CHAPTER 21

Option Valuation

• 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

Figure 21.1 Call Option Value before Expiration

Table 21.1 Determinants of Call Option Values

100

120

90

Stock Price

C

10

0

Call Option Value X = 110

Binomial Option Pricing: Text Example

Alternative Portfolio

Buy 1 share of stock at $100

Borrow $81.82 (10% Rate)

Net outlay $18.18

Payoff

Value of Stock 90 120

Repay loan - 90 - 90

Net Payoff 0 30

18.18

30

0

Payoff Structureis exactly 3 timesthe Call

Binomial Option Pricing: Text Example Continued

18.18

30

0

3C

30

0

3C = $18.18C = $6.06

Binomial Option Pricing: Text Example Continued

Alternative Portfolio - one share of stock and 3 calls written (X = 110)

Portfolio is perfectly hedgedStock Value 90 120Call Obligation 0 -30Net payoff 90 90

Hence 100 - 3C = 90/(1 + rf) = 90/(1.1) = 81.82

3C = 100 – 81.82 = 18.18 Thus C = 6.06

Replication of Payoffs and Option Values

Number of stocks per option (H)

H = (C+ - C-)/(S+ - S-) = (10 – 0)/(120 – 90) =1/3

Number of options per stock = 1/H = 3

Hedge Ratio (H)

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 100Possible outcomes:

Increase by 10% twiceDecrease by 5% twiceIncrease once and decrease once (2 paths)

Generalizing the Two-State Approach Continued

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 Consider Three Intervals

S

S +

S + +

S -S - -

S + -

S + + +

S + + -

S + - -

S - - -

Expanding to Consider Three Intervals Continued

Possible Outcomes with Three Intervals

Event Probability Final 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

Figure 21.5 Probability Distributions

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 distribution will be less than d

Black-Scholes Option Valuation

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 Continued

Figure 21.6 A Standard Normal Curve

So = 100 X = 95r = .10 T = .25 (quarter)= .50d1 = [ln(100/95) + (.10+(5 2/2))(0.25)]/(5.251/2)

= .43 d2 = .43 + ((5.251/2)

= .18

Call Option Example

N (.43) = .6664Table 21.2

d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700

Probabilities from Normal Distribution

N (.18) = .5714Table 21.2

d N(d) .16 .5636 .18 .5714 .20 .5793

Probabilities from Normal Distribution Continued

Table 21.2 Cumulative Normal Distribution

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

Spreadsheet 21.1 Spreadsheet to Calculate Black-Scholes Option Values

Figure 21.7 Using Goal Seek to Find Implied Volatility

Figure 21.8 Implied Volatility of the S&P 500 (VIX Index)

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 priceReplace S0 with S0 - PV (Dividends)

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

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

Hedging: Hedge ratio or delta The number of stocks required to hedge against

the price risk of holding one optionCall = 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

Figure 21.9 Call Option Value and Hedge Ratio

• 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

Figure 21.10 Profit on a Protective Put Strategy

Figure 21.11 Hedge Ratios Change as the Stock Price Fluctuates

Figure 21.12 S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals

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

Hedging and Delta

The appropriate hedge will depend on the deltaRecall 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

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

Table 21.3 Profit on a Hedged Put Portfolio

Table 21.4 Profits on Delta-Neutral Options Portfolio

Figure 21.13 Implied Volatility of the S&P 500 Index as a Function of Exercise Price