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### Chapter16

Option Valuation

Option Values

- Intrinsic value - payoff 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

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 yield decreases

A Simple Binomial Model

- A stock price is currently $20
- In three months it will be either $22 or $18

Stock Price = $22

Stock price = $20

Stock Price = $18

A Call Option

A 3-month call option on the stock has a strike price of 21.

Stock Price = $22

Option Price = $1

Stock price = $20

Option Price=?

Stock Price = $18

Option Price = $0

22D – 1

18D

Setting Up a Riskless Portfolio- Consider the Portfolio: long D shares short 1 call option
- Portfolio is riskless when 22D – 1 = 18D or
D = 0.25

Valuing the Portfolio(Risk-Free Rate is 12%)

- The riskless portfolio is:
long 0.25 shares short 1 call option

- The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50
- The value of the portfolio today is 4.5e– 0.12´0.25 = 4.3670

Valuing the Option

- The portfolio that is
long 0.25 shares short 1 option

is worth 4.367

- The value of the shares is 5.000 (= 0.25´20 )
- The value of the option is therefore 0.633 (= 5.000 – 4.367 )

Example:

- Suppose the stock now sells at $100, and the price will either double to $200 or fall in half to $50 by the year-end. A call option on the stock might specify an exercise price of $125 and a time to expiration of one year. The interest rate is 8%. What is the option price today?

Black-Scholes Option Valuation

Co= Soe-dTN(d1) - Xe-rTN(d2)

d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2)

d2 = d1 - (s 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

X = Exercise price.

d = Annual dividend yield of underlying stock

e = 2.71828, the base of the nat. 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

s = Standard deviation of annualized cont. compounded rate of return on the stock

Call Option Example

So = 100 X = 95

r = .10 T = .25 (quarter)

s = .50 d = 0

d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5.251/2)

= .43

d2 = .43 - ((.5)( .251/2)

= .18

Probabilities from Normal Dist.

N (.43) = .6664

Table 17.2

d N(d)

.42 .6628

.43 .6664 Interpolation

.44 .6700

Call Option Value

Co= Soe-dTN(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?

Put Option Value: Black-Scholes

P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)]

Using the sample data

P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)

P = $6.35

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

Exercise in class

The stock price of Ajax Inc. is currently $105. The stock price a year from now will be either $130 or $90 with equal probabilities. The interest rate at which investors can borrow is 10%. Using the binomial OPM, the value of a call option with an exercise price of $110 and an expiration date one year from now should be worth __________ today.

A) $11.60

B) $15.00

C) $20.00

D) $40.00

The stock price of Bravo Corp. is currently $100. The stock price a year from now will be either $160 or $60 with equal probabilities. The interest rate at which investors invest in riskless assets at is 6%. Using the binomial OPM, the value of a put option with an exercise price of $135 and an expiration date one year from now should be worth __________ today.

A) $34.09

B) $37.50

C) $38.21

D) $45.45

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