1 / 25

Chapter 21

Chapter 21. Options Valuation. Option Values. 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

yoshi
Download Presentation

Chapter 21

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 21 Options Valuation 21-1

  2. Option Values • 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 21-2

  3. Option value Value of Call Intrinsic Value Time value X Stock Price Time Value of Options: Call 21-3

  4. Factors Influencing Option Values: Calls FactorEffect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend Rate decreases 21-4

  5. Restrictions on Option Value: Call • Value cannot be negative • Value cannot exceed the stock value • Value of the call must be greater than the value of levered equity C > S0 - ( X + D ) / ( 1 + Rf )T C > S0 - PV ( X ) - PV ( D ) 21-5

  6. Allowable Range for Call Call Value Upper bound = S0 Lower Bound = S0 - PV (X) - PV (D) S0 PV (X) + PV (D) 21-6

  7. 75 C 0 Call Option Value X = 125 Binomial Option Pricing:Text Example 200 100 50 Stock Price 21-7

  8. Binomial Option Pricing:Text Example 150 Alternative Portfolio Buy 1 share of stock at $100 Borrow $46.30 (8% Rate) Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 0 150 53.70 0 Payoff Structure is exactly 2 times the Call 21-8

  9. 75 C 0 Binomial Option Pricing:Text Example 150 53.70 0 2C = $53.70 C = $26.85 21-9

  10. Another View of Replication of Payoffs and Option Values Alternative Portfolio - one share of stock and 2 calls written (X = 125) Portfolio is perfectly hedged Stock Value 50 200 Call Obligation 0-150 Net payoff 50 50 Hence 100 - 2C = 46.30 or C = 26.85 21-10

  11. Generalizing the Two-State Approach Assume that we can break the year into two six-month segments In each six-month segment the stock could increase by 10% or decrease by 5% Assume the stock is initially selling at 100 Possible outcomes Increase by 10% twice Decrease by 5% twice Increase once and decrease once (2 paths) 21-11

  12. 121 110 104.50 100 95 90.25 Generalizing the Two-State Approach 21-12

  13. Expanding to Consider Three Intervals • Assume that we can break the year into three intervals • For each interval the stock could increase by 5% or decrease by 3% • Assume the stock is initially selling at 100 21-13

  14. S + + + S + + S + + - S + S + - S S + - - S - S - - S - - - Expanding to Consider Three Intervals 21-14

  15. Possible Outcomes with Three Intervals Event Probability Stock Price 3 up 1/8 100 (1.05)3 =115.76 2 up 1 down 3/8 100 (1.05)2 (.97) =106.94 1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79 3 down 1/8 100 (.97)3 = 91.27 21-15

  16. Black-Scholes Option Valuation Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r + 2/2)T] / (T1/2) d2 = d1 + (T1/2) where Co = Current call option value. So = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d. 21-16

  17. Black-Scholes Option Valuation X = Exercise price. e = 2.71828, the base of the nat. log. r = Risk-free interest rate (annualizes 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 annualized cont. compounded rate of return on the stock 21-17

  18. Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) = .50 d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = .43 + ((5.251/2) = .18 21-18

  19. Probabilities from Normal Dist N (.43) = .6664 Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700 21-19

  20. Probabilities from Normal Dist. N (.18) = .5714 Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793 21-20

  21. Call Option Value Co = SoN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? 21-21

  22. Put Option Valuation: Using Put-Call Parity P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 X .25 - 100 P = 6.35 21-22

  23. Adjusting the Black-Scholes Model for Dividends • The call option formula applies to stocks that pay dividends • One approach is to replace the stock price with a dividend adjusted stock price Replace S0 with S0 - PV (Dividends) 21-23

  24. Using the Black-Scholes Formula Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Call = N (d1) Put = N (d1) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock 21-24

  25. Portfolio Insurance - Protecting Against Declines in Stock Value • Buying Puts - results in downside protection with unlimited upside potential • Limitations • Tracking errors if indexes are used for the puts • Maturity of puts may be too short • Hedge ratios or deltas change as stock values change 21-25

More Related