1. Option Valuation

2. Intrinsic value - profit that could be made if the option was immediately exercised –Call: stock price - exercise price –Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value Option Values

3. Call Option Price Boundaries 1. Basic boundaries –C t ≥ 0,Why? – –C t ≥ S t – X,Why? –Thus C t  Max (0, S t – X) where: C t =Price paid for a call option at time t. t = 0 is today, T =Immediately before the option's expiration. P t =Price paid for a put option at time t. S t =Stock price at time t. X = Exercise or Strike Price (X or E)

4. Call Option Price Boundaries 2. A tighter boundary Suppose we consider two different portfolios: Portfolio 1: Long position in stock at S 0 Portfolio 2: Buy 1 call option (C 0 ) and buy a T-bill with a face value = X.

5. Call Option Price Boundaries Possible values of the two portfolios at contract expiration: STST X + X + T-Bill S T - X0CallSTST STST S T > XS T < XS T > XS T < X Portfolio 2 (C 0 + P.V. of X)Portfolio 1 (S 0 ) At expiration, the value of Portfolio 2 is always ___ Portfolio 1 so initially the value of Portfolio 2 must also be ____ the value of Portfolio 1. Thus:   C 0 + (P.V. of X)  S 0 OR C 0  S 0 - (P.V. of X) C 0  Max[0, S 0 - (P.V. of X)]

6. Put Option Price Boundaries Suppose we consider two different portfolios: Portfolio 1: Long position in stock at S 0 Portfolio 2: Sell 1 put option (P 0 ) and buy a T-bill with a face value = X.

7. Put Option Price Boundaries Possible values of the two portfolios at contract expiration: XSTST + X + T-Bill 0-(X - S T )PutSTST STST S T > XS T < XS T > XS T < X Portfolio 2 (P.V. of X - P 0 )Portfolio 1 (S 0 ) At expiration, the value of Portfolio 1 is always ___ Portfolio 2 so initially the value of Portfolio 1 must also be ____ the value of Portfolio 2. Thus:   S 0  (P.V. of X) - P 0 OR P 0  (P.V. of X) - S 0 P 0  Max[0, (P.V. of X - S 0 )]

8. Value of a Call Option X Stock Price t Value $0 Prior to expiration Value at expiration or “Exercise” or “Intrinsic” Value 6 mo 2 mo Difference is the “Time Value” of the option The time value of a call incorporates the probability that S will be in the money at period T given S 0, time to T,  2 stock,X, and the level of interest rates  

9. Table: Determinants of Call Option Values

10. Restrictions on Option Value: Call Call value cannot be negative. The option payoff is zero at worst, and highly positive at best. Call value cannot exceed the stock value. Lower bound = adjusted intrinsic value: C > S 0 - PV (X) - PV (D) (D=dividend)

11. Figure: Range of Possible Call Option Values

12. Figure: Call Option Value as a Function of the Current Stock Price

13. Early Exercise: Calls The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires. The value of American and European calls is therefore identical. The call gains value as the stock price rises. Since the price can rise infinitely, the call is “worth more alive than dead.”

14. Early Exercise: Puts American puts are worth more than European puts, all else equal. The possibility of early exercise has value because: –The value of the stock cannot fall below zero. –Once the firm is bankrupt, it is optimal to exercise the American put immediately because of the time value of money.

15. Figure: Put Option Values as a Function of the Current Stock Price

16. 100 120 90 Stock Price C 10 0 Call Option Value X = 110 Binomial Option Pricing: Example

17. Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 - 90 Net Payoff 0 30 18.18 30 0 Payoff Structure is exactly 3 times the Call Binomial Option Pricing: Example

18. 18.18 30 0 3C 30 0 3C = $18.18 C = $6.06 Binomial Option Pricing: Example

19. Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged: Stock Value90120 Call Obligation0 -30 Net payoff90 90 Hence 100 - 3C = $81.82 or C = $6.06 Replication of Payoffs and Option Values

20. Hedge Ratio The number of stocks required to hedge against the price risk of holding one option In the example, the hedge ratio = 1 share to 3 calls or 1/3. Generally, the hedge ratio is:

21. Assume that we can break the year into three intervals. For each interval the stock could increase by 20% or decrease by 10%. Assume the stock is initially selling at $100. Expanding to Consider Three Intervals

22. S S + S + + S - S - - S + - S + + + S + + - S + - - S - - - Expanding to Consider Three Intervals

23. Possible Outcomes with Three Intervals EventProbabilityFinal Stock Price 3 up1/8100 (1.20) 3 = $172.80 2 up 1 down3/8100 (1.20) 2 (.90) = $129.60 1 up 2 down3/8100 (1.20) (.90) 2 = $97.20 3 down1/8100 (.90) 3 = $72.90

24. C o = S o N(d 1 ) - Xe -rT N(d 2 ) d 1 = [ln(S o /X) + (r +  2 /2)T] / (  T 1/2 ) d 2 = d 1 - (  T 1/2 ) where C o = Current call option value S o = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d Black-Scholes Option Valuation

25. X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualized, continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function  Standard deviation of the stock Black-Scholes Option Valuation

26. Figure: A Standard Normal Curve

27. S o = 100X = 95 r = 0.10T = 0.25 (quarter)  = 0.50 (50% per year) Thus: Example: Black-Scholes Valuation

28. Using a table or the NORMDIST function in Excel, we find that N (0.43) =0.6664 and N (0.18) = 0.5714. Therefore: C o = S o N(d 1 ) - Xe -rT N(d 2 ) C o = 100 (0.6664) - 95 e - 0.10 ( 0.25) (0.5714) C o = $13.70 Probabilities from Normal Distribution

29. Implied Volatility Implied volatility is volatility for the stock implied by the option price. Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? Call Option Value

30. Black-Scholes Model with Dividends The Black Scholes call option formula applies to stocks that do not pay dividends. What if dividends ARE paid? One approach is to replace the stock price with a dividend adjusted stock price Replace S 0 with S 0 - PV (Dividends)

31. Example: Black-Scholes Put Valuation P = Xe -rT [1-N(d 2 )] - S 0 [1-N(d 1 )] Using Example 18.2 data: S = 100, r =.10, X = 95, σ =.5, T =.25 We compute: $95e -10x.25 (1-.5714)-$100(1-.6664) = $6.35

32. Put Call Parity The value of a put can be found from the value of a call with similar terms as follows: The initial cost of this portfolio is C 0 – P 0 – S 0 + X(e -rT ) At expiration of the options the portfolio’s value will be $0 Net value – X Loan repay –(S T – X)0Write Call 0X – S T Long Put STST STST Stock S T > XS T < X Value at expirationPortfolio Consequently initial cost must = 0 and we have: 0 = C 0 – P 0 – S 0 + X(e -rT ) or: P 0 = C 0 – S 0 + X(e -rT ) Suppose we buy stock today at S 0, buy a put, write a call, and borrow the present value of the exercise price of the options until the options expire.

33. P = C + PV (X) - S o = C + Xe -rT - S o Using the example data P = 13.70 + 95 e -0.10 (0.25) - 100 P = $6.35 Put Option Valuation: Using Put- Call Parity

34. Empirical Evidence on Option Pricing The Black-Scholes formula performs worst for options on stocks with high dividend payouts. The implied volatility of all options on a given stock with the same expiration date should be equal. –Empirical test show that implied volatility actually falls as exercise price increases. –This may be due to fears of a market crash.

35. Problem 1a

36. Problem 1b

37. Problem 2

38. Problem 3