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