1. Financial Options and Applications in Corporate Finance 1

2. Topics in Chapter Financial Options Terminology Option Price Relationships Black-Scholes Option Pricing Model Put-Call Parity 2

3. What is a financial option? 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. 3

4. What is the single most important characteristic of an option? It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset. 4

5. Option Terminology Call option: An option to buy a specified number of shares of a security within some future period. Put option: An option to sell a specified number of shares of a security within some future period. 5

6. Option Terminology Strike (or exercise) price: The price stated in the option contract at which the security can be bought or sold. Expiration date: The last date the option can be exercised. 6

7. Option Terminology (Continued) Exercise value: The value of a call option if it were exercised today = Max[0, Current stock price - Strike price] Note: The exercise value is zero if the stock price is less than the strike price. Option price: The market price of the option contract. 7

8. Option Terminology (Continued) Time value: Option price minus the exercise value. It is the additional value because the option has remaining time until it expires. Stock option contracts are always written on 100 shares of stock. When the exercise price of an option is equal to the current price of a stock, it is at-the-money. If the payoff from exercising an option immediately is positive, it is in- the-money. If the payoff from exercising an option immediately is negative, it is out- of-the-money. Options where the strike price and the stock price are very far apart are referred to as deep in-the-money or deep out-of-the-money. 8

9. Option Terminology (Continued) Writing a call option: For every new option, there is an investor who “writes” the option. A writer creates the contract, sells it to another investor, and must fulfill the option contract if it is exercised. For example, the writer of a call must be prepared to sell a share of stock to the investor who owns the call. Covered option: A call option written against stock held in an investor’s portfolio. Naked (uncovered) option: An option written without the stock to back it up. 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 ½ years. 9

10. Consider the following data: Strike price = $25. Stock PriceCall Option Price $25$3.00 30 7.50 3512.00 4016.50 4521.00 5025.50 10

11. Exercise Value of Option Price of stock (a) Strike price (b) Intrinsic value of option (a)–(b) $25.00 $0.00 30.0025.00 5.00 35.0025.0010.00 40.0025.0015.00 45.0025.0020.00 50.0025.00 11

12. Market Price of Option Price of stock (a) Strike price (b) IV val. (c) Mkt. Price of opt. (d) $25.00 $0.00 $3.00 30.0025.00 5.00 7.50 35.0025.0010.0012.00 40.0025.0015.0016.50 45.0025.0020.0021.00 50.0025.00 25.50 12

13. Time Value of Option Price of stock (a) Strike price (b) IV Val. (c) Mkt. P of opt. (d) Time value (d) – (c) $25.00 $0.00 $3.00 30.0025.00 5.00 7.50 2.50 35.0025.0010.0012.00 2.00 40.0025.0015.0016.50 1.50 45.0025.0020.0021.00 1.00 50.0025.00 25.50 0.50 13

14. Call Time Value Diagram 14 5 10 15 20 25 30 35 40 Stock Price Option value 30 25 20 15 10 5 Market price Exercise value

15. Option Time Value Versus Intrinsic Value The time value, which is the option price less its 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. 15

16. Assumptions of the Black-Scholes Option Pricing Model 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. Risk-free rate, r RF, is known and constant during the option’s life. 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. 16

17. What are the three equations that make up the OPM? 17 V C = P[N(d 1 )] – X e -r RF t [N(d 2 )] d 1 =  t 0.5 d 2 = d 1 -  t 0.5 ln(P/X) + [r RF + (  2 /2)]t

18. What is the value of the following call option according to the OPM? Assume: P = $27 X = $25 r RF = 6% t = 0.5 years σ = 0.49 18

19. First, find d 1 and d 2. 19 d 1 = {ln($27/$25) + [(0.06 + 0.49 2 /2)](0.5)} ÷ {(0.49)(0.7071)} d 1 = 0.4819 d 2 = 0.4819 - (0.49)(0.7071) d 2 = 0.1355

20. Second, find N(d 1 ) and N(d 2 ) N(d 1 ) = N(0.4819) = 0.6851 N(d 2 ) = N(0.1355) = 0.5539 Note: Values obtained from Excel using NORMSDIST function. For example: N(d 1 ) = NORMSDIST(0.4819) 20

21. Third, find value of option. 21 V C = $27(0.6851) - $25 e -(0.06)(0.5) (0.5539) = $19.3536 - $25(0.97045)(0.6327) = $5.06

22. What impact do the following parameters have on a call option’s value? Current stock price: Call option value increases as the current stock price increases. Strike price: As the exercise price increases, a call option’s value decreases. Option period: As the expiration date is lengthened, a call option’s value increases. Longer time to expiration increases probability of very high stock price, which has big payoff. Also increases the probability of a very low stock price, but payoff is zero for any price below the strike price. Risk-free rate: Call option’s value tends to increase as r RF increases (reduces the PV of the exercise price). Stock return variance: Option value increases with variance of the underlying stock. Higher variance increases probability of very high stock price, which has big payoff. Also increases the probability of a very low stock price, but payoff is zero for any price below the strike price. 22

23. The Black-Scholes Formula (cont'd) Dividend-Paying Stocks If PV(Div) is the present value of any dividends paid prior to the expiration of the option, then: Where S x is the price of the stock excluding any dividends

24. The Black-Scholes Formula (cont'd) Dividend-Paying Stocks Because a European call option is the right to buy the stock without these dividends, it can be evaluated by using the Black-Scholes formula with S x in place of S.

25. The Black-Scholes Formula (cont'd) Dividend-Paying Stocks A special case is when the stock will pay a dividend that is proportional to its stock price at the time the dividend is paid. If q is the stock’s (compounded) dividend yield until the expiration date, then:

26. Put Options A put option gives its holder the right to sell a share of stock at a specified stock on or before a particular date. 26

27. Put-Call Parity Portfolio 1: Put option, Share of stock, P Portfolio 2: Call option, V C PV of exercise price, X 27

28. Portfolio Payoffs at Expiration Date T for P T <X and P T ≥X P T <XP T ≥X Port. 1Port. 2Port. 1Port. 2 Stock P T PTPT PutX-P T 0 Call0P T -X CashX X TotalXXPTPT PTPT 28

29. Put-Call Parity Relationship Portfolio payoffs are equal, so portfolio values also must be equal. Put + Stock = Call + PV of Exercise Price 29 Put + P = V C + Xe -r RF t Put = V C – P + Xe -r RF t V C = Put P+ Xe -r RF t

30. Implied Volatility Of the five required inputs in the Black-Scholes formula, only  is not observable directly. – Practitioners use two strategies to estimate the value of . Use historical data “Back out” the implied volatility – Implied Volatility » The volatility of an asset’s returns that is consistent with the quoted price of an option on the asset

31. 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 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.. Explain why equity of a levered firm can be thought of as a call option

32. The Value of Risky Debt I.Risky Debt=Assets–Equity Call option =– EEE

33. 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.

34. Application to Equity Valuation: A simple example The parameters of equity as a call option are as follows: Value of the underlying asset = V = 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%

35. 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%

36. 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%

37. 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

38. 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.

39. 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.

40. 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%.

41. Valuing Equity after the Project Assume the following inputs: Value of the underlying asset = V = 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%

42. 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.