1 / 33

Chapter 21

Chapter 21. Option 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.

astra-gallegos
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 Option 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. Time Value of Options: Call Option value Value of Call Intrinsic Value Time value X Stock Price

  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. Binomial Option Pricing:Text Example 200 75 100 C 50 0 Call Option Value X = 125 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. Binomial Option Pricing:Text Example 150 75 53.70 C 0 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. Generalizing the Two-State Approach 121 110 104.50 100 95 90.25

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

  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 natural 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 Value Using Black-Scholes P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using the sample call data S = 100 r = .10 X = 95 g = .5 T = .25 95e-10x.25(1-.5714)-100(1-.6664) = 6.35

  23. 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

  24. 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)

  25. 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.

  26. 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.

  27. Hedging On Mispriced Options Option value is positively related to volatility: • If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible. • Profit must be hedged against a decline in the value of the stock. • Performance depends on option price relative to the implied volatility.

  28. Hedging and Delta The appropriate hedge will depend on the delta. Recall the delta is the change in the value of the option relative to the change in the value of the stock. Change in the value of the option Change of the value of the stock Delta =

  29. Mispriced Option: Text Example Implied volatility = 33% Investor believes volatility should = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate r = 4% Delta = -.453

  30. Hedged Put Portfolio Cost to establish the hedged position 1000 put options at $4.495 / option $ 4,495 453 shares at $90 / share 40,770 Total outlay 45,265

  31. Profit Position on Hedged Put Portfolio Value of put option as function of stock price: implied vol. = 35% Stock Price 89 90 91 Put Price $5.254 $4.785 $4.347 Profit (loss) for each put .759 .290 (.148) Value of and profit on hedged portfolio Stock Price 89 90 91 Value of 1,000 puts $ 5,254 $ 4,785 $ 4,347 Value of 453 shares 40,317 40,770 41,223 Total 45,571 45,555 45,570 Profit 306 290 305

  32. Home Assignment Required: • problems 2, 7 (3rd ed). • problems 2, 7 (5th ed). • an additional problem (see next slide) • closely follow financial news!

  33. Home Assignment Assume that each year there are only 2 possible states of the economy: "up" and "down". If the economy goes "up" an "ABC" share's price, which is today $100, increases by 20%, and if the economy goes "down" it decreases by 30%. What should be the price of a 2 years call option on ABC's share with an exercise price of $105, if the riskless rate of return is 8% per year?

More Related