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


 Buddy Wilson
 2 years ago
 Views:
Transcription
1 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 on yen with a strike price of $0.008 sells for $ (iii) The continuously compounded riskfree interest rate on dollars is 3%. (iv) The continuously compounded riskfree interest rate on yen is.5%. Calculate the price of a fouryear yendenominated European put option on dollars with a strike price of 25. Solution. By the Put Call Parity (PCP) for currency options with x 0 as a spot exchange rate: C(x 0, K, T ) P (x 0, K, T ) = x 0 e r f T Ke r dt C = 0.0 e e = By the CallPut relationship in foreign and domestic currency: P f ( x 0, K, T ) = K x 0 C d (x 0, K, T ) Hence, the price of a fouryear yendenominated European put option on dollars with a strike price of = 25 is P = Problem = (i) C(K, T) denotes the current price of a Kstrike Tyear European call option on a nondividendpaying stock. (ii) P(K, T) denotes the current price of a Kstrike Tyear European put option on the same stock. (iii) S denotes the current price of the stock. (iv) The continuously compounded riskfree interest rate is r. Which of the following is (are) correct? (I) 0 C(50, T ) C(55, T ) 5e rt (II) 50e rt P (45, T ) C(50, T ) + S 55e rt (III) 45e rt P (45, T ) C(50, T ) + S 50e rt Solution. Since the maximum possible value of the difference between the two calls is: using K = 50 and K 2 = 55, we have: Since by direction property : if K K 2 then max(c(k ) C(K 2 )) = e rt (K 2 K ) C(50, T ) C(55, T ) 5e rt C(K ) C(K 2 )
2 2 again using K = 50 and K 2 = 55, we have C(50, T ) C(55, T ) 0 Therefore, I is true. By the Put Call Parity for a nondividend paying stock: C(S, K, T ) P (S, K, T ) = S Ke rt P (S, K, T ) C(S, K, T ) + S = Ke rt For K = 50, the equality becomes: Since by direction property : if K K 2 then using K = 45 and K 2 = 50, we have: P (S, 50, T ) C(S, 50, T ) + S = 50e rt P (K ) P (K 2 ) P (45) P (50) P (S, 45, T ) C(S, 50, T ) + S 50e rt On the other hand, for K = 45, the Put Call Parity becomes: Since by direction property : if K K 2 then using K = 45 and K 2 = 50, we have: Therefore, III is true. Thus, II is not true. P (S, 45, T ) C(S, 45, T ) + S = 45e rt C(K ) C(K 2 ) C(50) C(45) P (S, 45, T ) C(S, 50, T ) + S 45e rt Problem 3 Assume the BlackScholes framework. Eight months ago, an investor borrowed money at the riskfree interest rate to purchase a oneyear 75strike European call option on a nondividendpaying stock. At that time, the price of the call option was 8. Today, the stock price is 85. The investor decides to close out all positions. (i) The continuously compounded riskfree interest rate is 5%. (ii) The stock s volatility is 26%. Calculate the eightmonth holding profit. Solution. 8 months after purchasing the option, the remaining time to expiration is 4 months. We are given: Eur. call, t = 3, S = 85, K = 75, σ = 0.26, r = 0.05, δ = 0. C = Se δt N(d ) Ke rt N(d 2 ), where d = ln (S/K) + (r δ + 2 σ2 )t σ t N(d ) = N(.02) = = ln 85 d 2 = d σ t = = N(d 2 ) = N(0.87) = Ke rt = 75e = Se δt = ( ) C = = = = 0.53 =
3 3 Therefore, the holding profit is: Problem e = = Assume the BlackScholes framework holds. Consider an option on a stock. You are given the following information at time 0: (i) The stock price is S(0), which is greater than 80. (ii) The option price is (iii) The option delta is 0.8 (iv) The option gamma is The stock price changes to Using the deltagamma approximation, you find that the option price changes to 2.2. Determine S(0). Solution. Deltagamma approximation C(S t+h ) = C(S t ) + ɛ + 2 Γɛ2 C(S t+h ) C(S t ) = ɛ + 2 Γɛ2, ɛ = S t+h S t Thus: = 0.8ɛ ɛ ɛ 2 0.8ɛ = 0 D = = = ɛ = = or ɛ 2 = Since S(0) > 80, ɛ = 86 S(0) < = 6. Therefore, ɛ = and S(0) = = = Problem 5 A nondividend paying stock s price is currently 40. It is known that at the end of one month the stock s price will be either 42 or 38. The continuously compounded riskfree rate is Assume that the expected return on the stock is 0% as opposed to the riskfree rate. What is the true discount rate (rate of return) for the onemonth European call option with a strike price of 39? Solution. We are given: T = 2, n =, S = 40, K = 39, S u = 42, S d = 38, α = 0., r = 0.08, δ = 0. We need to find γ. The stock tree: S = 40 us = 42 ds = 38 The call tree:
4 4 C u = 3 C C d = 0 Note that u = Using equation that connects p and p : Since C d = 0, Using formulas for p and p : Therefore, Problem 6 =.05 and d = = 0.95 e γh (pc u + ( p)c d ) = e rh (p C u + ( p )C d ) e γh p = e rh p γ = h ln e rh p p = e(α δ)h d u d p = e(r δ)h d u d = e = e = = e γ = 2 ln = A stock s price follows a lognormal model. (i) The initial price is 90. (ii) α = 0.2. (iii) δ = (iv) σ = 0.3. Construct a 92% confidence interval for the price of the stock at the end of five years. Solution. The lognormal parameters m and v for the distribution of the stock price after five years are m = (α δ 0.5σ 2 )t = ( )5 = = v = σ t = = If Z N(m, v 2 ), then the 00( α)% confidence interval is defined to be ( m zα/2 v, m + z α/2 v ), where z α/2 is the number such that P [ ] α Z > z α/2 = 2 In our case α = 0.08 α 2 = 0.04 and P [Z > z 0.04 ] = 0.04 P [Z < z 0.04 ] = 0.04 P [Z < z 0.04 ] = 0.96 z 0.04 =.75 The confidence interval is ( m zα/2 v, m + z α/2 v ) = ( , ) = (.0989,.2489) p
5 5 Exponentiating, we get ( e.0989, e.2489) = (0.3332, ) Multiplying by the initial stock price of 90, the answer is (29.99, 33.79). Problem 7 For a twoyear 00strike American call option on a stock is modeled with a 2period binomial tree. (i) The initial stock price is 00. (ii) The tree is constructed based on forward prices. (iii) The continuously compounded riskfree interest rate is 5%. (iv) The stock s volatility is 30%. (v) The stock pays dividends at the rate of 5%. Determine the option premium. Solution. We are given: T = 2, n = 2, r = 0.05, γ = 0.05, S = 00, K = 00, σ = 0.3. We need to find C. Since the binomial tree is constructed using forward prices, we have: The stock tree: u = e (r δ)h+σ h = e ( ) +0.3 = e 0.3 =.3499 d = e (r δ)h σ h = e ( ) 0.3 = e 0.3 = us = u 2 S = 82.2 S = 00 uds = 00 ds = d 2 S = The call tree: C uu = 82.2 C u C C ud = 0 C d C dd = 0 Using the formula for risk neutral probability: p = + e σ h = + e 0.3 = , p =
6 6 We now use the formula for pricing an option with risk neutral probability: C u = e rh (p C uu + ( p )C ud ) = e = C d = e rh (p C ud + ( p )C dd ) = 0 Note that if the call is exercised at node u, the payoff will be = This is greater than obtained above, so it is optimal to exercise the call at that point and we ll use C u = in further calculations. C = e rh (p C u + ( p )C d ) = e = 4.64
Institutional 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 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 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 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 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 informationLecture 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 RiskNeutral Valuation 2 RiskNeutral World 3 TwoSteps Binomial
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 informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 11. The BlackScholes Model: Hull, Ch. 13.
Week 11 The BlackScholes Model: Hull, Ch. 13. 1 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2 The BlackScholes Model 1.
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 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 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 informationSection 4, PutCall Parity
HCMSAF1033B, Financial Economics, 6/30/10, age 49 Section 4, utcall arity The value of an otherwise similar call and put option on the same asset are related. Adam and Eve Example: XYZ stock pays no
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 informationFigure 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 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 information1 The BlackScholes Formula
1 The BlackScholes Formula In 1973 Fischer Black and Myron Scholes published a formula  the BlackScholes formula  for computing the theoretical price of a European call option on a stock. Their paper,
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 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 informationOption 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 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 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 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 information1 The BlackScholes model: extensions and hedging
1 The BlackScholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More 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 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 informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
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 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 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 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 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 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 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 informationChapter 15. Option on Stock Indices and Currencies. Promble Promble Promble Promble Finger 15.1 Tree for Promble 15.
Chapter 15 Option on Stock Indices and Currencies Promble 15.1. When the index goes down to 7, the value of the portfolio can be expected to be 1*(7/8)=$8.75 million.(this assumes that the dividend yield
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 informationMAT 3788 Lecture 8 Forwards for assets with dividends Currency forwards
MA 3788 Lecture 8 Forwards for assets with dividends Currency forwards Prof. Boyan Kostadinov, City ech of CUNY he Value of a Forward Contract as ime Passes Recall from last time that the forward price
More informationP1. Preface. Thank you for choosing ACTEX.
Preface P Preface Thank you for choosing ACTEX. Since Exam MFE was introduced in May 007, there have been quite a few changes to its syllabus and its learning objectives. To cope with these changes, ACTEX
More informationTrading Strategies Involving Options. Chapter 11
Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A riskfree bond and an option to create a principalprotected note A stock and an option Two or more options of the same type
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 informationProperties 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 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 informationHedging. 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
More informationWeek 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 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 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 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 informationChapter 13 The BlackScholesMerton Model
Chapter 13 The BlackScholesMerton Model March 3, 009 13.1. The BlackScholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
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 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 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 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 information14 Greeks Letters and Hedging
ECG590I Asset Pricing. Lecture 14: Greeks Letters and Hedging 1 14 Greeks Letters and Hedging 14.1 Illustration We consider the following example through out this section. A financial institution sold
More informationOptions: 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 informationCS 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 information4. 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 informationContents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM
Contents 1 PutCall Parity 1 1.1 Review of derivative instruments................................................ 1 1.1.1 Forwards............................................................ 1 1.1.2 Call
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 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 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 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 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 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 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 informationBeyond Black Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles
Beyond Black Scholes: Smile & Exo6c op6ons Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles 1 What is a Vola6lity Smile? Rela6onship between implied
More informationCURRENCY OPTION PRICING II
Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility BlackScholes Moel for Currency Options Properties of the BS Moel Option Sensitivity
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 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 informationCHAPTER 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 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 informationOPTIONS PRICING EXERCISE
William L. Silber Foundations of Finance (B01.2311) OPTIONS PRICING EXERCISE Minnesota Mining and Manufacturing (3M) is awarding a yearend bonus to its Senior VicePresident of Marketing in the form of
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 informationQuantitative Strategies Research Notes
Quantitative Strategies Research Notes December 1992 Valuing Options On PeriodicallySettled Stocks Emanuel Derman Iraj Kani Alex Bergier SUMMARY In some countries, for example France, stocks bought or
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 information(1) Consider a European call option and a European put option on a nondividendpaying stock. You are given:
(1) Consider a European call option and a European put option on a nondividendpaying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more
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 informationStudy on the Volatility Smile of EUR/USD Currency Options and Trading Strategies
Prof. Joseph Fung, FDS Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies BY CHEN Duyi 11050098 Finance Concentration LI Ronggang 11050527 Finance Concentration An Honors
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 informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
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 informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More informationSensitivity Analysis of Options. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 264
Sensitivity Analysis of Options c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 264 Cleopatra s nose, had it been shorter, the whole face of the world would have been changed. Blaise Pascal
More informationHedging of Financial Derivatives and Portfolio Insurance
Hedging of Financial Derivatives and Portfolio Insurance Gasper Godson Mwanga African Institute for Mathematical Sciences 6, Melrose Road, 7945 Muizenberg, Cape Town South Africa. email: gasper@aims.ac.za,
More informationDetermination of Forward and Futures Prices. Chapter 5
Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets
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 informationb. June expiration: 9523 = 95 + 23/32 % = 95.71875% or.9571875.9571875 X $100,000 = $95,718.75.
ANSWERS FOR FINANCIAL RISK MANAGEMENT A. 24 Value of Tbond Futures Contracts a. March expiration: The settle price is stated as a percentage of the face value of the bond with the final "27" being read
More informationCHAPTER 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 informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Financial Economics
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Financial Economics June 2014 changes Questions 130 are from the prior version of this document. They have been edited to conform
More informationCHAPTER 22 Options and Corporate Finance
CHAPTER 22 Options and Corporate Finance Multiple Choice Questions: I. DEFINITIONS OPTIONS a 1. A financial contract that gives its owner the right, but not the obligation, to buy or sell a specified asset
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
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 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 informationChapter 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 informationn(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3
ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric
More informationLecture 11: The Greeks and Risk Management
Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.
More informationCHAPTER 5 OPTION PRICING THEORY AND MODELS
1 CHAPTER 5 OPTION PRICING THEORY AND MODELS In general, the value of any asset is the present value of the expected cash flows on that asset. In this section, we will consider an exception to that rule
More informationBond 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 informationValuing Options / Volatility
Chapter 5 Valuing Options / Volatility Measures Now that the foundation regarding the basics of futures and options contracts has been set, we now move to discuss the role of volatility in futures and
More information