# Convenient Conventions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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

2 Payoff, Mathematically Speaking The payoff of a call at expiration is C =max(0,s X). The payoff of a put at expiration is P =max(0,x S). A call will be exercised only if the stock price is higher than the strike price. A put will be exercised only if the stock price is less than the strike price. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 169

3 Payoff 40 Long a call Payoff Short a call Price Price -40 Payoff 50 Long a put Payoff Short a put Price Price -50 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 170

4 Payoff, Mathematically Speaking (continued) At any time t before the expiration date, we call max(0,s t X) the intrinsic value of a call. At any time t before the expiration date, we call max(0,x S t ) the intrinsic value of a put. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 171

5 Payoff, Mathematically Speaking (concluded) A call is in the money if S>X, at the money if S = X, and out of the money if S<X. A put is in the money if S<X, at the money if S = X, and out of the money if S>X. Options that are in the money at expiration should be exercised. a Finding an option s value at any time before expiration is a major intellectual breakthrough. a 11% of option holders let in-the-money options expire worthless. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 172

6 Call value Stock price Put value Stock price c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 173

7 Cash Dividends Exchange-traded stock options are not cash dividend-protected (or simply protected). The option contract is not adjusted for cash dividends. The stock price falls by an amount roughly equal to the amount of the cash dividend as it goes ex-dividend. Cash dividends are detrimental for calls. The opposite is true for puts. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 174

8 Stock Splits and Stock Dividends Options are adjusted for stock splits. After an n-for-m stock split, the strike price is only m/n times its previous value, and the number of shares covered by one contract becomes n/m times its previous value. Exchange-traded stock options are adjusted for stock dividends. Options are assumed to be unprotected. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 175

9 Example Consider an option to buy 100 shares of a company for \$50 per share. A 2-for-1 split changes the term to a strike price of \$25 per share for 200 shares. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 176

10 Short Selling Short selling (or simply shorting) involves selling an assetthatisnot owned with the intention of buying it back later. If you short 1,000 XYZ shares, the broker borrows them from another client to sell them in the market. This action generates proceeds for the investor. The investor can close out the short position by buying 1,000 XYZ shares. Clearly, the investor profits if the stock price falls. Not all assets can be shorted. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 177

11 Payoff of Stock Payoff 80 Long a stock Payoff Short a stock Price Price -80 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 178

12 Covered Position: Hedge A hedge combines an option with its underlying stock in such a way that one protects the other against loss. Covered call: A long position in stock with a short call. a It is covered because the stock can be delivered to the buyer of the call if the call is exercised. Protective put: A long position in stock with a long put. Both strategies break even only if the stock price rises, so they are bullish. a A short position has a payoff opposite in sign to that of a long position. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 179

13 Profit Protective put Stock price Profit Covered call Stock price Solid lines are profits of the portfolio one month before maturity assuming the portfolio is set up when S = 95 then. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 180

14 Covered Position: Spread A spread consists of options of the same type and on the same underlying asset but with different strike prices or expiration dates. We use X L, X M,and X H to denote the strike prices with X L <X M <X H. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 181

15 Covered Position: Spread (continued) A bull call spread consists of a long X L call with the same expiration date. X H The initial investment is C L C H. call and a short The maximum profit is (X H X L ) (C L C H ). When both are exercised at expiration. The maximum loss is C L C H. When neither is exercised at expiration. If we buy (X H X L ) 1 units of the bull call spread and X H X L 0, a (Heaviside) step function emerges as the payoff. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 182

16 Profit Bull spread (call) 4 2 Stock price c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 183

17 Covered Position: Spread (continued) Writing an X H put and buying an X L put with identical expiration date creates the bull put spread. A bear spread amounts to selling a bull spread. It profits from declining stock prices. Three calls or three puts with different strike prices and the same expiration date create a butterfly spread. The spread is long one X L call, long one X H call, and short two X M calls. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 184

18 Profit Butterfly Stock price c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 185

19 Covered Position: Spread (continued) A butterfly spread pays a positive amount at expiration only if the asset price falls between X L and X H. Take a position in (X M X L ) 1 units of the butterfly spread. When X H X L 0, it approximates a state contingent claim, a which pays \$1 only when the state S = X M happens. b a Alternatively, Arrow security. b See Exercise of the textbook. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 186

20 Covered Position: Spread (concluded) The price of a state contingent claim is called a state price. The (undiscounted) state price equals 2 C X 2. Recall that C is the call s price. a In fact, the PV of 2 C/ X 2 is the probability density of the stock price S T = X at option s maturity. b a One can also use the put (see Exercise of the textbook). b Breeden and Litzenberger (1978). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 187

21 Covered Position: Combination A combination consists of options of different types on the same underlying asset. These options must be either all bought or all written. Straddle: A long call and a long put with the same strike price and expiration date. Since it profits from high volatility, a person who buys a straddle is said to be long volatility. Selling a straddle benefits from low volatility. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 188

22 Profit 10 Straddle Stock price c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 189

23 Covered Position: Combination (concluded) Strangle: Identical to a straddle except that the call s strike price is higher than the put s. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 190

24 Profit Strangle Stock price c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 191

25 Arbitrage in Option Pricing c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 192

26 All general laws are attended with inconveniences, when applied to particular cases. David Hume ( ) c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 193

27 Arbitrage The no-arbitrage principle says there is no free lunch. It supplies the argument for option pricing. A riskless arbitrage opportunity is one that, without any initial investment, generates nonnegative returns under all circumstances and positive returns under some. In an efficient market, such opportunities do not exist (for long). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 194

28 Portfolio Dominance Principle Consider two portfolios A and B. A should be more valuable than B if A s payoff is at least as good as B s under all circumstances and better under some. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 195

29 Two Simple Corollaries A portfolio yielding a zero return in every possible scenario must have a zero PV. Short the portfolio if its PV is positive. Buy it if its PV is negative. In both cases, a free lunch is created. Two portfolios that yield the same return in every possible scenario must have the same price. a a Aristotle, those who are equal should have everything alike. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 196

30 The PV Formula (p. 32) Justified Theorem 1 For a certain cash flow C 1,C 2,...,C n, P = n C i d(i). i=1 Suppose the price P <P. Short the n zeros that match the security s n cash flows. The proceeds are P dollars. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 197

31 P P C C 2 1 C 3 C 1 C2 C 3 C n C n security zeros c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 198

32 The Proof (concluded) Then use P of the proceeds to buy the security. The cash inflows of the security will offset exactly the obligations of the zeros. Arisklessprofitof P P dollars has been realized now. If P >P, just reverse the trades. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 199

33 Two More Examples An American option cannot be worth less than the intrinsic value. a Suppose the opposite is true. So the American option is cheaper than its intrinsic value. For the call: Short the stock and lend X dollars. For the put: Borrow X dollars and buy the stock. In either case, the payoff is the intrinsic value. a max(0,s t X) for the call and max(0,x S t ) for the put. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 200

34 (continued) Two More Examples (continued) At the same time, buy the option, promptly exercise it, and close the stock position. For the call, call the lent money to exercise it. For the put, deliver the stock and use the received strike price to settle the debt. The cost of buying the option is less than the intrinsic value. So there is an immediate arbitrage profit. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 201

35 Two More Examples (concluded) A put or a call must have a nonnegative value. Suppose otherwise and the option has a negative price. Buy the option for a positive cash flow now. It will end up with a nonnegative amount at expiration. So an arbitrage profit is realized now. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 202

36 Relative Option Prices These relations hold regardless of the model for stock prices. Assume, among other things, that there are no transactions costs or margin requirements, borrowing and lending are available at the riskless interest rate, interest rates are nonnegative, and there are no arbitrage opportunities. Let the current time be time zero. PV(x) stands for the PV of x dollars at expiration. Hence PV(x) =xd(τ) where τ is the time to expiration. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 203

37 Consider the portfolio of: Put-Call Parity a One short European call; One long European put; One share of stock; A loan of PV(X). C = P + S PV(X). (22) All options are assumed to carry the same strike price X and time to expiration, τ. The initial cash flow is therefore a Castelli (1877). C P S +PV(X). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 204

38 The Proof (continued) At expiration, if the stock price S τ X, the put will be worth X S τ and the call will expire worthless. The loan is now X. The net future cash flow is zero: 0+(X S τ )+S τ X =0. On the other hand, if S τ >X, the call will be worth S τ X and the put will expire worthless. The net future cash flow is again zero: (S τ X)+0+S τ X =0. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205

39 The Proof (concluded) The net future cash flow is zero in either case. The no-arbitrage principle (p. 196) implies that the initial investment to set up the portfolio must be nil as well. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 206

40 Consequences of Put-Call Parity There is only one kind of European option. The other can be replicated from it in combination with stock and riskless lending or borrowing. Combinations such as this create synthetic securities. S = C P +PV(X): A stock is equivalent to a portfolio containing a long call, a short put, and lending PV(X). C P = S PV(X): A long call and a short put amount to a long position in stock and borrowing the PV of the strike price (buying stock on margin). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 207

41 Intrinsic Value Lemma 2 An American call or a European call on a non-dividend-paying stock is never worth less than its intrinsic value. The put-call parity implies C =(S X)+(X PV(X)) + P S X. Recall C 0 (p. 202). It follows that C max(s X, 0), the intrinsic value. An American call also cannot be worth less than its intrinsic value (p. 200). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 208

42 Intrinsic Value (concluded) A European put on a non-dividend-paying stock may be worth less than its intrinsic value. Lemma 3 For European puts, P max(pv(x) S, 0). Prove it with the put-call parity. Can explain the right figure on p. 173 why P<X S when S is small. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 209

43 Early Exercise of American Calls European calls and American calls are identical when the underlying stock pays no dividends. Theorem 4 (Merton (1973)) An American call on a non-dividend-paying stock should not be exercised before expiration. By an exercise in text, C max(s PV(X), 0). Ifthecallisexercised,thevalueis S X. But max(s PV(X), 0) S X. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 210

44 Remarks The above theorem does not mean American calls should be kept until maturity. What it does imply is that when early exercise is being considered, a better alternative is to sell it. Early exercise may become optimal for American calls on a dividend-paying stock, however. Stock price declines as the stock goes ex-dividend. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 211

45 Early Exercise of American Calls: Dividend Case Surprisingly, an American call should be exercised only at a few dates. a Theorem 5 An American call will only be exercised at expiration or just before an ex-dividend date. In contrast, it might be optimal to exercise an American put even if the underlying stock does not pay dividends. a See Theorem of the textbook. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 212

46 A General Result a Theorem 6 (Cox and Rubinstein (1985)) Any piecewise linear payoff function can be replicated using a portfolio of calls and puts. Corollary 7 Any sufficiently well-behaved payoff function can be approximated by a portfolio of calls and puts. a See Exercise of the textbook. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 213

47 Convexity of Option Prices a Lemma 8 For three otherwise identical calls or puts with strike prices X 1 <X 2 <X 3, C X2 ωc X1 +(1 ω) C X3 P X2 ωp X1 +(1 ω) P X3 Here ω (X 3 X 2 )/(X 3 X 1 ). (Equivalently, X 2 = ωx 1 +(1 ω) X 3.) a See Lemma of the textbook. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 214

48 The Intuition behind Lemma 8 a Set up the following portfolio: ωc X1 C X2 +(1 ω) C X3. This is a butterfly spread (p. 184). It has a nonnegative value as, for any S at maturity, ω max(s X 1, 0) max(s X 2, 0) + (1 ω) max(s X 3, 0) 0. Therefore, ωc X1 C X2 +(1 ω) C X3 0. a Contributed by Mr. Cheng, Jen-Chieh (B ) on March 17, c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 215

49 Option on a Portfolio vs. Portfolio of Options Consider a portfolio of non-dividend-paying assets with weights ω i. Let C i denote the price of a European call on asset i with strike price X i. All options expire on the same date. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 216

50 Option on a Portfolio vs. Portfolio of Options (concluded) An option on a portfolio is cheaper than a portfolio of options. a Theorem 9 The call on the portfolio with a strike price X i ω i X i has a value at most ω i C i. The same holds for European puts. i a See Theorem of the textbook. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 217

51 Option Pricing Models c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 218

52 If the world of sense does not fit mathematics, so much the worse for the world of sense. Bertrand Russell ( ) Black insisted that anything one could do with a mouse could be done better with macro redefinitions of particular keys on the keyboard. EmanuelDerman, My Life as a Quant (2004) c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 219

53 The Setting The no-arbitrage principle is insufficient to pin down the exact option value. Need a model of probabilistic behavior of stock prices. One major obstacle is that it seems a risk-adjusted interest rate is needed to discount the option s payoff. Breakthrough came in 1973 when Black ( ) and Scholes with help from Merton published their celebrated option pricing model. a Known as the Black-Scholes option pricing model. a The results were obtained as early as June c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 220

54 Terms and Approach C: call value. P : put value. X: strikeprice S: stockprice ˆr >0: the continuously compounded riskless rate per period. R eˆr : gross return. Start from the discrete-time binomial model. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 221

55 Binomial Option Pricing Model (BOPM) Timeisdiscreteandmeasuredinperiods. If the current stock price is S, it can go to Su with probability q and Sd with probability 1 q, where 0 <q<1andd<u. In fact, d<r<u must hold to rule out arbitrage. Six pieces of information will suffice to determine the option value based on arbitrage considerations: S, u, d, X, ˆr, and the number of periods to expiration. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 222

56 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 223

57 Call on a Non-Dividend-Paying Stock: Single Period The expiration date is only one period from now. C u Su. C d Sd. is the call price at time 1 if the stock price moves to is the call price at time 1 if the stock price moves to Clearly, C u = max(0,su X), C d = max(0,sd X). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 224

58 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 225

59 Call on a Non-Dividend-Paying Stock: Single Period (continued) Set up a portfolio of h shares of stock and B dollars in riskless bonds. This costs hs + B. We call h the hedge ratio or delta. The value of this portfolio at time one is hsu + RB hsd + RB (up move), (down move). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 226

60 Call on a Non-Dividend-Paying Stock: Single Period (continued) Choose h and B such that the portfolio replicates the payoff of the call, hsu + RB = C u, hsd + RB = C d. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 227

61 Call on a Non-Dividend-Paying Stock: Single Period (concluded) Solve the above equations to obtain h = C u C d Su Sd 0, (23) B = uc d dc u (u d) R. (24) By the no-arbitrage principle, the European call should cost the same as the equivalent portfolio, a C = hs + B. As uc d dc u < 0, the equivalent portfolio is a levered long position in stocks. a Or the replicating portfolio, as it replicates the option. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 228

62 American Call Pricing in One Period Have to consider immediate exercise. C =max(hs + B,S X). When hs + B S X, the call should not be exercised immediately. When hs + B<S X, the option should be exercised immediately. For non-dividend-paying stocks, early exercise is not optimal by Theorem 4 (p. 210). So C = hs + B. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 229

63 Put Pricing in One Period Puts can be similarly priced. The delta for the put is (P u P d )/(Su Sd) 0, where Let B = up d dp u (u d) R. P u = max(0,x Su), P d = max(0,x Sd). The European put is worth hs + B. The American put is worth max(hs + B,X S). Early exercise is always possible with American puts. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 230

64 Risk Surprisingly, the option value is independent of q. a Hence it is independent of the expected gross return of the stock, qsu +(1 q) Sd. It therefore does not directly depend on investors risk preferences. The option value depends on the sizes of price changes, u and d, which the investors must agree upon. Then the set of possible stock prices is the same whatever q is. a More precisely, not directly dependent on q. Thanks to a lively class discussion on March 16, c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 231

65 Pseudo Probability After substitution and rearrangement, ( ) ( R d u d C u + hs + B = R Rewrite it as u R u d ) C d. where hs + B = pc u +(1 p) C d R p R d u d. (25), As 0 <p<1, it may be interpreted as a probability. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 232

66 Risk-Neutral Probability The expected rate of return for the stock is equal to the riskless rate ˆr under p as psu +(1 p) Sd = RS. The expected rates of return of all securities must be the riskless rate when investors are risk-neutral. For this reason, p is called the risk-neutral probability. The value of an option is the expectation of its discounted future payoff in a risk-neutral economy. So the rate used for discounting the FV is the riskless rate in a risk-neutral economy. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 233

### Option Basics. c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153

Option Basics c 2012 Prof. Yuh-Dauh 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. Yuh-Dauh Lyuu,

### 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

### Figure 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

### 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

### 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

### 9 Basics of options, including trading strategies

ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European

### Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.

Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero

### 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

### 11 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

### CHAPTER 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

### CHAPTER 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

### 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

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

### 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

### 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

### 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

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008. Options

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the so-called plain vanilla options. We consider the payoffs to these

### 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)

### 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

### Lecture 12. Options Strategies

Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same

### CHAPTER 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

### 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

### Trading Strategies Involving Options. Chapter 11

Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A risk-free bond and an option to create a principal-protected note A stock and an option Two or more options of the same type

### 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

### 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

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

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

### Finance 400 A. Penati - G. Pennacchi. Option Pricing

Finance 400 A. Penati - G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary

### 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

### 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

### Underlying (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 Put-Call-Forward

### Session 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

### 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

### The Binomial Option Pricing Model André Farber

1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a non-dividend paying stock whose price is initially S 0. Divide time into small

### 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

### 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

### 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

### Lecture 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

### 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

### Computational 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

### 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.

### Lecture 4: Derivatives

Lecture 4: Derivatives School of Mathematics Introduction to Financial Mathematics, 2015 Lecture 4 1 Financial Derivatives 2 uropean Call and Put Options 3 Payoff Diagrams, Short Selling and Profit Derivatives

### CHAPTER 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

### Factors Affecting Option Prices

Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The risk-free interest rate r. 6. The

### 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

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

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

### 1 Introduction to Option Pricing

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

### Name Graph Description Payoff Profit Comments. commodity at some point in the future at a prespecified. commodity at some point

Name Graph Description Payoff Profit Comments Long Commitment to purchase commodity at some point in the future at a prespecified price S T - F S T F No premium Asset price contingency: Always Maximum

### Chapter 1: Financial Markets and Financial Derivatives

Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange

### Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS

HW 6 Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS 4. Cost Payoff Profit Call option, X = 85 3.82 5.00 +1.18 Put option, X = 85 0.15 0.00-0.15 Call option, X = 90 0.40 0.00-0.40

### 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

### 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

### CHAPTER 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

### CHAPTER 20: OPTIONS MARKETS: INTRODUCTION

CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X

### Example 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 Put-Call 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.

### Chapter 6 Arbitrage Relationships for Call and Put Options

Chapter 6 Arbitrage Relationships for Call and Put Options Recall that a risk-free arbitrage opportunity arises when an investment is identified that requires no initial outlays yet guarantees nonnegative

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

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

### 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

### 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

### 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

### 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:

### CHAPTER 8: TRADING STRATEGES INVOLVING OPTIONS

CHAPTER 8: TRADING STRATEGES INVOLVING OPTIONS Unless otherwise stated the options we consider are all European. Toward the end of this chapter, we will argue that if European options were available with

### Chapter 20 Understanding Options

Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interest-rate options (III) Commodity options D) I, II, and III

### Introduction to Binomial Trees

11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths

### 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

### Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined

### Forward 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

### Introduction to Options

Introduction to Options By: Peter Findley and Sreesha Vaman Investment Analysis Group What Is An Option? One contract is the right to buy or sell 100 shares The price of the option depends on the price

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

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

### Chapter 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

### Economics 1723: Capital Markets Lecture 20

Economics 1723: Capital Markets Lecture 20 John Y. Campbell Ec1723 November 14, 2013 John Y. Campbell (Ec1723) Lecture 20 November 14, 2013 1 / 43 Key questions What is a CDS? What information do we get

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

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

### 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

### CHAPTER 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,

### BINOMIAL OPTION PRICING

Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing

### FINANCIAL 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.

### Online 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

### INTRODUCTION TO OPTIONS MARKETS QUESTIONS

INTRODUCTION TO OPTIONS MARKETS QUESTIONS 1. What is the difference between a put option and a call option? 2. What is the difference between an American option and a European option? 3. Why does an option

### FIN 3710. Final (Practice) Exam 05/23/06

FIN 3710 Investment Analysis Spring 2006 Zicklin School of Business Baruch College Professor Rui Yao FIN 3710 Final (Practice) Exam 05/23/06 NAME: (Please print your name here) PLEDGE: (Sign your name

### Derivative Users Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.

OPTIONS THEORY Introduction The Financial Manager must be knowledgeable about derivatives in order to manage the price risk inherent in financial transactions. Price risk refers to the possibility of loss

### 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

### 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

### TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder

### t = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3

MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate

### Determination 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

### Other 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

### The Binomial Model for Stock Options

2 The Binomial Model for Stock Options 2.1 The Basic Model We now discuss a simple one-step binomial model in which we can determine the rational price today for a call option. In this model we have two

### 2. 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

### EC372 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

### 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

### 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

### CFA Level -2 Derivatives - I

CFA Level -2 Derivatives - I EduPristine www.edupristine.com Agenda Forwards Markets and Contracts Future Markets and Contracts Option Markets and Contracts 1 Forwards Markets and Contracts 2 Pricing and

### Model-Free Boundaries of Option Time Value and Early Exercise Premium

Model-Free Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 33124-6552 Phone: 305-284-1885 Fax: 305-284-4800

### ECMC49F 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

### 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

### Sensitivity Analysis of Options. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 264

Sensitivity Analysis of Options c 2008 Prof. Yuh-Dauh 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

### 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

### 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

### American and European. Put Option

American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example