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CHAPTER 21 Option Valuation Investments, 8 th edition Bodie, Kane and Marcus Slides by Susan Hine Mc. Graw-Hill/Irwin Copyright © 2009 by The Mc. Graw-Hill Companies, Inc. All rights reserved.

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

Figure 21. 1 Call Option Value before Expiration 21 -3

Table 21. 1 Determinants of Call Option Values 21 -4

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 > S 0 - ( X + D ) / ( 1 + rf ) T C > S 0 - PV ( X ) - PV ( D ) 21 -5

Figure 21. 2 Range of Possible Call Option Values 21 -6

Figure 21. 3 Call Option Value as a Function of the Current Stock Price 21 -7

Figure 21. 4 Put Option Values as a Function of the Current Stock Price 21 -8

Binomial Option Pricing: Text Example 120 10 C 90 Stock Price 0 Call Option Value X = 110 21 -9

Binomial Option Pricing: Text Example Continued Alternative Portfolio Buy 1 share of stock at $100 Borrow $81. 82 (10% Rate) 18. 18 Net outlay $18. 18 Payoff Value of Stock 90 120 Repay loan - 90 Net Payoff 0 30 30 0 Payoff Structure is exactly 3 times the Call 21 -10

Binomial Option Pricing: Text Example Continued 30 30 18. 18 C 0 0 3 C = $18. 18 C = $6. 06 21 -11

Replication of Payoffs and Option Values Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged Stock Value 90 120 Call Obligation 0 -30 Net payoff 90 90 Hence 100 - 3 C = 81. 82 or C = 6. 06 21 -12

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 -13

Generalizing the Two-State Approach Continued 121 110 104. 50 100 95 90. 25 21 -14

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 -15

Expanding to Consider Three Intervals Continued S+++ S++- S+ S+- S S- S+-S-- S--- 21 -16

Possible Outcomes with Three Intervals Event Probability Final 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 -17

Figure 21. 5 Probability Distributions 21 -18

Black-Scholes Option Valuation Co = So. N(d 1) - Xe-r. TN(d 2) d 1 = [ln(So/X) + (r + 2/2)T] / ( T 1/2) d 2 = d 1 + ( T 1/2) where Co = Current call option value So = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d 21 -19

Black-Scholes Option Valuation Continued 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 21 -20

Figure 21. 6 A Standard Normal Curve 21 -21

Call Option Example So = 100 X = 95 r =. 10 T =. 25 (quarter) =. 50 d 1 = [ln(100/95) + (. 10+( 5 2/2))] / ( 5. 251/2) =. 43 d 2 =. 43 + (( 5 . 251/2) =. 18 21 -22

Probabilities from Normal Distribution N (. 43) =. 6664 Table 21. 2 d N(d). 42. 6628. 43. 6664 Interpolation. 44. 6700 21 -23

Probabilities from Normal Distribution Continued N (. 18) =. 5714 Table 21. 2 d N(d). 16. 5636. 18. 5714. 20. 5793 21 -24

Table 21. 2 Cumulative Normal Distribution 21 -25

Call Option Value Co = So. N(d 1) - Xe-r. TN(d 2) 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 -26

Spreadsheet 21. 1 Spreadsheet to Calculate Black-Scholes Option Values 21 -27

Figure 21. 7 Using Goal Seek to Find Implied Volatility 21 -28

Figure 21. 8 Implied Volatility of the S&P 500 (VIX Index) 21 -29

Black-Scholes Model with 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 S 0 with S 0 - PV (Dividends) 21 -30

Put Value Using Black-Scholes P = Xe-r. T [1 -N(d 2)] - S 0 [1 -N(d 1)] Using the sample call data S = 100 r =. 10 X = 95 g =. 5 T =. 25 95 e-10 x. 25(1 -. 5714)-100(1 -. 6664) = 6. 35 21 -31

Put Option Valuation: Using Put-Call Parity P = C + PV (X) - So = C + Xe-r. T - 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 -32

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 (d 1) Put = N (d 1) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock 21 -33

Figure 21. 9 Call Option Value and Hedge Ratio 21 -34

Portfolio Insurance • 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 -35

Figure 21. 10 Profit on a Protective Put Strategy 21 -36

Figure 21. 11 Hedge Ratios Change as the Stock Price Fluctuates 21 -37

Figure 21. 12 S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals 21 -38

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 21 -39

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 Delta = Change in the value of the option Change of the value of the stock 21 -40

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 21 -41

Table 21. 3 Profit on a Hedged Put Portfolio 21 -42

Table 21. 4 Profits on Delta-Neutral Options Portfolio 21 -43

Figure 21. 13 Implied Volatility of the S&P 500 Index as a Function of Exercise Price 21 -44