quote information shown below to answer the following questions. The underlying stock is currently selling for $83.
Problems on the Basics of Options used in Finance 2. Understanding Option Quotes Use the option quote information shown below to answer the following questions. The underlying stock is currently selling
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationOption 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 informationManual 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
More informationMidTerm Spring 2003
MidTerm Spring 2003 1. (1 point) You want to purchase XYZ stock at $60 from your broker using as little of your own money as possible. If initial margin is 50% and you have $3000 to invest, how many shares
More information