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21-1

Option ValuationOption ValuationChapter 21

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21-2

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 Values

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21-3 Time Value of Options: Call

Option value

XStock Price

Value of Call Intrinsic Value

Time value

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21-4

Factor Effect on value

Stock price increases

Exercise price decreases

Volatility of stock price increases

Time to expiration increases

Interest rate increases

Dividend Rate decreases

Factors Influencing Option Values: Calls

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21-5 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 )

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21-6 Allowable Range for Call

Call Value

S0

PV (X) + PV (D)

Upper

bou

nd =

S 0

Lower Bound

= S0 - PV (X) - PV (D)

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21-7

100

200

50

Stock Price

C

75

0

Call Option Value X = 125

Binomial Option Pricing:Text Example

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21-8

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

150

0

Payoff Structureis exactly 2 timesthe Call

Binomial Option Pricing:Text Example

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21-9

53.70

150

0

C

75

0

2C = $53.70C = $26.85

Binomial Option Pricing:Text Example

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21-10

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

Another View of Replication of Payoffs and Option Values

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21-11Generalizing 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).

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21-12 Generalizing the Two-State Approach

100

110

121

9590.25

104.50

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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

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21-14

S

S +

S + +

S -S - -

S + -

S + + +

S + + -

S + - -

S - - -

Expanding to Consider Three Intervals

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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

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21-16

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.

Black-Scholes Option Valuation

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21-17

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

Black-Scholes Option Valuation

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21-18

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

Call Option Example

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21-19

N (.43) = .6664

Table 17.2

d N(d)

.42 .6628

.43 .6664 Interpolation

.44 .6700

Probabilities from Normal Dist

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21-20

N (.18) = .5714

Table 17.2

d N(d)

.16 .5636

.18 .5714

.20 .5793

Probabilities from Normal Dist.

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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

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21-22 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

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21-23

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

Put Option Valuation: Using Put-Call Parity

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21-24 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)

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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 Elasticity

Percentage change in the option’s value given a 1% change in the value of the underlying stock.

Using the Black-Scholes Formula

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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 - Protecting Against Declines in Stock Value

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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.

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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

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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

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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

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21-31 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

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21-32 Home Assignment

Required:

• problems 2, 7 (3rd ed).

• problems 2, 7 (5th ed).

• an additional problem (see next slide)

• closely follow financial news!