# Spot and Forward Rates under Continuous Compounding

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Spot and Forward Rates under Continuous Compounding The pricing formula: P = n Ce is(i) + F e ns(n). i=1 The market discount function: d(n) = e ns(n). The spot rate is an arithmetic average of forward rates, S(n) = f(0, 1) + f(1, 2) + + f(n 1, n). n c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 126

2 Spot and Forward Rates under Continuous Compounding (concluded) The formula for the forward rate: f(i, j) = The one-period forward rate: js(j) is(i). j i f(t ) f(j, j + 1) = ln d(j + 1). d(j) S lim f(t, T + T ) = S(T ) + T T 0 T. f(t ) > S(T ) if and only if S/ T > 0. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 127

3 Unbiased Expectations Theory Forward rate equals the average future spot rate, f(a, b) = E[ S(a, b) ]. (14) Does not imply that the forward rate is an accurate predictor for the future spot rate. Implies the maturity strategy and the rollover strategy produce the same result at the horizon on the average. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 128

4 Unbiased Expectations Theory and Spot Rate Curve Implies that a normal spot rate curve is due to the fact that the market expects the future spot rate to rise. f(j, j + 1) > S(j + 1) if and only if S(j + 1) > S(j) from Eq. (12) on p So E[ S(j, j + 1) ] > S(j + 1) > > S(1) if and only if S(j + 1) > > S(1). Conversely, the spot rate is expected to fall if and only if the spot rate curve is inverted. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 129

5 More Implications The theory has been rejected by most empirical studies with the possible exception of the period prior to Since the term structure has been upward sloping about 80% of the time, the theory would imply that investors have expected interest rates to rise 80% of the time. Riskless bonds, regardless of their different maturities, are expected to earn the same return on the average. That would mean investors are indifferent to risk. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 130

6 A Bad Expectations Theory The expected returns on all possible riskless bond strategies are equal for all holding periods. So (1 + S(2)) 2 = (1 + S(1)) E[ 1 + S(1, 2) ] (15) because of the equivalency between buying a two-period bond and rolling over one-period bonds. After rearrangement, 1 E[ 1 + S(1, 2) ] = 1 + S(1) (1 + S(2)) 2. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 131

7 A Bad Expectations Theory (continued) Now consider two one-period strategies. Strategy one buys a two-period bond and sells it after one period. The expected return is E[ (1 + S(1, 2)) 1 ] (1 + S(2)) 2. Strategy two buys a one-period bond with a return of 1 + S(1). The theory says the returns are equal: [ 1 + S(1) (1 + S(2)) 2 = E S(1, 2) ]. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 132

8 A Bad Expectations Theory (concluded) Combine this with Eq. (15) on p. 131 to obtain [ ] 1 1 E = 1 + S(1, 2) E[ 1 + S(1, 2) ]. But this is impossible save for a certain economy. Jensen s inequality states that E[ g(x) ] > g(e[ X ]) for any nondegenerate random variable X and strictly convex function g (i.e., g (x) > 0). Use g(x) (1 + x) 1 to prove our point. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 133

9 Local Expectations Theory The expected rate of return of any bond over a single period equals the prevailing one-period spot rate: E [ (1 + S(1, n)) (n 1) ] (1 + S(n)) n = 1 + S(1) for all n > 1. This theory is the basis of many interest rate models. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 134

10 Duration Revisited To handle more general types of spot rate curve changes, define a vector [ c 1, c 2,..., c n ] that characterizes the perceived type of change. Parallel shift: [ 1, 1,..., 1 ]. Twist: [ 1, 1,..., 1, 1,..., 1 ]. Let P (y) i C i/(1 + S(i) + yc i ) i be the price associated with the cash flow C 1, C 2,.... Define duration as P (y)/p (0) y. y=0 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 135

11 Fundamental Statistical Concepts c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 136

12 There are three kinds of lies: lies, damn lies, and statistics. Benjamin Disraeli ( ) One death is a tragedy, but a million deaths are a statistic. Josef Stalin ( ) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 137

13 Moments The variance of a random variable X is defined as Var[ X ] E [ (X E[ X ]) 2 ]. The covariance between random variables X and Y is Cov[ X, Y ] E [ (X µ X )(Y µ Y ) ], where µ X and µ Y are the means of X and Y, respectively. Random variables X and Y are uncorrelated if Cov[ X, Y ] = 0. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 138

14 Correlation The standard deviation of X is the square root of the variance, σ X Var[ X ]. The correlation (or correlation coefficient) between X and Y is Cov[ X, Y ] ρ X,Y, σ X σ Y provided both have nonzero standard deviations. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 139

15 Variance of Sum Variance of a weighted sum of random variables equals [ n ] Var a i X i = i=1 n i=1 n a i a j Cov[ X i, X j ]. j=1 It becomes n a 2 i Var[ X i ] i=1 when X i are uncorrelated. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 140

16 Conditional Expectation X I denotes X conditional on the information set I. The information set can be another random variable s value or the past values of X, say. The conditional expectation E[ X I ] is the expected value of X conditional on I; it is a random variable. The law of iterated conditional expectations: E[ X ] = E[ E[ X I ] ]. If I 2 contains at least as much information as I 1, then E[ X I 1 ] = E[ E[ X I 2 ] I 1 ]. (16) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 141

17 The Normal Distribution A random variable X has the normal distribution with mean µ and variance σ 2 if its probability density function is 1 σ 2 2π e (x µ) /(2σ 2). This is expressed by X N(µ, σ 2 ). The standard normal distribution has zero mean, unit variance, and the distribution function Prob[ X z ] = N(z) 1 2π z e x2 /2 dx. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 142

18 Moment Generating Function The moment generating function of random variable X is θ X (t) E[ e tx ]. The moment generating function of X N(µ, σ 2 ) is [ θ X (t) = exp µt + σ2 t 2 ]. (17) 2 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 143

19 Distribution of Sum If X i N(µ i, σi 2 ) are independent, then ( X i N µ i, ) σi 2. i i i Let X i N(µ i, σ 2 i ), which may not be independent. Then n t i X i N n t i µ i, n n t i t j Cov[ X i, X j ]. i=1 i=1 i=1 j=1 X i are said to have a multivariate normal distribution. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 144

20 Generation of Univariate Normal Distributions Let X be uniformly distributed over (0, 1 ] so that Prob[ X x ] = x for 0 < x 1. Repeatedly draw two samples x 1 and x 2 from X until ω (2x 1 1) 2 + (2x 2 1) 2 < 1. Then c(2x 1 1) and c(2x 2 1) are independent standard normal variables where c 2(ln ω)/ω. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 145

21 A Dirty Trick and a Right Attitude Let ξ i (0, 1). are independent and uniformly distributed over A simple method to generate the standard normal variable is to calculate 12 i=1 ξ i 6. But this is not a highly accurate approximation and should only be used to establish ballpark estimates. a a Jäckel, Monte Carlo Methods in Finance (2002). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 146

22 A Dirty Trick and a Right Attitude (concluded) Always blame your random number generator last. a Instead, check your programs first. a The fault, dear Brutus, lies not in the stars but in ourselves that we are underlings. William Shakespeare ( ), Julius Caesar. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 147

23 Generation of Bivariate Normal Distributions Pairs of normally distributed variables with correlation ρ can be generated. X 1 and X 2 be independent standard normal variables. Set U ax 1, V ρu + 1 ρ 2 ax 2. U and V are the desired random variables with Var[ U ] = Var[ V ] = a 2 and Cov[ U, V ] = ρa 2. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 148

24 The Lognormal Distribution A random variable Y is said to have a lognormal distribution if ln Y has a normal distribution. Let X N(µ, σ 2 ) and Y e X. The mean and variance of Y are ( ) µ Y = e µ+σ2 /2 and σy 2 = e 2µ+σ2 e σ2 1, (18) respectively. They follow from E[ Y n ] = e nµ+n2 σ 2 /2. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 149

25 Option Basics c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 150

26 The shift toward options as the center of gravity of finance [... ] Merton H. Miller ( ) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 151

27 Calls and Puts A call gives its holder the right to buy a number of the underlying asset by paying a strike price. A put gives its holder the right to sell a number of the underlying asset for the strike price. How to price options? c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 152

28 Exercise When a call is exercised, the holder pays the strike price in exchange for the stock. When a put is exercised, the holder receives from the writer the strike price in exchange for the stock. An option can be exercised prior to the expiration date: early exercise. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153

29 American and European American options can be exercised at any time up to the expiration date. European options can only be exercised at expiration. An American option is worth at least as much as an otherwise identical European option because of the early exercise feature. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 154

30 C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 155

31 Payoff 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. 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). 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 156

32 Payoff (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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 157

33 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 158

34 Call value Stock price Put value Stock price c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 159

35 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 160

36 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 161

37 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 162

38 Short Selling Short selling (or simply shorting) involves selling an asset that is not 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 163

39 Payoff of Stock Payoff 80 Long a stock Payoff Short a stock Price Price -80 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 164

40 Covered Position: Hedge A hedge combines an option with its underlying stock in such a way that one protects the other against loss. Protective put: A long position in stock with a long put. Covered call: A long position in stock with a short call. a 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 165

41 Profit Protective put Stock price Profit Covered call Stock price Solid lines are current value of the portfolio. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 166

42 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 with X L < X M < X H. to denote the strike prices A bull call spread consists of a long X L call and a short call with the same expiration date. X H The initial investment is C L C H. The maximum profit is (X H X L ) (C L C H ). The maximum loss is C L C H. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 167

43 Profit Bull spread (call) 4 2 Stock price c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 168

44 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 169

45 Profit Butterfly Stock price c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 170

46 Covered Position: Spread (concluded) A butterfly spread pays off a positive amount at expiration only if the asset price falls between X L X H. and A butterfly spread with a small X H X L approximates a state contingent claim, which pays \$1 only when a particular price results. The price of a state contingent claim is called a state price. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 171

47 Covered Position: Combination A combination consists of options of different types on the same underlying asset, and they are 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. Strangle: Identical to a straddle except that the call s strike price is higher than the put s. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 172

48 Profit 10 Straddle Stock price c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 173

49 Profit Strangle Stock price c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 174

50 Arbitrage in Option Pricing c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 175

51 All general laws are attended with inconveniences, when applied to particular cases. David Hume ( ) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 176

52 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). The portfolio dominance principle says portfolio A should be more valuable than B if A s payoff is at least as good under all circumstances and better under some. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 177

53 A Corollary 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. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 178

54 The PV Formula Justified P = n i=1 C id(i) for a certain cash flow C 1, C 2,..., C n. If the price P < P, short the zeros that match the security s n cash flows and use P of the proceeds P to buy the security. Since the cash inflows of the security will offset exactly the obligations of the zeros, a riskless profit of P P dollars has been realized now. If the price P > P, a riskless profit can be realized by reversing the trades. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 179

55 P P C C 2 1 C 3 C 1 C2 C 3 C n C n security zeros c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 180

56 Two More Examples An American option cannot be worth less than the intrinsic value. Otherwise, one can buy the option, promptly exercise it and sell the stock with a profit. A put or a call must have a nonnegative value. Otherwise, one can buy it for a positive cash flow now and end up with a nonnegative amount at expiration. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 181

57 Relative Option Prices These relations hold regardless of the probabilistic 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 182

58 Put-Call Parity (Castelli, 1877) C = P + S PV(X). (19) Consider the portfolio of one short European call, one long European put, one share of stock, and a loan of PV(X). All options are assumed to carry the same strike price and time to expiration, τ. The initial cash flow is therefore C P S + PV(X). At expiration, if the stock price S τ X, the put will be worth X S τ and the call will expire worthless. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 183

59 The Proof (concluded) After the loan, now X, is repaid, 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. After the loan, now X, is repaid, the net future cash flow is again zero: (S τ X) S τ X = 0. The net future cash flow is zero in either case. The no-arbitrage principle implies that the initial investment to set up the portfolio must be nil as well. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 184

60 Consequences of Put-Call Parity There is only one kind of European option because the other can be replicated from it in combination with the underlying stock and riskless lending or borrowing. Combinations such as this create synthetic securities. S = C P + PV(X) says a stock is equivalent to a portfolio containing a long call, a short put, and lending PV(X). C P = S PV(X) implies 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 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 185

61 Intrinsic Value Lemma 1 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. It follows that C max(s X, 0), the intrinsic value. An American call also cannot be worth less than its intrinsic value. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 186

62 Intrinsic Value (concluded) A European put on a non-dividend-paying stock may be worth less than its intrinsic value (p. 159). Lemma 2 For European puts, P max(pv(x) S, 0). Prove it with the put-call parity. Can explain the right figure on p. 159 why P < X S when S is small. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 187

63 Early Exercise of American Calls European calls and American calls are identical when the underlying stock pays no dividends. Theorem 3 (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). If the call is exercised, the value is the smaller S X. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 188

64 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. Stock price declines as the stock goes ex-dividend. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 189

65 Early Exercise of American Calls: Dividend Case Surprisingly, an American call should be exercised only at a few dates. Theorem 4 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. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 190

66 Convexity of Option Prices Lemma 5 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.) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 191

67 Option on Portfolio vs. Portfolio of Options An option on a portfolio of stocks is cheaper than a portfolio of options. Theorem 6 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. Then the call on the portfolio with a strike price X i ω ix i has a value at most i ω ic i. All options expire on the same date. The same result holds for European puts. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 192

### 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

### Bond Price Volatility. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 71

Bond Price Volatility c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 71 Well, Beethoven, what is this? Attributed to Prince Anton Esterházy c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University

### 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,

### 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

### Chapter 9 Parity and Other Option Relationships

Chapter 9 Parity and Other Option Relationships Question 9.1. This problem requires the application of put-call-parity. We have: Question 9.2. P (35, 0.5) C (35, 0.5) e δt S 0 + e rt 35 P (35, 0.5) \$2.27

### 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.

### This question is a direct application of the Put-Call-Parity [equation (3.1)] of the textbook. Mimicking Table 3.1., we have:

Chapter 3 Insurance, Collars, and Other Strategies Question 3.1 This question is a direct application of the Put-Call-Parity [equation (3.1)] of the textbook. Mimicking Table 3.1., we have: S&R Index S&R

### 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

### Chapter 3 Insurance, Collars, and Other Strategies

Chapter 3 Insurance, Collars, and Other Strategies Question 3.1. This question is a direct application of the Put-Call-Parity (equation (3.1)) of the textbook. Mimicking Table 3.1., we have: S&R Index

### Finance 350: Problem Set 8 Alternative Solutions

Finance 35: Problem Set 8 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. All payoff

### Underlying Pays a Continuous Dividend Yield of q

Underlying Pays a Continuous Dividend Yield of q The value of a forward contract at any time prior to T is f = Se qτ Xe rτ. (33) Consider a portfolio of one long forward contract, cash amount Xe rτ, and

### 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

### 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

### A Report On Pricing and Technical Analysis of Derivatives

A Report On Pricing and Technical Analysis of Derivatives THE INDIAN INSTITUTE OF PLANNING AND MANAGEMENT EXECUTIVE SUMMARY The emergence of Derivatives market especially Futures and Options can be traced

### 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

### Zero-Coupon Bonds (Pure Discount Bonds)

Zero-Coupon Bonds (Pure Discount Bonds) The price of a zero-coupon bond that pays F dollars in n periods is F/(1 + r) n, where r is the interest rate per period. Can meet future obligations without reinvestment

### 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

### Chapter 2 Questions Sample Comparing Options

Chapter 2 Questions Sample Comparing Options Questions 2.16 through 2.21 from Chapter 2 are provided below as a Sample of our Questions, followed by the corresponding full Solutions. At the beginning of

### 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

### 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

### This paper is not to be removed from the Examination Halls

~~FN3023 ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON FN3023 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,

### 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

### 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 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 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

### 1.2 Solutions to Chapter 1 Exercises

8 CHAPTER. CALCULUS. OPTIONS. PUT CALL PARITY.. Solutions to Chapter Exercises Problem : Compute lnx) dx. Solution: Using integration by parts, we find that lnx) dx = x) lnx) dx = x lnx) xlnx)) dx = x

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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

### 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

### UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Final Exam. December 2012

UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2012 MBA Final Exam December 2012 Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open notes.

### 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

### Call Option: An option, but not an obligation, to buy at a specified price. o European: Can be exercised only at expiration.

I. Introduction: A derivative is a financial instrument that has a value determined by an underlying asset. Uses of derivatives: o Risk management o Speculation on prices o Reducing transaction costs o

### Determination of Forward and Futures Prices

Determination of Forward and Futures Prices 3.1 Chapter 3 3.2 Consumption vs Investment Assets Investment assets assets held by significant numbers of people purely for investment purposes Examples: gold,

### Contents. iii. MFE/3F Study Manual 9 th edition 10 th printing Copyright 2015 ASM

Contents 1 Put-Call Parity 1 1.1 Review of derivative instruments................................. 1 1.1.1 Forwards........................................... 1 1.1.2 Call and put options....................................

### Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

### 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

### 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

### 1 Strategies Involving A Single Option and A Stock

Chapter 5 Trading Strategies 1 Strategies Involving A Single Option and A Stock One of the attractions of options is that they can be used to create a very wide range of payoff patterns. In the following

### 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

### 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

### Chapter 7: Option pricing foundations Exercises - solutions

Chapter 7: Option pricing foundations Exercises - solutions 1. (a) We use the put-call parity: Share + Put = Call + PV(X) or Share + Put - Call = 97.70 + 4.16 23.20 = 78.66 and P V (X) = 80 e 0.0315 =

### 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)

### Lecture 4 Options & Option trading strategies

Lecture 4 Options & Option trading strategies * Option strategies can be divided into three main categories: Taking a position in an option and the underlying asset; A spread which involved taking a position

### 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

### 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

### Chapter 9 Trading Strategies Involving Options

Chapter 9 Trading Strategies Involving Options Introduction Options can be used to create a wide range of different payoff functions Why? /loss lines involving just underlying assets (forward) and money

### 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

### 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

### 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

### 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

### 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

### CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT

CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT PROBLEM SETS 1. In formulating a hedge position, a stock s beta and a bond s duration are used similarly to determine the expected percentage gain or loss

### 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

### Solutions to practice questions Study Session 11

BPP Professional Education Solutions to practice questions Study Session 11 Solution 11.1 A mandatory convertible bond has a payoff structure that resembles a written collar, so its price can be determined

### 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

### Options. The Option Contract: Puts. The Option Contract: Calls. Options Markets: Introduction. A put option gives its holder the right to sell

Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims (A claim that can be made when certain specified

### 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

### 15 Solution 1.4: The dividend growth model says that! DIV1 = \$6.00! k = 12.0%! g = 4.0% The expected stock price = P0 = \$6 / (12% 4%) = \$75.

1 The present value of the exercise price does not change in one hour if the risk-free rate does not change, so the change in call put is the change in the stock price. The change in call put is \$4, so

### 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

### 1 Introduction to Derivative Instruments

1 Introduction to Derivative Instruments The past few decades have witnessed a revolution in the trading of derivative securities in world financial markets. A financial derivative may be defined as a

### Lecture 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)

### 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

### 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

### Option strategies. Stock Price Payoff Profit. The butterfly spread leads to a loss when the final stock price is greater than \$64 or less than \$56.

Option strategies Problem 11.20. Three put options on a stock have the same expiration date and strike prices of \$55, \$60, and \$65. The market prices are \$3, \$5, and \$8, respectively. Explain how a butterfly

### 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

### 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:

### Forward contracts and futures

Forward contracts and futures A forward is an agreement between two parties to buy or sell an asset at a pre-determined future time for a certain price. Goal To hedge against the price fluctuation of commodity.

### 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

### 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

### End-of-chapter Questions for Practice (with Answers)

MFIN6003 Derivative Securities Dr. Huiyan Qiu End-of-chapter Questions for Practice (with Answers) Following is a list of selected end-of-chapter questions for practice from McDonald s Derivatives Markets.

### Lecture 17. Options trading strategies

Lecture 17 Options trading strategies Agenda: I. Basics II. III. IV. Single option and a stock Two options Bull spreads Bear spreads Three options Butterfly spreads V. Calendar Spreads VI. Combinations:

### 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,

### 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

### Week 1: Futures, Forwards and Options derivative three Hedge: Speculation: Futures Contract: buy or sell

Week 1: Futures, Forwards and Options - A derivative is a financial instrument which has a value which is determined by the price of something else (or an underlying instrument) E.g. energy like coal/electricity

### CFA Examination DERIVATIVES - FUTURES Page 1 of 5

DERIVATIVES FUTURES A derivative is a contract or agreement whose value depends upon the price of some other (underlying) commodity, security or index. A. FORWARD CONTRACTS Forward: an agreement between

### Lecture 14 Option pricing in the one-period binomial model.

Lecture: 14 Course: M339D/M389D - Intro to Financial Math Page: 1 of 9 University of Texas at Austin Lecture 14 Option pricing in the one-period binomial model. 14.1. Introduction. Recall the one-period

### 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

### 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

### Options and Derivative Pricing. U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University.

Options and Derivative Pricing U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University. e-mail: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,

### Reading 28 Risk Management Applications of Option Strategies

Reading 28 Risk Management Applications of Option Strategies A Compare the use of covered calls and protective puts to manage risk exposure to individual securities. Covered calls 1. Covered Call = Long

### 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

### Lecture Quantitative Finance Spring Term 2015

3.1: s, Swap Lecture Quantitative Finance Spring Term 2015 Guest Lecture: Dr. Robert Huitema : March 12, 2015 1 / 51 3.1: s, Swap 1 2 3.1: s, Swap 3 2 / 51 3.1: s, Swap 1 2 3.1: s, Swap 3 3 / 51 3.1: s,

### Expected payoff = 1 2 0 + 1 20 = 10.

Chapter 2 Options 1 European Call Options To consolidate our concept on European call options, let us consider how one can calculate the price of an option under very simple assumptions. Recall that the

### The Delta Method and Applications

Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

### 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