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
Top Related