Options ValuationOptions Valuation

Chapter 21Chapter 21

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 ValuesOption Values

Time Value of Options: CallTime Value of Options: Call

Option value

XStock Price

Value of Call Intrinsic Value

Time value

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: Factors Influencing Option Values: CallsCalls

Restrictions on Option Value: CallRestrictions 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 equityC > S0 - ( X + D ) / ( 1 + Rf )T

C > S0 - PV ( X ) - PV ( D )

Allowable Range for CallAllowable Range for Call

Call Value

S0

PV (X) + PV (D)

Upper

bou

nd =

S 0

Lower Bound

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

100

200

50

Stock Price

C

75

0

Call Option Value X = 125

Binomial Option Pricing:Binomial Option Pricing:Text ExampleText Example

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:Binomial Option Pricing:Text ExampleText Example

53.70

150

0

C

75

0

2C = $53.70C = $26.85

Binomial Option Pricing:Binomial Option Pricing:Text ExampleText Example

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 Another View of Replication of Payoffs and Option ValuesPayoffs and Option Values

Generalizing the Generalizing the Two-State ApproachTwo-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)

Generalizing the Generalizing the Two-State ApproachTwo-State Approach

100

110

121

9590.25

104.50

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 Expanding to Consider Three IntervalsConsider Three Intervals

S

S +

S + +

S -S - -

S + -

S + + +

S + + -

S + - -

S - - -

Expanding to Expanding to Consider Three IntervalsConsider Three Intervals

Possible Outcomes with Possible Outcomes with Three IntervalsThree 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

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 price

N(d) = probability that a random draw from a normal dist. will be less than d.

Black-Scholes Option ValuationBlack-Scholes Option Valuation

X = Exercise price.

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

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

Black-Scholes Option ValuationBlack-Scholes Option Valuation

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 ExampleCall Option Example

N (.43) = .6664

Table 17.2

d N(d)

.42 .6628

.43 .6664 Interpolation

.44 .6700

Probabilities from Normal DistProbabilities from Normal Dist

N (.18) = .5714

Table 17.2

d N(d)

.16 .5636

.18 .5714

.20 .5793

Probabilities from Normal Dist.Probabilities from Normal Dist.

Co = SoN(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?

Call Option ValueCall Option Value

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: Put Option Valuation: Using Put-Call ParityUsing Put-Call Parity

Adjusting the Black-Scholes Adjusting the Black-Scholes Model for DividendsModel 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 priceReplace S0 with S0 - PV (Dividends)

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 FormulaUsing the Black-Scholes Formula

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 Portfolio Insurance - Protecting Against Declines in Stock ValueAgainst Declines in Stock Value