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21-3 Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value
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McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 21Chapter 21
Option ValuationOption Valuation
21-2
Intrinsic value - profit that could be made if the option was immediately exercised.
Call: stock price - exercise pricePut: exercise price - stock price
Time value - the difference between the option price and the intrinsic value.
Option Values
21-3
Time Value of Options: Call
Option value
XStock Price
Value of Call Intrinsic Value
Time value
21-4
Factor Effect on valueStock price increasesExercise price decreasesVolatility of stock price increasesTime to expiration increasesInterest rate increasesDividend Rate decreases
Factors Influencing Option Values: Calls
21-5
Restrictions on Option Value: Call
Value cannot be negativeValue cannot exceed the stock valueValue of the call must be greater than the value of levered equityC > S0 - ( X + D ) / ( 1 + Rf )T
C > S0 - PV ( X ) - PV ( D )
21-6
Allowable Range for Call
Call Value
S0
PV (X) + PV (D)
Upper
boun
d = S 0
Lower Bound
= S0 - PV (X) - PV (D)
21-7
100
200
50
Stock Price
C
75
0
Call Option Value X = 125
Binomial Option Pricing: Text Example
21-8
Alternative PortfolioBuy 1 share of stock at $100Borrow $46.30 (8% Rate)Net outlay $53.70PayoffValue of Stock 50 200Repay loan - 50 -50Net Payoff 0 150
53.70
150
0Payoff Structureis exactly 2 timesthe Call
Binomial Option Pricing: Text Example
21-9
53.70
150
0
C
75
0
2C = $53.70C = $26.85
Binomial Option Pricing: Text Example
21-10
Alternative Portfolio - one share of stock and 2 calls written (X = 125)
Portfolio is perfectly hedgedStock Value 50 200Call Obligation 0 -150Net payoff 50 50
Hence 100 - 2C = 46.30 or C = 26.85
Replication of Payoffs and Option Values
21-11
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% twiceDecrease by 5% twiceIncrease once and decrease once (2 paths).
21-12
Generalizing the Two-State Approach
100
110
121
9590.25
104.50
21-13
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.
Expanding to Consider Three Intervals
21-14
S
S +
S + +
S -S - -
S + -
S + + +
S + + -
S + - -
S - - -
Expanding to Consider Three Intervals
21-15
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
21-16
Co = SoN(d1) - Xe-rTN(d2)d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)d2 = d1 + (T1/2)whereCo = Current call option value.So = Current stock priceN(d) = probability that a random draw from a
normal dist. will be less than d.
Black-Scholes Option Valuation
21-17
X = Exercise pricee = 2.71828, the base of the natural logr = Risk-free interest rate (annualizes
continuously compounded with the same maturity as the option)
T = time to maturity of the option in yearsln = Natural log functionStandard deviation of annualized cont.
compounded rate of return on the stock
Black-Scholes Option Valuation
21-18
So = 100 X = 95r = .10 T = .25 (quarter)= .50d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2)
= .43 d2 = .43 + ((5.251/2)
= .18
Call Option Example
21-19
N (.43) = .6664Table 17.2
d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700
Probabilities from Normal Dist
21-20
N (.18) = .5714Table 17.2
d N(d) .16 .5636 .18 .5714 .20 .5793
Probabilities from Normal Dist.
21-21
Co = SoN(d1) - Xe-rTN(d2)Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70Implied VolatilityUsing Black-Scholes and the actual price
of the option, solve for volatility.Is the implied volatility consistent with the
stock?
Call Option Value
21-22
Put Value Using Black-Scholes
P = Xe-rT [1-N(d2)] - S0 [1-N(d1)]
Using the sample call dataS = 100 r = .10 X = 95 g = .5 T = .2595e-10x.25(1-.5714)-100(1-.6664) = 6.35
21-23
P = C + PV (X) - So = C + Xe-rT - So
Using the example dataC = 13.70 X = 95 S = 100r = .10 T = .25P = 13.70 + 95 e -.10 X .25 - 100P = 6.35
Put Option Valuation: Using Put-Call Parity
21-24
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 S0 with S0 - PV (Dividends)
21-25
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 ElasticityPercentage change in the option’s value given a 1% change in the value of the underlying stock.
Using the Black-Scholes Formula
21-26
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.
Portfolio Insurance
21-27
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.
21-28
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.
Delta = Change in the value of the option
Change of the value of the stock
21-29
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
21-30
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
21-31
Profit Position on Hedged Put PortfolioValue of put option: 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 5,570
Profit 306 290 305