1. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 1 Chapter 15: Markets for Options and Contingent Claims Objective Options Pricing relationships Pricing models Financial decisions analyzed through options

2. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 2 Chapter 15 Contents 15.1 How Options Work 15.2 Investing with Options 15.3 The Put-Call Parity Relationship 15.4 Volatility & Option Prices 15.5 Two-State (Binomial) Option Pricing 15.6 Dynamic Replication & the Binomial Model 15.7 The Black-Scholes Model 15.8 Implied Volatility 15.9 Contingent Claims Analysis of Corporate Debt and Equity 15.10 Credit Guarantees 15.11 Other Applications of Option-Pricing Methodology

3. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 3 Objectives How to use options to modify one’s exposure to investment risk. To understand the pricing relationships that exist among calls, puts, stocks, and bonds. To explain the binomial and Black-Scholes option-pricing models and apply them to the valuation of corporate bonds and other contingent claims. To explore the range of financial decisions that can be fruitfully analyzed in terms of options.

4. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 4 Introduction This chapter explores how option prices are affected by the volatility of the underlying securityThis chapter explores how option prices are affected by the volatility of the underlying security Exchange traded options appeared in 1973, enabling us to determine the market’s estimate of future volatility, rather than relying on historical valuesExchange traded options appeared in 1973, enabling us to determine the market’s estimate of future volatility, rather than relying on historical values

5. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 5 Definition of an Option Recall that an American {European} call (put) option is the right, but not the obligation to buy (sell) an asset at a specified price any time before its expiration date {on its expiration date}Recall that an American {European} call (put) option is the right, but not the obligation to buy (sell) an asset at a specified price any time before its expiration date {on its expiration date}

6. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 6 Ubiquitous Options This chapter focuses on traded options, but it would be a mistake to believe that the tools we will be developing are restricted to traded optionsThis chapter focuses on traded options, but it would be a mistake to believe that the tools we will be developing are restricted to traded options Some examples of options are given on the next few slidesSome examples of options are given on the next few slides

7. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 7 Government Price Supports Governments sometimes provide assistance to farmers by offering to purchase agricultural products at a specified support priceGovernments sometimes provide assistance to farmers by offering to purchase agricultural products at a specified support price If the market price is lower than the support, then a farmer will exercise her right to ‘put’ her crop to the government at the higher priceIf the market price is lower than the support, then a farmer will exercise her right to ‘put’ her crop to the government at the higher price

8. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 8 Old Mortgage Traditional US mortgages give the householder the right to call the mortgage at a strike equal to the outstanding principleTraditional US mortgages give the householder the right to call the mortgage at a strike equal to the outstanding principle If interest rates have fallen below the note’s rate, then the home owner will consider refinancing the mortgageIf interest rates have fallen below the note’s rate, then the home owner will consider refinancing the mortgage

9. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 9 New Mortgage You pay some ‘points’ to lock-in an interest rate on a mortgageYou pay some ‘points’ to lock-in an interest rate on a mortgage –If rates fall, you may renegotiate the mortgage, and then pay more points to lock in the new lower rate –If rates rise, then you will go to the settlement table with a lower-than-market interest rate

10. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 10 Tenure and Seniority In a company that has a policy of last-in first-out, a worker with seniority may forego a higher salary in another company because of the loss of job securityIn a company that has a policy of last-in first-out, a worker with seniority may forego a higher salary in another company because of the loss of job security The worker has been given the right, but not the obligation, to have work under a set of adverse economic conditionsThe worker has been given the right, but not the obligation, to have work under a set of adverse economic conditions

11. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 11 Copper Pennies, Silver Coin Silver and copper coinage has been replaced by zinc and cooper alloys/ composites to reduce their minting costsSilver and copper coinage has been replaced by zinc and cooper alloys/ composites to reduce their minting costs The old coins are often legal currency, and so contain an option feature:The old coins are often legal currency, and so contain an option feature: –If the price of the underlying metal falls below its legal value, I have the right to return the coin into circulation

12. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 12 Insurance Insurance policies often give you the right, but not the obligation to do something, it is therefore option-likeInsurance policies often give you the right, but not the obligation to do something, it is therefore option-like –The renewable rider on a term life policy is an option –If somebody: is terminally ill, then the rider is very valuableis terminally ill, then the rider is very valuable remains in good health, then it is not valuableremains in good health, then it is not valuable

13. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 13 Supply Contracts A nuclear power plant supplier once got into serious trouble by guaranteeing to supply enriched uranium at a fixed priceA nuclear power plant supplier once got into serious trouble by guaranteeing to supply enriched uranium at a fixed price The market price of enriched uranium rose precipitouslyThe market price of enriched uranium rose precipitously

14. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 14 Technological Leases A computer leasing company had a clause in its lease stating that the customer had the right to cancelA computer leasing company had a clause in its lease stating that the customer had the right to cancel The computer manufacturer introduced next generation of computers, and the leasing company’s customer’s canceled their leases, resulting in a massive inventory of obsolete computersThe computer manufacturer introduced next generation of computers, and the leasing company’s customer’s canceled their leases, resulting in a massive inventory of obsolete computers

15. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 15 Limited Liability The owners of a limited liability corporation have the right, but not the obligation, to ‘put’ the company to the corporation’s creditors and bondholdersThe owners of a limited liability corporation have the right, but not the obligation, to ‘put’ the company to the corporation’s creditors and bondholders Limited liability is, in effect, a put optionLimited liability is, in effect, a put option

16. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 16 Trading on Commission You are a trader with a contract giving you a commission of 20% of each months trading profitsYou are a trader with a contract giving you a commission of 20% of each months trading profits If you make a loss, then you walk away, but if you make a profit, you stayIf you make a loss, then you walk away, but if you make a profit, you stay –You may be tempted to increase your volatility to boost the value of your option

17. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 17 15.1 How Options Work The Language of OptionsThe Language of Options – –Contingent Claim: Any asset whose future pay-off depends upon the outcome of an uncertain event – –Call: an option to buy – –Put: an option to sell – –Strike or Exercise Price: the fixed price specified in an option contract – –Expiration or Maturity Date: The date after which an option can’t be exercised – –American Option: an option that can be exercised at any time up to and including maturity date

18. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 18 – –European Option: an option that can only be exercised on the maturity date – –Tangible Value: The hypothetical value of an option if it were exercised immediately – –At-the-Money: an option with a strike price equal to the value of the underlying asset – –Out-of-the-Money: an option that’s not at-the-money, but has no tangible value – –In-the-Money: an option with a tangible value – –Time Value: the difference between an option’s market value and its tangible value – –Exchange-Traded Option: A standardized option that an exchange stands behind in the case of a default – –Over the Counter Option: An option on a security that is not an exchange-traded option

19. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 19

20. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 20

21. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 21 15.2 Investing with Options The payoff diagram (terminal conditions, boundary conditions) for a call and a put option, each with a strike (exercise price) of $100, is derived nextThe payoff diagram (terminal conditions, boundary conditions) for a call and a put option, each with a strike (exercise price) of $100, is derived next

22. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 22 Option Payoff Diagrams The value of an option at expiration follows immediately from its definitionThe value of an option at expiration follows immediately from its definition –In the case of a call option with strike of $100, if the stock price is $90 ($110), then exercising the option results purchasing the share for $100, which is $10 more expensive ($10 less expensive) than buying it, so you wouldn't (would) exercise your right

23. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 23 Call Option Payoff Diagram

24. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 24 Put Option Payoff Diagram

25. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 25 Payoff Diagrams for Alternative Bullish Stock Strategies

26. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 26

27. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 27 15.3 The Put-Call Parity Relation Consider the following two strategiesConsider the following two strategies –Purchase a put with a strike price of $100, and the underlying share –Purchase a call with a strike price of $100 and a bond that matures at the same date with a face of $100 The maturity values are tabulated and plotted against share price:The maturity values are tabulated and plotted against share price:

28. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 28 Payoff Diagram for Pure Discount Bond Plus Call

29. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 29

30. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 30

31. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 31 Observation The most important point to observe is that the value of the “call + bond” strategy, is identical (at maturity) with the protective-put strategy “put + share”The most important point to observe is that the value of the “call + bond” strategy, is identical (at maturity) with the protective-put strategy “put + share” So, if the put and the call have the same strike price, we obtain the put-call parity relationship: put + share = call + bondSo, if the put and the call have the same strike price, we obtain the put-call parity relationship: put + share = call + bond

32. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 32 Technical Note –The above relationship is true, in general, for dividend-less European options, but the actual proof requires taking expectations of the option and security boundary conditions –The argument is therefore a heuristic for remembering and ‘seeing’ the relationship –The full proof is left for your investment class

33. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 33 Put-Call Parity for American and European Options A European option that pays no dividend during its life fully satisfies the requirements of put-call parityA European option that pays no dividend during its life fully satisfies the requirements of put-call parity In the case of American options, the relationship is fully accurate only at maturity, because American puts are sometimes exercised earlyIn the case of American options, the relationship is fully accurate only at maturity, because American puts are sometimes exercised early

34. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 34 Put-Call Parity Equation

35. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 35 Synthetic Securities The put-call parity relationship may be solved for any of the four security variables to create synthetic securities:The put-call parity relationship may be solved for any of the four security variables to create synthetic securities: C=S+P-B C=S+P-B S=C-P+B S=C-P+B P=C-S+B P=C-S+B B=S+P-C B=S+P-C

36. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 36 Synthetic Securities C=S+P-B and P=C-S+B may be used by floor traders to flip between a call and a put C=S+P-B and P=C-S+B may be used by floor traders to flip between a call and a put S=C-P+B may be used by short-term traders wishing to take advantage of lower transaction costs S=C-P+B may be used by short-term traders wishing to take advantage of lower transaction costs B=S+P-C may be used to create a synthetic bond said to pay a slightly higher return than the physical bond B=S+P-C may be used to create a synthetic bond said to pay a slightly higher return than the physical bond

37. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 37 Options and Forwards We saw in the last chapter that the discounted value of the forward was equal to the current spotWe saw in the last chapter that the discounted value of the forward was equal to the current spot The relationship becomesThe relationship becomes

38. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 38 Implications for European Options If the forward price of the underlying stock is equal to the strike price, then the value of the call is equal to the value of the putIf the forward price of the underlying stock is equal to the strike price, then the value of the call is equal to the value of the put This relationship is so important, that some option traders define ‘at-the- money’ in terms of the forward rather than the spotThis relationship is so important, that some option traders define ‘at-the- money’ in terms of the forward rather than the spot

39. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 39 Implications for European Options If (F > E) then (C > P)If (F > E) then (C > P) If (F = E) then (C = P)If (F = E) then (C = P) If (F < E) then (C < P)If (F < E) then (C < P) E is the common strike priceE is the common strike price F is the forward price of underlying shareF is the forward price of underlying share C is the call priceC is the call price P is the put priceP is the put price

40. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 40 Strike = Forward Call = Put

41. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 41

42. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 42 PV Strike Strik e

43. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 43 15.4 Volatility and Option Prices We next explore what happens to the value of an option when the volatility of the underlying stock increasesWe next explore what happens to the value of an option when the volatility of the underlying stock increases –We assume a world in which the stock price moves during the year from $100 to one of two new values at the end of the year when the option matures –Assume risk neutrality

44. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 44

45. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 45 Illustration Explained The stock volatility in the second scenario is higher, and the expected payoffs for both the put and the call are also higherThe stock volatility in the second scenario is higher, and the expected payoffs for both the put and the call are also higher –This is the result of truncation, and holds in all empirically reasonable cases Conclusion: Volatility increases all option pricesConclusion: Volatility increases all option prices

46. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 46 15.5 Two-State (Binomial) Option-Pricing –We are now going to derive a relatively simple model for evaluating options The assumptions will at first appear totally unrealistic, but using some underhand mathematics, the model may be made to price options to any desired level of accuracyThe assumptions will at first appear totally unrealistic, but using some underhand mathematics, the model may be made to price options to any desired level of accuracy The advantage of the method is that it does not require learning stochastic calculus, and yet it illustrates all the key steps necessary to derive any option evaluation modelThe advantage of the method is that it does not require learning stochastic calculus, and yet it illustrates all the key steps necessary to derive any option evaluation model

47. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 47 Binary Model Assumptions Assume:Assume: –the exercise price is equal to the forward price of the underlying stock option prices then depend only on the volatility and time to maturity, and do not depend on interest ratesoption prices then depend only on the volatility and time to maturity, and do not depend on interest rates the put and call have the same pricethe put and call have the same price

48. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 48 Binary Model Assumptions More specifically we assume:More specifically we assume: –share price = strike price = $100 –time to maturity = 1 year –dividend rate = interest rate = 0 –stock prices either rise or fall by 20% in the year, and so are either $80 or $120 at yearend

49. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 49 Binary Model: Call Strategy:Strategy: –replicate the call using a portfolio of the underlying stockthe underlying stock the riskless bondthe riskless bond –by the law of one price, the price of the actual call must equal the price of the synthetic call

50. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 50 Binary Model: Call Implementation:Implementation: –the synthetic call, C, is created by buying a fraction x of shares, of the stock, S, and simultaneously selling short risk free bonds with a market value ybuying a fraction x of shares, of the stock, S, and simultaneously selling short risk free bonds with a market value y the fraction x is called the hedge ratiothe fraction x is called the hedge ratio

51. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 51 Binary Model: Call Specification:Specification: –We have an equation, and given the value of the terminal share price, we know the terminal option value for two cases: –By inspection, the solution is x=1/2, y = 40

52. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 52 Binary Model: Call Solution:Solution: –We now substitute the value of the parameters x=1/2, y = 40 into the equation –to obtain:

53. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 53 Binary Model: Put Strategy:Strategy: –replicate the put using a portfolio of the underlying stock and riskless bond –by the law of one price, the price of the actual put must equal the price of the synthetic put replicated above Minor changes to the call argument are made in the next few slides for the putMinor changes to the call argument are made in the next few slides for the put

54. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 54 Binary Model: Put Implementation:Implementation: –the synthetic put, P, is created by sell short a fraction x of shares, of the stock, S, and simultaneously buy risk free bonds with a market value ysell short a fraction x of shares, of the stock, S, and simultaneously buy risk free bonds with a market value y the fraction x is called the hedge ratiothe fraction x is called the hedge ratio

55. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 55 Decision Tree for Dynamic Replication of Call Option

56. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 56 Binary Model: Put Specification:Specification: –We have an equation, and given the value of the terminal share price, we know the terminal option value for two cases: –By inspection, the solution is x=1/2, y = 60

57. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 57 Binary Model: Put Solution:Solution: –We now substitute the value of the parameters x=1/2, y = 60 into the equation –to obtain:

58. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 58 15.6 Dynamic Replication and the Binomial Model –We now take the next step towards greater realism by dividing the year into 2 sub- periods of half a year each. This gives 3 possible outcomes –Our first task is to find a self-financing investment strategy that does not require injection or withdrawal of new funds during the life of the option –We first create a decision tree:

59. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 59 Decision Tree for Dynamic Replication of a Call Option ($120*100%) + (-$100) = $20

60. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 60 Reading the Decision Tree The tree is constructed backwards because we know only the future contingent call pricesThe tree is constructed backwards because we know only the future contingent call prices –For Example, when constructing the weights for time 6-months, the option prices for 12- months are used For consistency with the next model, the discrete stock prices are usually fixed ratios, i.e. 121, 110, 100, 90.91, 82.64For consistency with the next model, the discrete stock prices are usually fixed ratios, i.e. 121, 110, 100, 90.91, 82.64

61. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 61 The Power of Lattice Models –Lattice models, of which the binary model is the simplest, are very important to traders because they may be modified to handle different distributions, the possibility of early exercise, and discrete dividend payments –To see how easy it is to change the distributional assumption, the above illustration results in stock prices being normally distributed, and the modification results in a lognormal distribution of prices

62. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 62 15.7 The Black-Scholes Model The most widely used model for pricing options is the Black-Scholes modelThe most widely used model for pricing options is the Black-Scholes model –This model is completely consistent with the binary model as the interval between stock prices decreases to zero –The model provides theoretical insights into option behavior –The assumptions are elegant, simple, and quite realistic

63. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 63 The Black-Scholes Model We will work with the generalized form of the model because the small additional complexity results in considerable additional power and flexibilityWe will work with the generalized form of the model because the small additional complexity results in considerable additional power and flexibility First, notation:First, notation:

64. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 64 The Black-Scholes Model: Notation C = price of callC = price of call P = price of putP = price of put S = price of stockS = price of stock E = exercise priceE = exercise price T = time to maturityT = time to maturity ln(.) = natural logarithmln(.) = natural logarithm e = 2.71828...e = 2.71828... N(.) = cum. norm. dist’n The following are annual, compounded continuously: r = domestic risk free rate of interest d = foreign risk free rate or constant dividend yield σ = volatility

65. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 65 The Normal Problem It is not unusual for a student to have a problem computing the cumulative normal distribution using tablesIt is not unusual for a student to have a problem computing the cumulative normal distribution using tables –table structures vary, so be careful –using standard-issue normal tables degrades computed option values because of errors caused by catastrophic subtraction –{Many professionals use Hasting’s formula as reported in Abramowitz and Stegun as equation 26.2.19 (never, never use 26.2.18). Its certificate valid in 0<=x<Inf, so use symmetry to get -Inf<x<0}

66. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 66 The Normal Problem The functions that come with Excel have adequate accuracy, so consider using ‘Normsdist()’ in the statistical functions (note the s in Normsdist)The functions that come with Excel have adequate accuracy, so consider using ‘Normsdist()’ in the statistical functions (note the s in Normsdist)

67. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 67 The Black-Scholes Model: What’s missing There are no expectations about future returns in the modelThere are no expectations about future returns in the model The model is preference-freeThe model is preference-free σ-risk, not  -risk, is the relevant riskσ-risk, not  -risk, is the relevant risk

68. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 68 The Black-Scholes Model: Equations

69. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 69 The Black-Scholes Model: Equations (Forward Form)

70. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 70 The Black-Scholes Model: Equations (Simplified)

71. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 71 So What Does it Mean? You can now obtain the value of non- dividend paying European optionsYou can now obtain the value of non- dividend paying European options With a little skill, you can widen this to obtain approximate values of European options on shares paying a dividend, and to some American optionsWith a little skill, you can widen this to obtain approximate values of European options on shares paying a dividend, and to some American options

72. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 72

73. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 73

74. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 74 Implied Volatility SPX (July 2005-June 2006)

75. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 75

76. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 76 Observable Variables All the variables are directly observable, excepting the volatility, σ, and possibly, the next cash dividend, dAll the variables are directly observable, excepting the volatility, σ, and possibly, the next cash dividend, d We do not have to delve into the psyche of investors to evaluate optionsWe do not have to delve into the psyche of investors to evaluate options We do not forecast future prices to obtain option valuesWe do not forecast future prices to obtain option values

77. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 77 15.8 Implied Volatility The following slides show how to estimate volatility using ExcelThe following slides show how to estimate volatility using Excel –The option most commonly used to estimate volatility is the one closest to the present value of the strike price: That is, the option that is has a strike closest to the forward price –This option has the most “oomph”

78. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 78 Insert any number to start Formula for option value minus the actual call value

79. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 79

80. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 80 Pat’s Plan Pat has a plan to get rich with no risk:Pat has a plan to get rich with no risk: –Set up special portfolio, (Pat calls this a “self financing, delta neutral portfolio with positive curvature,” but Pat has this thing with words)

81. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 81 Pat, the Strategist Short some shares, and off-set small price changes about the current price with some call options, then invest the difference in bondsShort some shares, and off-set small price changes about the current price with some call options, then invest the difference in bonds

82. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 82

83. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 83 Pat, the Cartographer Apparently, what Pat has done is to find the tangent (at today’s share price) of the call value curve, using bonds and stock in the right proportionsApparently, what Pat has done is to find the tangent (at today’s share price) of the call value curve, using bonds and stock in the right proportions This is what we did earlier when we constructed the binary pricing modelThis is what we did earlier when we constructed the binary pricing model At the current price of $103, the tangent mimics the call curveAt the current price of $103, the tangent mimics the call curve

84. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 84 Pat (Continued) Pat then went short the tangency portfolio, and long the call to create the thick black portfolioPat then went short the tangency portfolio, and long the call to create the thick black portfolio Observe that the minimum value of the portfolio is zero, and this occurs at the current price, so it is self-financingObserve that the minimum value of the portfolio is zero, and this occurs at the current price, so it is self-financing Pat makes money if share prices move up or downPat makes money if share prices move up or down

85. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 85 Pat Triumphant This clearly defeats the law of one price: There is no downside risk, no construction costs, and yet will yield a positive profit almost all the timeThis clearly defeats the law of one price: There is no downside risk, no construction costs, and yet will yield a positive profit almost all the time What’s wrong with Pat’s analysis?What’s wrong with Pat’s analysis?

86. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 86 Pat Dejected The answer is that it takes time for a price to move, and during that time, all other things being equal, the value of the option will decayThe answer is that it takes time for a price to move, and during that time, all other things being equal, the value of the option will decay Think of a downwards sloping, very slick, rain-gutter containing a critter:Think of a downwards sloping, very slick, rain-gutter containing a critter: –The critter may climb the walls of the gutter, but it is constantly sliding down the gutter

87. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 87 Pat in Despair The next diagram shows the value of the portfolio today and one week henceThe next diagram shows the value of the portfolio today and one week hence The construction lines have been removed, and the graph has been re- scaledThe construction lines have been removed, and the graph has been re- scaled

88. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 88

89. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 89 Pat Condemned to Poverty The diagram shows that, if next week’s share prices fall between about $97 and $105.5, Pat will enjoy a lossThe diagram shows that, if next week’s share prices fall between about $97 and $105.5, Pat will enjoy a loss As time passes, decay will make this strategy a very risky oneAs time passes, decay will make this strategy a very risky one –Another factor Pat did not take into account is that volatility is itself volatile, so the hedge may disintegrate

90. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 90 15.9 Contingent Claims Analysis of Corporate Debt and Equity The CCA approach uses a different set of informational assumptions than the discounted cash flow (DCF) method:The CCA approach uses a different set of informational assumptions than the discounted cash flow (DCF) method: –it uses the risk-free rate rather than a risk- adjusted discount rate –it uses knowledge of the prices of one or more related assets and their volatilities

91. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 91 Contingent-Claims Analysis of Stock and Bonds: Debtco Debtco is a real-estate holding company and has issuedDebtco is a real-estate holding company and has issued –1,000,000 common shares –80,000 pure discount bonds, face $,1000, maturity 1-year

92. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 92 Debtco, Continued –The total market value of Debtco is $100,000,000 –The risk-free rate, (and therefore, by the law of one price, Debtco’s bond rate,) is 4%

93. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 93 Debtco, Notation E the market value of the stock issueE the market value of the stock issue D the market value of the debt issueD the market value of the debt issue V the total current market value; V = E + DV the total current market value; V = E + D V 1 the total market value one year henceV 1 the total market value one year hence (The law of one price ensures that V = E + D must be true, otherwise there will be an arbitrage opportunity)(The law of one price ensures that V = E + D must be true, otherwise there will be an arbitrage opportunity)

94. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 94 Debtco, Security Valuation Value of the bondsValue of the bonds –By the rule of one price, the value of the bonds must equal their face value discounted at the risk-free rate for a year D = 80,000 * $1,000 / 1.04 = $76,923,077D = 80,000 * $1,000 / 1.04 = $76,923,077 –By the total value of the firm, V = E + D, the value of the stock is E = V - D = $100,000,000 - $76,923,077 = $23,076,923E = V - D = $100,000,000 - $76,923,077 = $23,076,923

95. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 95 Debtco, Payoff –A consequence of Debtco’s having bonds with a risk-free rate is that the company has either purchased bond default insurance from a third party, or that the firm’s assets have no (downside) risk –For many companies, a more realistic assumption is that the assets do have risk, and to evaluate such securities requires a payoff function for the bonds or stock:

96. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 96

97. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 97 Negative Firm Values –We have assumed that the value of the firm never falls below zero, but while unusual, it is possible for the market value of a firm’s assets to be less than zero –Consider Enviromess Inc., a firm that for years polluted the Hudson River with a byproduct of Lifecide ® The cost of cleaning up the river may well greatly exceed the firm’s financial resourcesThe cost of cleaning up the river may well greatly exceed the firm’s financial resources

98. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 98 Negative Firm Values –Negative values of limited liability companies is irrelevant to the construction of the payoff diagrams –It does influence the value of the firm’s debt and equity through the (truncated) distribution of the firms future values

99. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 99 Debtco, Probabilities –In addition to the payoff diagrams, we need information about the probabilities of the future values of the firm

100. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 100 Probability Density of a Firm's Value 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 020406080100120140160180200 Value of a Firm Probability Density

101. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 101 Debtco, Probabilities –Considerable effort is employed when estimating the probabilities used in CCA, but, to obtain a basic understanding of CCA it is enough to assume a very simple distribution –We will assume that the firm may take on only one of two possible values a year from now, when the bond matures, namely $70 or $140 million –(The two-state assumption can be generalized into a n- state lattice model with any specified degree of accuracy)

102. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 102 Debtco Security Payoff Table ($’000,000)

103. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 103 Debtco’s Replicating Portfolio LetLet –x be the fraction of the firm in replicator –Y be the borrowings at the risk-free rate in the replicator –In $’000,000 the following equations must be satisfied

104. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 104 Debtco’s Replicating Portfolio ($’000)

105. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 105 Debtco’s Replicating Portfolio Note that the above slide shows that (with the weights on the last-but-one slide) the value of the replicating portfolio and the stock are identical now and one-year henceNote that the above slide shows that (with the weights on the last-but-one slide) the value of the replicating portfolio and the stock are identical now and one-year hence By the law of one price, the value of Debtco’s Stock is $28.02 each (One million issued)By the law of one price, the value of Debtco’s Stock is $28.02 each (One million issued)

106. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 106 Debtco’s Replicating Portfolio We know value of the firm is $1,000,000, and the value of the total equity is $28,021,978, so the market value of the debt with a face of 80,000,000 is $71,978,022We know value of the firm is $1,000,000, and the value of the total equity is $28,021,978, so the market value of the debt with a face of 80,000,000 is $71,978,022 The yield on this debt is (80…/71…) - 1 = 11.14%The yield on this debt is (80…/71…) - 1 = 11.14%

107. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 107 Another View of Debtco’s Replicating Portfolio (‘$000)

108. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 108 Interpretation –The market value of the firm’s risky debt consists of about $58 million riskless debt and about $14 million of the risky firm There is a sense that holders of risky debt accept some of the firm-as-a-whole’s risky cash flow, just as shareholders doThere is a sense that holders of risky debt accept some of the firm-as-a-whole’s risky cash flow, just as shareholders do –Shareholders accept the remaining $85 million of the firm-as-a-whole, and finance it with the equivalent of about $58 million in default-free debt

109. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 109 Valuing the Bonds given the Price of the Stock There are three values that are in equilibrium with each other:There are three values that are in equilibrium with each other: –the value of the firm (  done) –the value of the bond (to be done next) –the value of the stock If we know one, the others may be deducedIf we know one, the others may be deduced

110. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 110 Valuing Bonds Given the Stock Price ($’000,000’s) Assume that, as before we have:Assume that, as before we have: –Scenario a: value of firm in 1-year = $70 –Scenario b: value of firm in 1-year = $140 –Risk-free 1-year bonds produce a 4% yield –Total face value of Debtco’s bonds is $80 Assume also:Assume also: –Debtco has a million shares outstanding, total market value $20

111. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 111 Valuing Bonds –We can replicate the firm’s equity using x = 6/7 of the firm, and about Y = $58 million riskless borrowing (earlier analysis) –The implied value of the bonds is then $90,641,026 - $20,000,000 = $70,641,026 & the yield is (80.00-70.64)/70.64 = 13.25%

112. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 112 Valuing Stock Given the Bond Yield Assume that, as before we have:Assume that, as before we have: –Scenario a: value of firm in 1-year = $70 –Scenario b: value of firm in 1-year = $140 –Risk-free 1-year bonds produce a 4% yield Assume also:Assume also: –Debtco’s bonds yield 10% (current value $909.09)

113. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 113 Bond Replication In order to replicate the bond, we will purchase a fraction, x, of the firm, and purchase the value Y of risk-free bondsIn order to replicate the bond, we will purchase a fraction, x, of the firm, and purchase the value Y of risk-free bonds At maturity, the value of the bonds isAt maturity, the value of the bonds is –Scenario a, V = $70 million: $70 million –Scenario b, V = $140million: $80 million

114. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 114 Replication Portfolio

115. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 115 Determining the Weight of Firm Invested in Bond, x, and the Value of the R.F.-Bond, Y

116. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 116 Valuing Stock –We can replicate the bond by purchasing 1/7 of the company, and $57,692,308 of default- free 1-year bonds –The market value of the bonds is $909.0909 * 80,000 = $72,727,273 –The value of the stock is therefore E=V -D = $105,244,753 - $72,727,273= $32,517,480

117. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 117 Convertible Bonds A convertible bond obligates the issuing firm to redeem the bond at par value upon maturity, or to allow the bond holder to convert the bond into a pre- specified number of share of common stockA convertible bond obligates the issuing firm to redeem the bond at par value upon maturity, or to allow the bond holder to convert the bond into a pre- specified number of share of common stock

118. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 118 Convertible Bonds: The Convertidebt Corporation Assume that Convertidebt is in every way like Debtco, but each bond is convertible to 20 common stock at maturityAssume that Convertidebt is in every way like Debtco, but each bond is convertible to 20 common stock at maturity –If all the debt is converted, then the number of common stock will rise from 1,000,000 to 1,000,000 + 80,000 * 20 = 2,600,000 shares

119. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 119

120. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 120 Bondholder Entitlements Given that a conversion occurs, the value of each common stock will beGiven that a conversion occurs, the value of each common stock will be –Value of firm / 2,600,000 The bond holders will receive 1,600,000 of these shares, so the bondholders will own 1.6/2.6 of the firm, leaving the shareholders with 1/2.6 of the firmThe bond holders will receive 1,600,000 of these shares, so the bondholders will own 1.6/2.6 of the firm, leaving the shareholders with 1/2.6 of the firm The critical value for conversion is firm’s value = 80 million*2.6/1.6 = $130 millionThe critical value for conversion is firm’s value = 80 million*2.6/1.6 = $130 million

121. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 121 Payoffs from Convertible Bonds Scenario a: The value of the firm is 70 million: The bondholders will own the company, so the value is 70 millionScenario a: The value of the firm is 70 million: The bondholders will own the company, so the value is 70 million Scenario b: The value of the firm is 140 million: The bondholders will own 1.6/2.6 of the company, or $86,153,846Scenario b: The value of the firm is 140 million: The bondholders will own 1.6/2.6 of the company, or $86,153,846 –The remaining analysis follows exactly the same recipe as the conventional bond

122. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 122 Dynamic Replication If you refer back to the convertible bond example, you will observe that only two points on the doubly-kinked payoff curve were sampled (clearly unrealistic)If you refer back to the convertible bond example, you will observe that only two points on the doubly-kinked payoff curve were sampled (clearly unrealistic) As in the case of binary option evaluation, by increasing the number of sample points, you may achieve any desired level of accuracyAs in the case of binary option evaluation, by increasing the number of sample points, you may achieve any desired level of accuracy

123. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 123 Outline of Method Bifurcate the year into two six-month intervalsBifurcate the year into two six-month intervals Starting at the present time at Node-A (value of the firm is $100 million) there are two scenariosStarting at the present time at Node-A (value of the firm is $100 million) there are two scenarios –the price rises to $115 MM (Node-B) –the price falls to $90 MM (Node-C)

124. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 124 Setup Assumptions (See Next Diagram) –Given Node-B occurred at month 6 ($115 MM), then after a further 6-months, there are two more scenarios the price rises to $140 MM (Node-D)the price rises to $140 MM (Node-D) the price falls all the way to $90 MM (Node-E)the price falls all the way to $90 MM (Node-E) –Given Node-C occurred at month 6 ($90 MM), then after a further 6-months, there are two more scenarios The price rises to $110 MM (Node-F)The price rises to $110 MM (Node-F) The price falls further to $70 MM (Node-G)The price falls further to $70 MM (Node-G)

125. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 125 Outline Decision Tree Node-B $115MM Node-C $90MM Node-D $140MM Node-F $110MM Node-E $90MM Node-G $70MM Node-A $100MM Month 0Month 6Month 12

126. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 126 Outline of Method There are three decision nodes A, B, CThere are three decision nodes A, B, C –At each node, a replicating decision is made First B (from D & E), and independently C (from F & G)First B (from D & E), and independently C (from F & G) Then, using the backwards induction, A (from B & C)Then, using the backwards induction, A (from B & C) –The steps are exactly as outlined in detail for a non-composite decisions The example is featured in the textbookThe example is featured in the textbook

127. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 127 Outline of Method The only detail that requires your attention is that the portfolio is completely self-financing at each nodeThe only detail that requires your attention is that the portfolio is completely self-financing at each node

128. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 128 Summary –The fundamental principle of CCA is that one may replicate the securities issued by a firm through the purchase and sale of the firm (as a whole) and the risk-free asset, and that this dynamic replication strategy is self- financing –The CCA is a direct consequence of the law of one price –Note that informational inputs are relatively few

129. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 129 Probabilities –The selection of the prices at which to replicate the portfolios is equivalent to selecting a probability distribution It is usual to construct lattices that re- associate at given values, forming a fishnetIt is usual to construct lattices that re- associate at given values, forming a fishnet The basis of the spacing, and the ratio of temporal spacing to spatial spacing, determine the distributionThe basis of the spacing, and the ratio of temporal spacing to spatial spacing, determine the distribution –As mentioned in the chapter, this step requires considerable skill

130. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 130 Valuing Pure State-Contingent Securities Recall: In Debtco and Convertidebt, there were only two possible values that the firm could take at maturityRecall: In Debtco and Convertidebt, there were only two possible values that the firm could take at maturity Define a pure state-contingent security to be a security that pays $1 in one of these states and $0 in the otherDefine a pure state-contingent security to be a security that pays $1 in one of these states and $0 in the other

131. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 131 Valuing Pure State-Contingent Securities

132. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 132 State-Contingent Security #1

133. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 133 State Contingent Securities to Evaluate Real Securities Price of Debtco stock is equivalent to 60 type 1 state contingent securities and no type 2, so stock price = 60 * 0.467033 = $28.02Price of Debtco stock is equivalent to 60 type 1 state contingent securities and no type 2, so stock price = 60 * 0.467033 = $28.02 The price of a Debtco bond is equivalent to 1000 type 1 SCS and 875 type 2, so bond price = 1000 * $0.467033 + 875 * $0.494505 = $899.73The price of a Debtco bond is equivalent to 1000 type 1 SCS and 875 type 2, so bond price = 1000 * $0.467033 + 875 * $0.494505 = $899.73

134. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 134 Advantages of Pure State Contingent Securities SCS enable us to evaluate any security that is dependent upon the value of the firm and the risk-free bondSCS enable us to evaluate any security that is dependent upon the value of the firm and the risk-free bond –For example, Convertidebt securities are priced quick using SCS prices (already computed) and the payoff schedule –Think of a SCS as the conditional probability of an event, weighted by the riskless time value of money

135. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 135 15.10 Pricing a Bond Guarantee Guarantees against credit risk commonGuarantees against credit risk common – –Parent corporations guarantee the debt of subsidiaries – –Commercial banks and insurance companies offer guarantees for a fee on a spectrum of financial instruments including swaps & letters of credit – –U.S. Government guarantees bank deposits, SBA loans, pensions, farm & student loans, mortgages, the debt of other sovereign countries, and huge strategic corporations – –They occur implicitly every time a risky loan is made

136. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 136 Example Earlier, we computed the value of the Debtco bond and found it to be $899.73, but the value of a risk-free bond was $961.54Earlier, we computed the value of the Debtco bond and found it to be $899.73, but the value of a risk-free bond was $961.54 A third-party might be willing to insure Debtco’s bonds against default, and the equilibrium price for this is $961.54 - $899.73 = $61.81A third-party might be willing to insure Debtco’s bonds against default, and the equilibrium price for this is $961.54 - $899.73 = $61.81

137. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 137 (We assume that the insuring company itself has no credit risk)(We assume that the insuring company itself has no credit risk) –We may compute the cost of this kind of insurance using our computed SCS values and a Bond Guarantee payoff table (next slide) –The insurance pays off only when the firm’s price is $70,000,000

138. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 138 Payoff for Debtco’s Bond Guarantee

139. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 139 SCS Conformation of Guarantee’s Price Guarantee’s price is 125 * $0.494505 = $61.81Guarantee’s price is 125 * $0.494505 = $61.81

140. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 140 15.11 Other Applications of Option-Pricing Methodology –This slide presentation started with a range of options that are embedded in products and contracts –Options not associated with financial instruments are called real options –The future is uncertain, so having flexibility to decide what to do after some of the uncertainty has been removed has value

141. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 141 Options in Project-Investment Valuations: –Option to initiate –Option to expand –Option to abandon –Option to reduce scale –Option to adjust timing –Option to exploit a future technology

142. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 142 Examples: –Choice of oil or gas to generate electricity –Product development of pharmaceuticals –Making a sequel to a movie –Vocational education –Litigation decisions –strategic decisions