1. 19 - 1 Options Option-Pricing Formula Options & Corporate Finance

2. 19 - 2 An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time. What is an option?

3. 19 - 3 It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset. What is the single most important characteristic of an option?

4. 19 - 4 Call option: Option to buy a specified number of shares of a security within some future period. Put option: Option to sell a specified number of shares of a security within some future period. In terms of an option contract such as GE July 18 call: Underlying asset - GE stock Option Terminology

5. 19 - 5 Definition and Terminology Strike (or exercise) price : the amount paid by the option buyer for the asset if he/she decides to exercise Exercise: the act of paying the strike price to buy the asset Expiration: the date by which the option must be exercised or become worthless Exercise style: specifies when the option can be exercised –European-style: can be exercised only at expiration date –American-style: can be exercised at any time before expiration –Bermudan-style: Can be exercised during specified periods

6. 19 - 6 Option price: The market price of the option contract. (isn’t stated) Expiration date: The date the option matures. (third Friday in December) Intrinsic value: The value of a call option if it were exercised today = Current stock price - Strike price. Exercise (or strike) price: The price stated in the option contract at which the security can be bought or sold. ($18)

7. 19 - 7 Covered option: A call option written against stock held in an investor’s portfolio. Naked (uncovered) option: An option sold without the stock to back it up. In-the-money call: A call whose exercise price is less than the current price of the underlying stock.

8. 19 - 8 Out-of-the-money call: A call option whose exercise price exceeds the current stock price. LEAPs: Long-term Equity AnticiPation securities that are similar to conventional options except that they are long-term options with maturities of up to 2 1/2 years.

9. 19 - 9 Stock options are written in 100- share increments, so the minimum purchase would be 100 shares. Stock options are written in 100- share increments, so the minimum purchase would be 100 shares. What is the minimum number of shares in a stock option contract?

10. 19 - 10 A visual tool used to show the many possible payoff scenarios with a particular option contract. A visual tool used to show the many possible payoff scenarios with a particular option contract. What is a payoff diagram?

11. 19 - 11 Option Basics Value of call at expiration (C) Value of common stock (S 1 ) 0 E If S 1 > E, then call option value = S 1 – E If S 1 < E, then call option value = 0

12. 19 - 12 Because S exceeds the call option’s exercise price (S>E), the call option is in the money. Because S exceeds the call option’s exercise price (S>E), the call option is in the money. P 0 = $27, 6-month call option sells for $4/share, 6-month put option sells for $2/share, and both options’ striking price = $25. Is the call option in or out of the money?

13. 19 - 13 Because S exceeds the put option’s exercise price, the put option is out of the money. Because S exceeds the put option’s exercise price, the put option is out of the money. Is the put option in or out of the money

14. 19 - 14 A payoff diagram for the call option: 25 27 2931 33 4 0 -4 Stock Price at Expiration ($) Profit per Option ($) Call Option Payoff Diagram Holder Seller

15. 19 - 15 A payoff diagram for the put option: 23 25 27 29 31 33 4 0 -4 Stock Price at Expiration ($) Profit per Option ($) Put Option Payoff Diagram Holder Seller

16. 19 - 16 Option Value Determinants CallPut 1.Stock price+– 2.Exercise price–+ 3.Interest rate+– 4.Variability in stock price++ 5.Expiration date++ Value of call Value of Upper bound (S 0 )Lower bound (S 0 – E) a call can never be Value of this high a call C 0 must be Value of here a call can never be this low Value of common E stocks (S 0 ) The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. The precise position will depend on the above five factors.

17. 19 - 17 Stock PriceCall Option Price $25$ 3.00 30 7.50 35 12.00 40 16.50 45 21.00 50 25.50 Exercise price = $25. Consider the following data:

18. 19 - 18 Create a table which shows (a) stock price, (b) strike price, (c) intrinsic value, (d) option price, and (e) time value of option price Price of StrikeIntrinsic Value Stock (a) Price (b) of Option (a) - (b) $25.00$25.00$0.00 30.00 25.00 5.00 35.00 25.00 10.00 40.00 25.0015.00 45.00 25.0020.00 50.00 25.0025.00

19. 19 - 19 Intrinsic Value Mkt. Price Time Value of Option (c) of Option (d) (d) - (c) $ 0.00 $ 3.00 $ 3.00 5.00 7.50 2.50 10.00 12.00 2.00 15.00 16.50 1.50 20.00 21.00 1.00 25.00 25.50 0.50 Table (Continued)

20. 19 - 20 What happens to the premium of the option price over the intrinsic value as the stock price rises? The premium of the option price over the intrinsic value declines as the stock price increases. This is due to the declining degree of leverage provided by options as the underlying stock price increases, and the greater loss potential of options at higher option prices.

21. 19 - 21 The stock underlying the call option provides no dividends during the call option’s life. There are no transactions costs for the sale/purchase of either the stock or the option. R F is known and constant during the option’s life. What are the assumptions of the Black-Scholes Option Pricing Model? (Cont...)

22. 19 - 22 Security buyers may borrow any fraction of the purchase price at the short-term risk-free rate. No penalty for short selling and sellers receive immediately full cash proceeds at today’s price. Call option can be exercised only on its expiration date. Security trading takes place in continuous time, and stock prices move randomly in continuous time.

23. 19 - 23 C = S[N(d 1 )] - Ee -R F t [N(d 2 )]. d 1 =.  t d 2 = d 1 -  t. What are the three equations that make up the OPM? ln(P/X) + [R F + (  2 /2)]t

24. 19 - 24 C = Current value of a call option with time t until expiration. S = Current price of the underlying stock. (More...) Explain the terms used in each equation.

25. 19 - 25 N(d 1 ) = Probability that a deviation less than d 1 will occur in a standard normal distribution. Thus, N(d 1 ) and N(d 2 ) represent areas under a standard normal distribution function. E = Exercise, or striking, price of the option. e = 2.7183

26. 19 - 26 R F = Risk-free interest rate. t = Time until the option expires (the option period). ln(S/E) = Natural logarithm of S/E.  2 = Variance of the rate of return on the stock.

27. 19 - 27 What is the value of the following call option according to the OPM? Assume: S = $27; E = $25; R F = 6%; t = 0.5 years:  2 = 0.11 C = $27[N(d 1 )] - $25e -(0.06)(0.5) [N(d 2 )]. ln($27/$25) + [(0.06 + 0.11/2)](0.5) (0.3317)(0.7071) = 0.5736. d 2 = d 1 - (0.3317)(0.7071) = d 1 - 0.2345 = 0.5736 - 0.2345 = 0.3391. d 1 =

28. 19 - 28 N(d 1 ) = N(0.5736) = 0.5000 + 0.2168 = 0.7168. N(d 2 ) = N(0.3391) = 0.5000 + 0.1327 = 0.6327. Note: Values obtained from Z-Table in text. C = $27(0.7168) - $25e -0.03 (0.6327) = $19.3536 - $25(0.97045)(0.6327) = $4.0036.

29. 19 - 29 A Call Option Pricing Problem Given the information below, calculate the 3-month call option price that is consistent with the Black-Scholes model. S 0 = $47E = $45r =.05  =.40 Solution

30. 19 - 30 Current stock price: Call option value increases as the current stock price increases. Exercise price: As the exercise price increases, a call option’s value decreases. What impact do the following para- meters have on a call option’s value?

31. 19 - 31 Option period: As the expiration date is lengthened, a call option’s value increases (more chance of becoming in the money.) Risk-free rate: Call option’s value tends to increase as R F increases (reduces the PV of the exercise price). Stock return variance: Option value increases with variance of the underlying stock (more chance of becoming in the money).

32. 19 - 32 Various option contract combinations that investors use as a means of reducing risk through diversification. Various option contract combinations that investors use as a means of reducing risk through diversification. Allow investors a wide variety of risk/return tradeoffs via options. Allow investors a wide variety of risk/return tradeoffs via options. Include long and short straddles, strips and straps. Include long and short straddles, strips and straps. What are combination positions?

33. 19 - 33 Debt represents sales of existing assets to creditors, who give the Shareholders a “call option” with an exercise price equal to the principal plus interest. Debt represents sales of existing assets to creditors, who give the Shareholders a “call option” with an exercise price equal to the principal plus interest. If the firm is profitable, shareholders will “exercise the call” and buy back the assets. If the firm is profitable, shareholders will “exercise the call” and buy back the assets. Explain why equity of a levered firm can be thought of as a call option.

34. 19 - 34 If unprofitable, shareholders will default on the loan. This is equivalent to not exercising the call and giving up the company’s assets to the creditors. If unprofitable, shareholders will default on the loan. This is equivalent to not exercising the call and giving up the company’s assets to the creditors.

35. 19 - 35 Risky Debt We can use the diagrams of option value at expiration to demonstrate that risky debt can be viewed as a combination of securities including either a call or put option.

36. 19 - 36 The Value of Risky Debt I.Risky Debt=Assets–Equity Call option =– EEE II.Risky Debt=Risk-free bonds–Put option =– EE E for example, Lockheed bonds = Risk-free bonds – Federal Loan Guarantee

37. 19 - 37 Example: in 1980 Chrysler Corp was granted $1.5 billion in government loan guarantees to keep them out of bankruptcy The value of the put option given to Chrysler (debt and equity) stakeholders by the government was equal to the value of the risk-free government-backed debt minus the value of comparable risky Chrysler bonds: Value of put = value of risk-free debt-value of risky debt

38. 19 - 38 Debt and Equity As Options Levered equity is a call option. The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond. If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call. They will pay the bondholders and “call in” the assets of the firm. If at the maturity of the debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire.

39. 19 - 39 Debt and Equity As Options Levered equity is a put option. The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond. If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in- the-money put. They will put the firm to the bondholders. If at the maturity of the debt the shareholders have an out-of-the-money put, they will not exercise the option (i.e. NOT declare bankruptcy) and let the put expire.

40. 19 - 40 Application to Equity Valuation: A simple example The parameters of equity as a call option are as follows: Value of the underlying asset = E = Value of the firm = $ 100 million Exercise price = B = Face Value of outstanding debt = $ 80 million Life of the option = t = Life of zero-coupon debt = 10 years Variance in the value of the underlying asset =  2 = Variance in firm value = 0.16 Riskless rate = R = Treasury bond rate corresponding to option life = 10%

41. 19 - 41 Solution d 2 =1.5994-0.40(10) 0.5 =0.3345 N(d1) =0.9451 N (d 2 ) =0.6310 B=100-75.94=24.06 Cost of debt=(1/T)Ln(B/B)=(1/10)(Ln(80/24.06)=12.0%

42. 19 - 42 Value of a troubled firm Assuming that we have the followings: V = Value of the firm = $ 50 million Exercise price = B = Face Value of outstanding debt = $ 80 million Life of the option = t = Life of zero-coupon debt = 10 years Variance in the value of the underlying asset =  2 = Variance in firm value = 0.16 Riskless rate = R = Treasury bond rate corresponding to option life = 10%

43. 19 - 43 Value of a troubled firm Based upon these inputs, the Black-Scholes model provides the following value for the call: d1 = 1.0515 N(d 1 ) = 0.8534 d2 = -0.2135 N(d 2 ) = 0.4155 Value of the call = 50 (0.8534) - 80 e (-0.10)(10) (0.4155) = $30.44 million Value of the bond= $50 - $30.44 = $19.56 million

44. 19 - 44 Valuing Equity in a Troubled Firm The equity in this firm has substantial value, because of the option characteristics of equity. This might explain why stock in firms, which are in Chapter 11 and essentially bankrupt, still has value.

45. 19 - 45 Conflict between bondholders and stockholders Stockholders and bondholders have different objective functions, and this can lead to agency problems, where stockholders can expropriate wealth from bondholders. The conflict can manifest itself in a number of ways - for instance, stockholders have an incentive to take riskier projects than bondholders do, and to pay more out in dividends than bondholders would like them to. This conflict between bondholders and stockholders can be illustrated using the option pricing model. Since equity is a call option on the value of the firm, an increase in the variance in the firm value, other things remaining equal, will lead to an increase in the value of equity. It is therefore conceivable that stockholders can take risky projects with negative net present values, which while making them better off, may make the bondholders and the firm less valuable. This is illustrated in the following example.

46. 19 - 46 Effect on value of the conflict between stockholders and bondholders Consider again the firm described before: value of assets of $100 million, a face value of zero- coupon ten-year debt of $80 million, a standard deviation in the value of the firm of 40%. The equity and debt in this firm were valued as follows: Value of Equity = $75.94 million Value of Debt = $24.06 million Value of Firm = $100 million Now assume that the stockholders have the opportunity to take a project with a negative net present value of -$2 million, but assume that this project is a very risky project that will push up the standard deviation in firm value to 50%.

47. 19 - 47 Valuing Equity after the Project Assume the following inputs: Value of the underlying asset = E = Value of the firm = $ 100 million - $2 million = $ 98 million (The value of the firm is lowered because of the negative net present value project) Exercise price = B = Face Value of outstanding debt = $ 80 million Life of the option = t = Life of zero-coupon debt = 10 years Variance in the value of the underlying asset =  2 = Variance in firm value = 0.25 Riskless rate = R = 10%

48. 19 - 48 Valuing Equity after the Project Based upon these inputs, the Black-Scholes model provides the following value for the equity and debt in this firm. Value of Equity = $77.71 Value of Debt = $20.29 Value of Firm = $98.00 The value of equity rises from $75.94 million to $ 77.71 million, even though the firm value declines by $2 million. The increase in equity value comes at the expense of bondholders, who find their wealth decline from $24.06 million to $20.19 million.