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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

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Chapter 21

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  1. Chapter 21 Options Valuation

  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

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

  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

  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 )

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

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

  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

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

  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

  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)

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

  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

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

  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

  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.

  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

  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

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

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

  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?

  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

  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)

  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

  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

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