CAS/CIA Exam 3 Spring 2007 Solutions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "CAS/CIA Exam 3 Spring 2007 Solutions"

Transcription

1 CAS/CIA Exam 3 Note: The CAS/CIA Spring 007 Exam 3 contained 16 questions that were relevant for SOA Exam MFE. The solutions to those questions are included in this document. Solution 3 B Chapter 1, PutCall Parity We are asked to find the arbitrage profit per share, but it is not clear whether the question means that one share is involved at the outset or that one share is involved at the end of 6 months. Fortunately, both interpretations lead to the same answer. We can use putcall parity to understand this problem: rt d T CEur ( K, T) + Ke = S0e + PEur ( K, T) Putcall parity tells us that the following strategies produce the same cash flow at the end of 6 months: Purchase 1 call option and lend the present value of the $50 strike price. The cost of this strategy is: rt 0.03(0.5) CEur ( K, T) + Ke = + 50e = (0.5) Purchase e strategy is: shares of stock and purchase 1 put option. The cost of this d T 0.0(0.5) S0 e + PEur ( K, T) = 49.70e +.35 = Since the first strategy costs less than the second strategy, arbitrage profits can be earned by going long the first strategy and shorting the second strategy (buy low, sell high). The net payoff in 6 months is zero, and the arbitrage profit now is: = Based on this result, Choice B is the correct answer. Note that the answer above is based on one share at the end of 6 months, but at the 0.0(0.5) outset, only e shares are sold. If we want to determine the arbitrage profits based on the sale of one share at the outset, then we must gross up the strategies above 0.0(0.5) by e. This results in an arbitrage profit of: 0.0(0.5) e = The correct answer remains B. ActuarialBrew.com 014 Page 1

2 Solution 4 E Chapter 1, PutCall Parity We can use putcall parity to find the value of the put option: rt CEur ( K, T) + Ke = S0 PV0, T ( Div) + PEur ( K, T) 0.10(0.5) 0.10( /1) 0.10(5 /1) + 30e = 9 0.5e 0.5 e + PEur (30,0.5) PEur (30,0.5) =.508 Solution 1 A Chapter, Factors Affecting Premiums Statement I is true, because an American option provides all of the rights provided by a European option and more. Statement II is false, because a higher stock price produces a lower payoff for a put option. Statement III is false, because a higher strike price reduces the payoff of a call option. Solution 13 A Chapter 1, Synthetic Tbills We can use putcall parity to find the interest rate: C ( K, T) + Ke = S + P ( K, T) Eur r 0.5 r 0.5 rt e = e = r 0.5 = ln( ) r = Eur ActuarialBrew.com 014 Page

3 Solution 14 D Chapter 3, TwoPeriod Binomial Model The riskneutral probability of an upward move is constant since: ( )1 90 ( rd ) h e e d p* = = 100 = u d ( )1 99 ( rd ) h e e d p* = = 110 = u d ( )1 81 ( rd ) h e e d p* = = 90 = u d If the final stock price is $11, then the payoff of the call option is $11 $95 = $6. If the final stock price is $99, then the payoff of the call option is $99 $95 = $4. If the final stock price is $81, then the payoff of the call option is $0. The value of the call option is: n rhn ( ) n j n j j n j V( S0, K,0) e ( p*) (1 p*) V( S0u d, K, hn) j j () e [( ) (6) ( )( )4 0] Solution 15 C Chapter 3, Riskneutral Probability The values of u and d are: ( r d) h+ s h ( )(1 / 3) / 3 u = e = e = ( rd) hs h ( )(1 / 3) / 3 d = e = e = The riskneutral probability of an upward movement is: ( rd ) h ( )(1 / 3) e d e p* = = = ud ActuarialBrew.com 014 Page 3

4 Solution 16 C Chapter 3, Replication The option pays 6 in the upstate and 0 in the downstate: V V u d Max[0,18 1] 6 Max[0,4 1] 0 The dividend rate is zero, so the number of shares needed to replicate the option is: h Vu Vd e e Su ( d) 10(18 /10 4 /10) Solution 17 D Chapter 3, TwoPeriod Binomial Model Since no dividend is mentioned, we assume that the dividend rate is zero. The values of u and d are: ( r d) h+ s h ( )(1) u = e = e = ( rd) hs h ( )(1) d = e = e = The riskneutral probability of an upward movement is: ( rd ) h ( )(1) e d e p* = = = ud The stock price tree and its corresponding tree of option prices are: Stock European Put The put option can be exercised only if the stock price falls to $ At that point, the put option has a payoff of $ Although the entire table of put prices is filled in above, we can directly find the value of the put option as follows: n rhn ( ) ÈÊnˆ j nj j nj 0 = Â ÍÁ 0 j j = 0 ÎË 0.05(1)() VS (, K,0) e ( p*) (1 p*) VSud (, Khn, ) = e È ( ) (1.7916) = Î ActuarialBrew.com 014 Page 4

5 Solution 0 B Chapter 7, BlackScholes Call Price The BlackScholes Formula uses the riskfree interest rate, not the expected annual return. The first step is to calculate d 1 and d : We have: S K + r d + s T = = ln( / ) ( 0.5 ) ln(58.96 / 60) ( ) 0.5 d s T = d d s T = 1 = = Nd ( 1 ) N( ) N( 0.10) 1 N(0.10) Nd ( ) N( ) N( 0.0) 1 N(0.0) The value of the European call option is: T rt CEur Se N( d1) Ke N( d) 0.05(0.5) 0.06(0.5) 58.96e e Solution 1 A Chapter 7, BlackScholes Put Price Since no dividend is mentioned, we assume that the dividend rate is zero. The first step is to calculate d 1 and d : We have: ln( S/ K) + ( r d s ) T ln(9.67 / 8.75) + ( ) 0.5 d1 = = s T = d = d1 s T = = Nd ( 1 ) N( ) N(0.70) Nd ( ) N( ) N(0.50) The value of the European put option is: rt T PEur Ke N( d) Se N( d1) 0.08(0.5) 0.00(0.5) 8.75 e ( ) 9.67 e ( ) ActuarialBrew.com 014 Page 5

6 Solution 3 C Chapter 9, DeltaHedging We are not told the interval of time over which the stock price increases. Instead, we are told that the stock price quickly jumps. We can take this to mean that the stock price increase is instantaneous. Therefore there is no need to consider the impact from borrowing or lending at the riskfree rate. The delta of a call option covering 1 share is: d T Call e N( d1 ) e D = = = The marketmaker sold 100 call options, each of which covers 100 shares. Therefore, from the perspective of the marketmaker, the delta of the position to be hedged is: , To deltahedge the position, the marketmaker purchases 5, shares of stock. When the stock price increases, the marketmaker has a gain on the stock and a loss on the call options: Gain on Stock: 5, (41 40) = 5, Gain on Calls: = 5, Net Profit: The marketmaker has a net loss of Solution 33 D Chapter 9, MarketMaker s Profit If a marketmaker writes an option and deltahedges the position, then the marketmaker s profit is: e Marketmaker's profit ª Gt hqt rh( StDt Vt) Statement I is false, since the change in the stock price, e, is squared in the formula above. Statement II is true since time decay reduces the value of the option over time. Statement III is poorly worded. The first sentence of Statement III is not always true, so it would be reasonable for a student to conclude that Statement III is false. The ambiguity in Statement III does not prevent us from obtaining the correct answer. Since Statement I is false, the only possible correct answer is D. Since Statement III is poorly worded, so let s consider a clearer version: If in order to hedge, the marketmaker must purchase stock, then the net carrying cost is a component of the overall cost. This statement is true, because the marketmaker must pay interest on any funds that are borrowed. ActuarialBrew.com 014 Page 6

7 The exam s version of Statement III appears at the bottom of page 40 in the textbook. Solution 34 E Chapter 10, Barrier Options All of the options have the same strike price. We can use the parity relationship for barrier options to find the value of a regular option. Making use of the downandout and downandin options that have a barrier of $30,000, we have: Knockin option + Knockout option = Ordinary option = 55 Therefore, the value of the ordinary option is $55. We can use the parity relationship again, this time with the upandin and upandout options that have a barrier of $50,000: Knockin option + Knockout option = Ordinary option X + 15 = 55 X = 40 The value of the upandin option with a barrier of $50,000 is $40. Solution 35 B Chapter 15, Itô s Lemma This question contains a typo. The correct answer to the question as worded is: 1 ds(t) = ( a d s )S(t)dt + sdz(t) This correct answer does not appear as one of the choices, though. Based on the fact that Choice B is listed as the correct answer, it seems clear that a parenthesis was misplaced in the first bullet point in the question. The corrected expression is: ( a d 1 s s ) S(t ) = S(0) exp ( )t + Z(t ) The solution below is based on the corrected expression for S(t ) shown above. A lognormal stock price implies that changes in the stock price follow geometric Brownian motion: + ( a d 0.5 s ) t sz( t) St () = S(0) e dst () = ( a d) Stdt () + sstdzt () () The expression in Choice B is equal to ds( t ) : dst () = ( a d) Stdt () + sstdzt () () 1 1 = ( a d) S() t dt + ss() t dz() t + s S() t dt s S() t dt 1 1 = ( a d s ) S() t dt + ss() t dz() t + s S() t dt ActuarialBrew.com 014 Page 7

8 Solution 36 E Chapter 19, Vasicek Model This question contains a typo. Before we answer it, let s make 3 modifications: In the first line, Vasieck should be Vasicek. The third bullet point should be: r = 0.05 The final line should be, Calculate the expected change in the interest rate, expressed as an annual rate. The process can be written as: dr = 0.15(0.10 r) dt dZ The expected change in the interest rate is: Edr [ ] = E[0.15(0.10 rdt ) dz] = ( rdt ) = ( rdt ) Since r = 0.05, we have: [ ] E[ dr] = ( r) dt = (0.05) dt = dt To convert the expected change in the interest rate into an annual rate, we must divide by the increment of time: dt = dt Solution 37 B Chapter 18, BlackDermanToy Model This question contains a typo. Just under the table, the phrase should be, where volatility refers to the volatility of the bond yield one year from today. The yield volatility for the 3year bond is: È 1/( T 1) 1/ P(1, T, ru) 1 ÈP(1,3, ru) 1 Yield volatility = 0.5 ln Í = 0.5 ln Í Í 1/( T 1) 1/ Î P(1, T, rd) 1 ÍÎP(1,3, rd) 1 È 1/ = 0.5 ln Í = / ÍÎ ActuarialBrew.com 014 Page 8

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

More information

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald) Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

More information

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 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 information

More 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 More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

More information

Option Valuation. Chapter 21

Option Valuation. Chapter 21 Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

More information

Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies

Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies No-arbitrage bounds on option prices Binomial option pricing Black-Scholes-Merton

More information

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

More information

Contents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM

Contents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM Contents 1 Put-Call Parity 1 1.1 Review of derivative instruments................................................ 1 1.1.1 Forwards............................................................ 1 1.1.2 Call

More information

Options/1. Prof. Ian Giddy

Options/1. Prof. Ian Giddy Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

More information

Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

More information

S 1 S 2. Options and Other Derivatives

S 1 S 2. Options and Other Derivatives Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating

More information

Pricing Formula for 3-Period Discrete Barrier Options

Pricing Formula for 3-Period Discrete Barrier Options Pricing Formula for 3-Period Discrete Barrier Options Chun-Yuan Chiu Down-and-Out Call Options We first give the pricing formula as an integral and then simplify the integral to obtain a formula similar

More information

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

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs) Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A four-year dollar-denominated European put option

More information

Chapter 21 Valuing Options

Chapter 21 Valuing Options Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher

More information

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

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13. Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

More information

UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:

UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date: UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,

More information

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

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

More information

One Period Binomial Model

One Period Binomial Model FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

More information

Properties of Stock Options. Chapter 10

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

More information

P-1. Preface. Thank you for choosing ACTEX.

P-1. 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 information

Lecture 5: Put - Call Parity

Lecture 5: Put - Call Parity Lecture 5: Put - Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible

More information

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

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

More information

Lecture 21 Options Pricing

Lecture 21 Options Pricing Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call

More information

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

More information

Put-Call Parity. chris bemis

Put-Call Parity. chris bemis Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain

More information

Option Pricing Basics

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

More information

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

1. Assume that a (European) call option exists on this stock having on exercise price of $155. MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on

More information

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. 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

More information

On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

More information

Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9

Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9 Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying

More information

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

More information

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common

More information

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1 Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. Black-Scholes-Merton formula 2. Binomial trees

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

Use the option quote information shown below to answer the following questions. The underlying stock is currently selling for $83.

Use the option quote information shown below to answer the following questions. The underlying stock is currently selling for $83. Problems on the Basics of Options used in Finance 2. Understanding Option Quotes Use the option quote information shown below to answer the following questions. The underlying stock is currently selling

More information

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

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

More information

Chapter 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 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 information

where N is the standard normal distribution function,

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

More information

The Black-Scholes Model

The Black-Scholes Model The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

More information

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. Put-Call Parity II. One-Period Binomial Option Pricing III. Adding Periods to the Binomial Model IV. Black-Scholes

More information

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

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

More information

Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing. This is sometimes referred to as the intrinsic value of the option. Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

More information

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 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

More information

Chapter 14 Review Note Sample Excerpt

Chapter 14 Review Note Sample Excerpt Chapter 14 Review Note Sample Excerpt Exotic Options: I Derivatives Markets (2 nd Edition) Online Excerpt of Section 14.5 with hree Questions and Solutions Introduction his document provides a sample excerpt

More information

Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

More information

Lecture 3: Put Options and Distribution-Free Results

Lecture 3: Put Options and Distribution-Free Results OPTIONS and FUTURES Lecture 3: Put Options and Distribution-Free Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distribution-free results? option

More information

Introduction to Options. Derivatives

Introduction to Options. Derivatives Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

More information

European Options Pricing Using Monte Carlo Simulation

European 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 information

Exam MFE/Exam 3F Review

Exam MFE/Exam 3F Review Exam MFE/Exam 3F Review Johnew Zhang April 22, 2012 Contents 1 Put-Call Parity 4 1.1 Review....................................... 4 1.1.1 Forwards................................. 4 1.1.2 Call and put

More information

Stock. Call. Put. Bond. Option Fundamentals

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

More information

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option

More information

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

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13 Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

More information

Financial Options: Pricing and Hedging

Financial Options: Pricing and Hedging Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial

More information

Exam MFE/3F Sample Questions and Solutions #1 to #76

Exam MFE/3F Sample Questions and Solutions #1 to #76 Exam MFE/3F Sample Questions and Solutions #1 to #76 In this version, standard normal distribution values are obtained by using the Cumulative Normal Distribution Calculator and Inverse CDF Calculator

More information

(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given:

(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more

More information

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

Option Payoffs. Problems 11 through 16: Describe (as I have in 1-10) the strategy depicted by each payoff diagram. #11 #12 #13 #14 #15 #16 Option s Problems 1 through 1: Assume that the stock is currently trading at $2 per share and options and bonds have the prices given in the table below. Depending on the strike price (X) of the option

More information

Options: Valuation and (No) Arbitrage

Options: Valuation and (No) Arbitrage Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

More information

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

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

More information

FIN 411 -- Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices

FIN 411 -- Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices FIN 411 -- Investments Option Pricing imple arbitrage relations s to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but

More information

Martingale Pricing Applied to Options, Forwards and Futures

Martingale Pricing Applied to Options, Forwards and Futures IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the

More information

CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the put-call parity theorem as follows: P = C S + PV(X) + PV(Dividends)

More information

Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

More information

An Introduction to Exotic Options

An Introduction to Exotic Options An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor

More information

VALUATION IN DERIVATIVES MARKETS

VALUATION IN DERIVATIVES MARKETS VALUATION IN DERIVATIVES MARKETS September 2005 Rawle Parris ABN AMRO Property Derivatives What is a Derivative? A contract that specifies the rights and obligations between two parties to receive or deliver

More information

Week 12. Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14.

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

More information

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

Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12 Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond

More information

Chapter 2 Introduction to Option Management

Chapter 2 Introduction to Option Management Chapter 2 Introduction to Option Management The prize must be worth the toil when one stakes one s life on fortune s dice. Dolon to Hector, Euripides (Rhesus, 182) In this chapter we discuss basic concepts

More information

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

More information

a. What is the portfolio of the stock and the bond that replicates the option?

a. What is the portfolio of the stock and the bond that replicates the option? Practice problems for Lecture 2. Answers. 1. A Simple Option Pricing Problem in One Period Riskless bond (interest rate is 5%): 1 15 Stock: 5 125 5 Derivative security (call option with a strike of 8):?

More information

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 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

More information

b. June expiration: 95-23 = 95 + 23/32 % = 95.71875% or.9571875.9571875 X $100,000 = $95,718.75.

b. June expiration: 95-23 = 95 + 23/32 % = 95.71875% or.9571875.9571875 X $100,000 = $95,718.75. ANSWERS FOR FINANCIAL RISK MANAGEMENT A. 2-4 Value of T-bond 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 information

Option pricing. Vinod Kothari

Option pricing. Vinod Kothari Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

More information

CHAPTER 5 OPTION PRICING THEORY AND MODELS

CHAPTER 5 OPTION PRICING THEORY AND MODELS 1 CHAPTER 5 OPTION PRICING THEORY AND MODELS In general, the value of any asset is the present value of the expected cash flows on that asset. In this section, we will consider an exception to that rule

More information

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

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

More information

Part V: Option Pricing Basics

Part V: Option Pricing Basics erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, put-call parity introduction

More information

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

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

More information

FINANCIAL OPTION ANALYSIS HANDOUTS

FINANCIAL OPTION ANALYSIS HANDOUTS FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any

More information

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the

More information

Options (1) Class 19 Financial Management, 15.414

Options (1) Class 19 Financial Management, 15.414 Options (1) Class 19 Financial Management, 15.414 Today Options Risk management: Why, how, and what? Option payoffs Reading Brealey and Myers, Chapter 2, 21 Sally Jameson 2 Types of questions Your company,

More information

Options Markets: Introduction

Options Markets: Introduction Options Markets: Introduction Chapter 20 Option Contracts call option = contract that gives the holder the right to purchase an asset at a specified price, on or before a certain date put option = contract

More information

1 The Black-Scholes model: extensions and hedging

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

More information

Consider a European call option maturing at time T

Consider a European call option maturing at time T Lecture 10: Multi-period Model Options Black-Scholes-Merton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T

More information

4. Options Markets. 4.6. Applications

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

More information

Derivative Securities: Lecture 3 Options. Sources: J. Hull Instructor s notes Yahoo Finance Bloomberg.com

Derivative Securities: Lecture 3 Options. Sources: J. Hull Instructor s notes Yahoo Finance Bloomberg.com Derivative Securities: Lecture 3 Options Sources: J. Hull Instructor s notes Yahoo Finance Bloomberg.com Options A call option is a contract that allows the holder (long) to purchase an underlying asset

More information

The Black-Scholes pricing formulas

The Black-Scholes pricing formulas The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

More information

Chapter 21: Options and Corporate Finance

Chapter 21: Options and Corporate Finance Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the

More information

Chapter 13 The Black-Scholes-Merton Model

Chapter 13 The Black-Scholes-Merton Model Chapter 13 The Black-Scholes-Merton Model March 3, 009 13.1. The Black-Scholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)

More information

UCLA 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 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 information

CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given

More information

Convenient Conventions

Convenient Conventions C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff

More information

Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?

Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions

More information

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 1 Exam FM Questions Practice Exam 1 1. Consider the following yield curve: Year Spot Rate 1 5.5% 2 5.0% 3 5.0% 4 4.5% 5 4.0% Find the four year forward rate. A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 2.

More information

Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values

Option 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 information

Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

More information

Options. + Concepts and Buzzwords. Readings. Put-Call Parity Volatility Effects

Options. + Concepts and Buzzwords. Readings. Put-Call Parity Volatility Effects + Options + Concepts and Buzzwords Put-Call Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options

More information

The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models 780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon

More information

Options 1 OPTIONS. Introduction

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

More information