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


 Griffin Cole
 2 years ago
 Views:
Transcription
1 Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated. Even though I will focus on the slope and convexity restrictions, the idea is the same for the violation of any arbitrage bound. 1 Notation: I will use the following notation throughout these notes: K Strike price T Maturity r Riskfree rate C (K) The price of an American call at time t < T c (K) The price of a European call at time t < T P (K) The price of an American put at time t < T p (K) The price of a European put at time t < T r(t t) B B (t, T ) e Also, I will assume that there are options trading at three strike prices K 1, K 2 and K 3 such that: K 1 < K 2 < K 3 2 The slope restriction: All that you need to know to check the slope restriction (and to implement arbitrage opportunities, when it doesn t hold) are 4 simple but powerful inequalities: C (K 2 ) C (K 1 ) + (K 2 K 1 ) 0 (for both American and European calls) (1) P (K 1 ) P (K 2 ) + (K 2 K 1 ) 0 (for both American and European puts) (2) c (K 2 ) c (K 1 ) + (K 2 K 1 ) B 0 (for European calls only) (3) p (K 1 ) p (K 2 ) + (K 2 K 1 ) B 0 (for European puts only) (4) 1 Please send any comments, suggestions and corrections to 1
2 In this section, I will show why the inequalities above should hold and how they lead to the so called slope restrictions. I start by showing that the first two inequalities are easily derived from the following result: Result 1 The price of a call (put) option is an decreasing (increasing) function of the strike price. In other words, K 1 < K 2 = C (K 1 ) C (K 2 ) (5) This holds for both American and European Options. K 1 < K 2 = P (K 1 ) P (K 2 ) (6) In words, calls with a larger strike price are worth less than otherwise identical calls, and puts with a larger strike price are worth more than otherwise identical puts To show that Result 1 is true, consider the payoff of two (otherwise identical) European calls with different strikes. A quick look at graph 1 is enough to conclude that the investor is always better off (at maturity) holding the call with the lowest strike. Now, consider an American call option: at any time before maturity exercising the option with the lowest strike will be more profitable than exercising the option with the highest strike (the payoff graph is basically the same). A similar argument shows that puts with the highest strike is worth more than the put with the lowest price. Put Options with Different Strike Prices Call Options with Different Strike Prices K 2 At any given stock price, we are better off holding the put with the largest strike price K 2 At any given stock price, we are better off holding the call with the smallest strike price K 1 K 1 This simple observation about the sensitivity of calls and puts leads to a powerful result: Result 2 The sensitivity of calls and puts to the strike price is bounded (from above for calls and from below for puts). Specifically, dc dk 0 dp dk This holds for both American and European options. 2
3 As you probably know, increasing (decreasing) functions have the property that the derivative (the slope of the graph) evaluated at any point is positive (negative) 2. An easy way to see that is the following: note that Result 1 implies: C (K 2 ) C (K 1 ) K 2 K 1 0 for every values of K 2 and K 1, such that K 2 K 1 > 0 (if you are still not convinced, take another look at graph 1). Loosely speaking, making the difference (K 2 K 1 ) smaller and smaller in the inequality above leads to dc dk 0. A similar argument can be made for the puts. Thus, the slope of the graph of a call (as a function of the strike price) is never negative and the slope of the graph of a put (as a function of the strike price) is never positive. But we can do better than that: not only we can tell the sign of these slopes, but also we can find lower and upper bounds for calls and puts, respectively. The following result makes this idea precise. Result 3 The sensitivity of the calls and puts to their strike prices is bounded from above and from below, respectively. Specifically, 1 dc dk 0 dp dk 1 This holds for both American and European options// In particular, for European Options only, we have that: B dc dk 0 dp dk B To show that the above result is true, consider the following strategy: Buy a call with strike price K 2 and sell an otherwise identical call with strike price K 1. What would be the payoff of such strategy if the options were European? What would be the payoff of such strategy if the options were American and we exercise them both today? Graph 3 answers both questions. 2 Actually, this is not quite right. The correct theorem is that strictly increasing (decreasing) continuous functions have positive (negative) derivatives. But as mentioned before, we will abstain from such formalisms 3
4 Payoff of the Strategy: C(K2)  C(K1) C(K2) The payoff of this strategy is never positive! 0 C(K2)  C(K1) C(K1) Since the price of the call with the highest strike is worth more (Result 1), there will be an outflow of money at maturity. In the case of an American call, this outflow will occur at the time of exercise. Of course, there is no way someone would follow that strategy since it always leads to a negative payoff. But, suppose there was a way to assure the holder of such strategy an additional amount of money (K 2 K 1 ) > 0 at the time of exercise. We could then guarantee a positive payoff at maturity for any stock price. Graph 4 illustrates that idea. Payoff of the Strategy: C(K2)  C(K1) + (K2  K1) x B This is the payoff of the strategy after summing (K2  K1) to C(K2)  C(K1) C(K2) K 2  K 1 0 C(K2)  C(K1) C(K1) But how could someone guarantee a positive inflow of money of (K 2 K 1 ) at time of exercise? In the case of Europeans the time of exercise corresponds to the maturity of the option. So, in this case, you just have to lend the present value of (K 2 K 1 ), i.e. (K 2 K 1 ) B (t, T ). In the case of American options, you 4
5 would require that amount of money ( K 2 K 1 ) every period, since there is always the possibility of early exercise. Now, since the payoff of such strategy cannot be negative, its value today has to be positive. Thus: C (K 2 ) C (K 1 ) + (K 2 K 1 ) 0 c (K 2 ) c (K 1 ) + (K 2 K 1 ) B 0 (for both American and European calls) (for European calls only) or, equivalently: C (K 2 ) C (K 1 ) (K 2 K 1 ) c (K 2 ) c (K 1 ) (K 2 K 1 ) 1 B (for both American and European calls) (for European calls only) If we make the difference K 2 K 1 smaller and smaller, in the limit we will have slope restrictions 1 dc dc 0 for American and European and B 0 for European options dk dk only3.// As an exercise, do the same for the put options. 3 The Convex Restriction Again, all that you have to know to derive the convex restriction is that: C (K 3 ) C (K 2 ) K 3 K 2 C (K 2) C (K 1 ) K 2 K 1 0 As before, lets show that the above inequality is true and then how it leads to a convex restriction.// By definition, the inequality above says that the price of a call is a convex function of the strike price. Loosely speaking, we say that a function y = y (x) is convex when the sensitivity of y to variations in x increases as we increase x. In sum, the following result holds: Result 4 The price of a call or put option is a convex function of the strike price. In other words, for K 1 < K 2 : K 1 < K 2 = dc dk < dc K1 dk (7) K2 K 1 < K 2 = dp dk < dp K1 dk (8) K2 where dx dk K1 represents the derivative (slope or sensitivity) of the function X with respect to K, when evaluated at the point K 1. This holds for both American and European Options. 3 Note that we are making use of Result 1 here. 5
6 In words, as the difference between the strike prices increases, the sensitivity to the strike price of two otherwise identical options also increases. To show that this is true, consider a butterfly spread using calls (i.e. buy one call with the highest strike, buy one call with the lowest strike and sell two calls with the intermediary strike). For K 1 < K 2 < K 3, the payoff of such strategy is depicted below: Payoff of a Butterfly Strategy  c(k 1 ) The Payoff is never negative! K 1  c(k 3 ) K 3  c(k 2 ) It is clear that the payoff of such strategy is positive (actually, nonnegative) everywhere. Thus, its price today have to be positive. Thus, it must be true that: C (K 3 ) C (K 2 ) [C (K 2 ) C (K 1 )] 0 (9) Now, assume that K 2 = K 1+K 3 2, which implies that (K 3 K 2 ) = (K 2 K 1 ). We can divide both sides of (9) above by (K 3 K 2 ) = (K 2 K 1 ) and get: or, equivalently, C (K 3 ) C (K 2 ) K 3 K 2 C (K 2) C (K 1 ) K 2 K 1 0 (10) C (K 3 ) C (K 2 ) C (K 2) C (K 1 ) K 3 K 2 K 2 K 1 /bigskip Because this restriction actually defines a convex function, it is called convexity restriction.// Now, if we make the difference (K 3 K 2 ) = (K 2 K 1 ) go to zero, we have: K 3 K 1 = dc dk dc K3 dk This holds for both American and European calls.// As an exercise, do the same for puts. K1 6
7 4 How to Implement Arbitrage Opportunities Arbitrages opportunities are very easy to construct when some inequality restriction is violated. All that you have to do is to use the inequality that describes the violation to construct a portfolio that gives you some money today. Then, you just have show that there is no way to lose money anytime in the future. We will look at two examples corresponding to violations of the slope and convexity restrictions for European options. 4.1 Violation of the slope restriction for European Options As mentioned above, all that we need to check for a violation of the slope restriction are the two slope inequalities : c (K 2 ) c (K 1 ) + (K 2 K 1 ) B 0 p (K 1 ) p (K 2 ) + (K 2 K 1 ) B 0 (for European calls only) (for European puts only) Now, say that the slope restriction is violated for calls. In this case, we have that: or, equivalently, c (K 2 ) c (K 1 ) + (K 2 K 1 ) B < 0 c (K 2 ) + c (K 1 ) (K 2 K 1 ) B > 0 (11) Here is how you have to interpret (11): since the elements in this inequality are amounts of money, I can make money today by replicating what is happening on the left hand side of (11). More specifically, just copy the numbers above to the column of the table that represents the transactions today: Transaction Today Maturity c (K 2 ) +c (K 1 ) (K 2 K 1 ) B Total > 0 Now, figure out what you have to do to get the right signs (which will lead to a positive amount of money today) Transaction Today Maturity Buy call with strike price K 2 c (K 2 ) Sell call with strike price K 1 +c (K 1 ) Lend (K 2 K 1 ) B dollars (K 2 K 1 ) B Total > 0 7
8 The final step is to check that we do not lose money at maturity. Obviously, this will always be the case if some arbitrage bound is being violated: Transaction Today Maturity Buy call with strike price K 2 c (K 2 ) max (0, S (T ) K 2 ) Sell call with strike price K 1 +c (K 1 ) max (0, S (T ) K 1 ) Lend (K 2 K 1 ) B dollars (K 2 K 1 ) B K 2 K 1 Total > 0? Since there is uncertainty about the behavior of the asset at maturity, we need to check each case: Transaction Maturity: If S (T ) < K 1 < K 2 If K 1 < S (T ) < K 2 If K 1 < K 2 < S (T ) Buy call with strike price K 2 0 S (T ) K 2 S (T ) K 2 Sell call with strike price K (S (T ) K 1 ) Lend (K 2 K 1 ) B dollars K 2 K 1 Total K 2 K 1 > 0 S (T ) K 1 > 0 0 And we are done! Note that the approach taken here is quite general: whenever you have an arbitrage bound being violated, you have an inequality. Play around with this inequality to make it look like X + Y + Z > 0. Then, put what is on the left side in the column describing the transactions (remember, X, Y and Z are the values today). By construction, you will always make money today. All that is left is to show is that you will not lose money in the future. Once again, this will always be the case (by definition) if an arbitrage restriction is being violated. 4.2 Violation of the convex restriction for European calls: Lets do the same thing for the convex restriction for calls. Since the convex restriction can be translated into the inequality (9), a violation of the convex restriction is the same as 4 : or, equivalently: C (K 3 ) C (K 2 ) [C (K 2 ) C (K 1 )] < 0 [C (K 3 ) C (K 2 )] + [C (K 2 ) C (K 1 )] > 0 But what is on the left side? A butterfly spread, right? So you know you will never lose money at maturity (take another look at the graph above). But even if you didn t know that, the same conclusion could be reached by just repeating the argument above: Write down the inflows and outflows of money today and the corresponding transactions. 4 There is a catch here: I am assuming that (K 3 K 2 ) = (K 2 K 1 ). If that is not the case you have to work with inequality (10). The argument is pretty much the same, though. 8
9 Transaction Today Maturity Buy call with strike price K 1 c (K 1 ) Sell call with strike price K 2 c (K 2 ) Buy call with strike price K 3 c (K 3 ) Sell call with strike price K 2 c (K 2 ) Total: > 0 Since this is just a butterfly, its payoff is always positive, so that I am guaranteed not to lose money in the future. You can actually show that by plugging in the payoffs: Transaction Today Maturity Buy call with strike price K 1 c (K 1 ) max (0, S (T ) K 1 ) Sell call with strike price K 2 c (K 2 ) max (0, S (T ) K 2 ) Buy call with strike price K 3 c (K 3 ) max (0, S (T ) K 3 ) Sell call with strike price K 2 c (K 2 ) max (0, S (T ) K 2 ) Total: > 0? Once again, we have to check each possible case. A quick tip: if you are unsure about all the cases you have to consider, plot the payoff of the entire strategy on a graph. Transaction Maturity: (note: K 2 = K 1+K 3 ) 2 If S (T ) < K 1 If K 1 < S (T ) < K 2 If K 2 < S (T ) < K 3 If K 3 < S (T ) Buy call (strike K 1 ) 0 S (T ) K 1 S (T ) K 1 S (T ) K 1 Sell call (strike K 2 ) 0 0 (S (T ) K 2 ) (S (T ) K 2 ) Buy call (strike K 3 ) S (T ) K 3 Sell call (strike K 2 ) 0 0 (S (T ) K 2 ) (S (T ) K 2 ) Total: 0 S (T ) K 1 > 0 2K 2 K 1 S (T ) > 0 0 And we are done. What if you use puts instead of calls to construct the butterfly? Before redoing all the tables for the put case, draw a graph and compare to the one above... 9
Option Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)
Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.
More 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 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 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 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 information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More 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 informationUniversity of Texas at Austin. HW Assignment 7. Butterfly spreads. Convexity. Collars. Ratio spreads.
HW: 7 Course: M339D/M389D  Intro to Financial Math Page: 1 of 5 University of Texas at Austin HW Assignment 7 Butterfly spreads. Convexity. Collars. Ratio spreads. 7.1. Butterfly spreads and convexity.
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationA short note on American option prices
A short note on American option Filip Lindskog April 27, 2012 1 The setup An American call option with strike price K written on some stock gives the holder the right to buy a share of the stock (exercise
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
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 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 informationUnderlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)
INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the PutCallForward
More informationOptions. Moty Katzman. September 19, 2014
Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to
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 information11 Option. Payoffs and Option Strategies. Answers to Questions and Problems
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various
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 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 informationChapter 2 An Introduction to Forwards and Options
Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram
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 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 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 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 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 22: FUTURES MARKETS
CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support
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 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 informationChapter 3: Commodity Forwards and Futures
Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique
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 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 informationOnline Appendix: Payoff Diagrams for Futures and Options
Online Appendix: Diagrams for Futures and Options As we have seen, derivatives provide a set of future payoffs based on the price of the underlying asset. We discussed how derivatives can be mixed and
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 informationQuestions and Answers
MA3245 Financial Mathematics I Suggested Solutions of Tutorial 1 (Semester 2/0304) Questions and Answers 1. What is the difference between entering into a long forward contract when the forward price
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 informationECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005
ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005 Options: General [1] Define the following terms associated with options: a. Option An option is a contract which gives the holder
More informationEC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals
EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline Contents 1 Call options and put options 1 2 Payoffs on options
More 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 informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. YuhDauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More informationChapter 20 Understanding Options
Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interestrate options (III) Commodity options D) I, II, and III
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 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 informationComputational Finance Options
1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to
More 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 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 informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More informationDerivatives: Options
Derivatives: Options Call Option: The right, but not the obligation, to buy an asset at a specified exercise (or, strike) price on or before a specified date. Put Option: The right, but not the obligation,
More informationCHAPTER 14. Stock Options
CHAPTER 14 Stock Options Options have fascinated investors for centuries. The option concept is simple. Instead of buying stock shares today, you buy an option to buy the stock at a later date at a price
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationOptions 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 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 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 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 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 informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 PutCall Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More informationChapter 6 Arbitrage Relationships for Call and Put Options
Chapter 6 Arbitrage Relationships for Call and Put Options Recall that a riskfree arbitrage opportunity arises when an investment is identified that requires no initial outlays yet guarantees nonnegative
More informationRogue trading? So what is a derivative? by John Dickson. Rogue trading?
1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,
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 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 informationSolutions to Practice Questions (Bonds)
Fuqua Business School Duke University FIN 350 Global Financial Management Solutions to Practice Questions (Bonds). These practice questions are a suplement to the problem sets, and are intended for those
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 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 informationThe Binomial Model for Stock Options
2 The Binomial Model for Stock Options 2.1 The Basic Model We now discuss a simple onestep binomial model in which we can determine the rational price today for a call option. In this model we have two
More informationGoals. Options. Derivatives: Definition. Goals. Definitions Options. Spring 2007 Lecture Notes 4.6.1 Readings:Mayo 28.
Goals Options Spring 27 Lecture Notes 4.6.1 Readings:Mayo 28 Definitions Options Call option Put option Option strategies Derivatives: Definition Derivative: Any security whose payoff depends on any other
More informationChapter 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 informationBefore we discuss a Call Option in detail we give some Option Terminology:
Call and Put Options As you possibly have learned, the holder of a forward contract is obliged to trade at maturity. Unless the position is closed before maturity the holder must take possession of the
More informationLecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model Recall that the price of an option is equal to
More informationCHAPTER 11: ARBITRAGE PRICING THEORY
CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationForward Contracts and Forward Rates
Forward Contracts and Forward Rates Outline and Readings Outline Forward Contracts Forward Prices Forward Rates Information in Forward Rates Reading Veronesi, Chapters 5 and 7 Tuckman, Chapters 2 and 16
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
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 informationTwoState Options. John Norstad. jnorstad@northwestern.edu http://www.norstad.org. January 12, 1999 Updated: November 3, 2011.
TwoState Options John Norstad jnorstad@northwestern.edu http://www.norstad.org January 12, 1999 Updated: November 3, 2011 Abstract How options are priced when the underlying asset has only two possible
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION PROBLEM SETS 1. Options provide numerous opportunities to modify the risk profile of a portfolio. The simplest example of an option strategy that increases risk
More informationEXERCISES FROM HULL S BOOK
EXERCISES FROM HULL S BOOK 1. Three put options on a stock have the same expiration date, and strike prices of $55, $60, and $65. The market price are $3, $5, and $8, respectively. Explain how a butter
More informationFTS Real Time Client: Equity Portfolio Rebalancer
FTS Real Time Client: Equity Portfolio Rebalancer Many portfolio management exercises require rebalancing. Examples include Portfolio diversification and asset allocation Indexation Trading strategies
More informationLecture 3: Forward Contracts Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 3: Forward Contracts Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Derivatives Derivatives are financial
More informationPayoff (Riskless bond) Payoff(Call) Combined
ShortAnswer 1. Is the payoff to stockholders most similar to the payoff on a long put, a long call, a short put, a short call or some combination of these options? Long call 2. ebay s current stock price
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationFUNDING 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. PutCall Parity II. OnePeriod Binomial Option Pricing III. Adding Periods to the Binomial Model IV. BlackScholes
More informationDerivative: a financial instrument whose value depends (or derives from) the values of other, more basic, underlying values (Hull, p. 1).
Introduction Options, Futures, and Other Derivatives, 7th Edition, Copyright John C. Hull 2008 1 Derivative: a financial instrument whose value depends (or derives from) the values of other, more basic,
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 informationUnderstanding Profit and Loss Graphs
Understanding Profit and Loss Graphs The axis defined The Y axis, or the vertical, up/down axis, represents the profit or loss for the strategy. Anything on the Y axis above the X axis represents a gain.
More informationCHAPTER 22: FUTURES MARKETS
CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus $250 1329.78 = $332,445. The closing futures price for the March contract was 1364.00,
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 2. Forwards, Options, and Hedging This lecture covers the basic derivatives contracts: forwards (and futures), and call and put options. These basic contracts are
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
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 informationOther variables as arguments besides S. Want those other variables to be observables.
Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value
More informationCHAPTER 20 Understanding Options
CHAPTER 20 Understanding Options Answers to Practice Questions 1. a. The put places a floor on value of investment, i.e., less risky than buying stock. The risk reduction comes at the cost of the option
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 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 informationTrading around a position using covered calls
Trading around a position using covered calls June 23, 2011 1 Trading around a position using covered calls June 23, 2011 June 23, 2011 2 Disclaimer This presentation is the creation of Roger Manzolini
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 information1 Interest rates, and riskfree investments
Interest rates, and riskfree investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)
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 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 informationOptions: Definitions, Payoffs, & Replications
Options: Definitions, s, & Replications Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch) s Options Markets 1 / 34 Definitions and terminologies An option gives the
More informationInvesco 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 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 information