1. 1 Chapter 15: Option Pricing Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To show how the law of one price may be used to derive prices of options To show how to infer implied volatility from option prices

2. 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 Option Pricing 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. 3 Objectives To show how the Law of One Price can be used to derive prices of optionsTo show how the Law of One Price can be used to derive prices of options To show how to infer implied volatility from option pricesTo show how to infer implied volatility from option prices

4. 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. 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. 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. 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. 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. 9 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

10. 10 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

11. 11 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

12. 12 –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

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15. 15 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

16. 16 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

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18. 18 15.3 The Put-Call Parity Relationship 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

19. 19 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

20. 20 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

21. 21 Put-Call Parity Equation

22. 22 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

23. 23 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

24. 24 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

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26. 26 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

27. 27 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

28. 28 Binary Model Assumptions We assume:We assume: –share price = strike price = $100 –time to maturity = 1 year –interest rate = 5% –stock prices either rise or fall by 20% in the year, and so are either $80 or $120 at yearend

29. 29 Binary Model: Call Strategy:Strategy: –replicate the call using a portfolio of the underlying stockthe underlying stock a zero coupon riskless bond with a face value of $100a zero coupon riskless bond with a face value of $100 –by the law of one price, the price of the actual call must equal the price of the synthetic call

30. 30 Binary Model: Call Implementation:Implementation: –the synthetic call, C, is created by holding x number of shares of the stock, S, and y number of risk free bonds with a market value Bholding x number of shares of the stock, S, and y number of risk free bonds with a market value B C = xS + yBC = xS + yB C = x *100 + y * 95.2381C = x *100 + y * 95.2381

31. 31 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: 20 = x * 120 + y * 100 0 = x * 80 + y * 100 By inspection, the solution is x= 0.5, y = - 0.4

32. 32 Binary Model: Call Solution:Solution: –We now substitute the value of the parameters x= 0.5, y = - 0.4 into the equation in slide 30 to obtain: C = 0.5 *100 - 0.4 * 95.2381 C = $11.905

33. 33 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

34. 34 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:

35. 35 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

36. 36 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}

37. 37 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)

38. 38 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

39. 39 The Black-Scholes Model: Equations

40. 40 The Black-Scholes Model: Equations (Forward Form)

41. 41 The Black-Scholes Model: Equations (Simplified)

42. 42 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

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46. 46 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

47. 47 15.8 Implied Volatility Implied volatility is defined as the value of the volatility that makes the observed market price of the option equal to the value computed using the Black-Scholes option formula.Implied volatility is defined as the value of the volatility that makes the observed market price of the option equal to the value computed using the Black-Scholes option formula. Solver or Goal Seek function in Excel can be used to compute the implied volatilitySolver or Goal Seek function in Excel can be used to compute the implied volatility

48. 48 15.9 Contingent Claims Analysis (CCA)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

49. 49 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

50. 50 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%

51. 51 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)

52. 52 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

53. 53 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:

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55. 55 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

56. 56 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

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58. 58 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

59. 59 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

60. 60 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

61. 61 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

62. 62 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