# Chapter 16 ```16
Option Valuation
1
What is an Option Worth?
• At expiration, an option is worth its intrinsic value.
• Before expiration, put-call parity allows us to price
options. But,
• To calculate the price of a call, we need to know the put
price.
• To calculate the price of a put, we need to know the call
price.
• So, we want to know the value of a call option:
• Before expiration, and
• Without knowing the price of the put
16-2
A Simple Model to Value
Options Before Expiration, I.
• Suppose we want to know the price of a call
option with
•
•
•
•
One year to maturity
(T = 1.0)
A \$110 exercise price
(K = 110)
The current stock price is \$108 (S0 =108)
The one-year risk-free rate is 10 percent (r = 10%)
16-3
A Simple Model to Value
Options Before Expiration, I.
• We know what the stock price will be in one year.
• S1 = \$130 or \$115 (but no other values).
• S1 is still uncertain.
• We do not know the probabilities of these two values.
• Therefore, the call option value at expiration will be:
• \$130 – \$110 = \$20 or
• \$115 - \$110 = \$5
• This call option is certain to finish in the money.
• A similar put option is certain to finish out of the money.
16-4
A Simple Model to Value
Options Before Expiration, II.
• If you know the price of a similar put, you can use
put-call parity to price a call option before it expires.
C - P  S0 - K/(1  r)T
C - 0  \$108  \$110/(1.10 )
C  \$108 - \$100  \$8.
• The chosen pair of stock prices guarantees that the
call option finishes in the money.
• Suppose, however, we want to allow the call option
to expire in the money OR out of the money.
16-5
The One-Period Binomial Option Pricing
Model—The Assumptions
• S = the stock price today; the stock pays no
dividends.
• Assume that the stock price in one period is either
S &times; u or S &times; d, where:
• u (for “up” factor) is &gt; 1
• and d (for “down” factor) &lt; 1
• Suppose S = \$100, u = 1.1 and d = .95
• S1 = \$100 &times; 1.1 = \$110 or S1 = \$100 &times; .95 = \$95
• What is the call price today, if:
• K = 100
• R = 3%
16-6
The One-Period Binomial Option
Pricing Model—The Setup
• Consider the following portfolio:
• Buy a fractional share of the underlying asset• The Greek letter D (Delta) = this fraction
• Sell one call option
• Finance the difference by borrowing \$(DS – C)
• Key Question:
What is the value of this portfolio, today and
at option expiration?
16-7
The Value of this Portfolio
(long D Shares and short one call) is:
• Important: DS is NOT the change in S.
Rather, it is a dollar amount, DS = Δ * S.
DS&times;u - Cu
DS - C
Cu and Cd : The intrinsic value of the
call if the stock price increases to S&times;u
or decreases to S&times;d.
DS&times;d - Cd
Portfolio Value Today
Portfolio Value At Expiration
16-8
To Calculate Today’s Call Price,
C:
• There is one combination of a fractional share and
one call that makes this portfolio risk-less.
• The portfolio will have the same value when the
underlying asset increases as it does when the
underlying asset decreases in value.
• The portfolio is riskless if: DSu – Cu = DSd – Cd
• We know all values in this equation, except D.
• S = \$100; Su = \$110; Sd = \$95
• Cu = MAX(Su – K, 0) = MAX(\$110 – 100,0) = \$10
• Cd = MAX(Sd – K, 0) = MAX(\$95 – 100,0) = \$0
16-9
First Step: Calculate D
To make the portfolio riskless:
DSu – Cu = DSd – Cd
DSu – DSd = Cu – Cd
D(Su – Sd) = Cu – Cd,
We can calculate D:
D = (Cu – Cd) / (Su – Sd)
D = (10 – 0) / 110 – 95
D = 10 / 15
D=2/3
16-10
Sidebar: What is D?
• D, delta, is the riskless hedge ratio.
• D, delta, is the fractional share amount
needed to hedge one call.
• Therefore, the number of calls to hedge
one share is 1/D
16-11
The One-Period Binomial Option
Pricing Model—The Formula
• A riskless portfolio today should be worth
(DS – C)(1+r) in one period.
• So:
(DS – C)(1+r) = DSu – Cu
(which equals DSd – Cd because we chose the
“correct” D).
• Solve the equation above for C.
16-12
The One-Period Binomial Option
Pricing Model—The Formula
• Solving the equation above for C:
( DS - C)(1 r)  DSu - Cu
ΔS(1  r)  C(1  r)  ΔSu  Cu
ΔS(1  r)  ΔSu  C(1  r)  Cu
ΔS(1  r  u)  C(1  r)  Cu
ΔS(1  r  u)  Cu  C(1 r)
ΔS(1  r  u)  Cu
C
1 r
16-13
Calculate the Call Price, C.
C
C
C
C
ΔS(1  r  u)  C u
1 r
(2/3)(\$100 )(1  .03  1.10)  \$10
1  .03
(\$200/3)( .07)  \$10
1  .03
\$5.33
 \$5.18
1.03
16-14
Now We Can Calculate the Call Price, C.
If C= \$5.18, what is the price of a similar put?
Using Put-Call Parity:
P  S  C  K/(1  r)
P  \$100  \$5.18  \$100/1.03
P  \$5.18  \$100/1.03  100
P  \$2.27
16-15
The Two-Period Binomial Option Pricing
Model
• Suppose there are two periods to expiration instead
of one.
• Repeat much of the process used in the one-period
binomial option pricing model.
• This method can be used to price:
• European call options.
• European put options.
• American calls and puts (with a modification to allow for
early exercise).
• An exotic array of options (with the appropriate
modifications).
16-16
The Method
We can find binomial option prices for two (or more)
periods by using the following five steps:
1. Build a price “tree” for stock prices through time.
2. Use the intrinsic value formula to calculate the
possible option values at expiration.
3. Calculate the fractional share needed to form
each riskless portfolio at the next-to-last date.
4. Calculate all possible option prices at the next-tolast date.
5. Repeat this process by working back to today.
16-17
The
Binomial
Option
Pricing
Model
with Many
Periods
16-18
What Happens When the Number
of Periods Gets Really, Really Big?
• For European options on non-dividend paying
stocks, the binomial method converges to the
Black-Scholes option pricing formula.
• To calculate the prices of many other types of
options, however, we still need to use a
computer (and methods similar in spirit to the
binomial method).
16-19
The Black-Scholes Option Pricing Model
• The BSOPM calculates the price of a call option
before maturity
•
•
•
•
•
Dates from the early 1970s
Professors Fischer Black and Myron Scholes
Facilitated option pricing
CBOE was launched soon after BSOPM appeared
1997 Nobel Prize in Economics
• Important contributions by professor Robert
Merton
• The Black-Scholes-Merton option pricing model
16-20
Black-Scholes Option Valuation
Variables
C
P
S
K
r

T
ln
e
N(d)
= Current call option value
= Current put option value
= Current stock price
= Option strike or exercise price
= Risk-free interest rate
 Stock price volatility
= time to maturity of the option in years
= Natural log function; ln(x)
= 2.71828, the base of the natural log; exp()
= probability that a random draw from a
normal distribution will be less than d
Solution
Variables
Input
Variables
Functions
16-21
Black-Scholes Option Valuation
C  SN(d1)  Ke
P  Ke
 rT
 rT
N (d 2 )
N(d 2 )  SN(d1)
ln(S / K )  (r   2 / 2)T
d1 
 T
d 2  d1   T
16-22
Formula Functions
• e-rt = exp(-rt) = natural exponent of the value
of –rt (a discount factor)
• ln(S/K) = natural log of the &quot;moneyness&quot;
term, S/K
• N(d1) and N(d2) denotes the standard
normal probability for the values of d1
and d2.
• Formula makes use of the fact that:
N(-d1) = 1 - N(d1)
16-23
Example: Computing Prices
for Call and Put Options
Suppose you are given the following inputs:
S = \$50
K = \$45
T = 3 months (or 0.25 years)
 = 25% (stock volatility)
r = 6%
16-24
Step 1: Calculating d1 and d2
ln(S K   (r  σ 2 2 T
d1 
σ T
2
(
(

ln 50 45  0.06  0.25 2 0.25

0.25 0.25
0.10536  0.09125  0.25

 1.02538
0.125
d2  d1  σ T  1.02538  0.25 0.25  0.90038
16-25
Step 2: Excel’s “=NORMSDIST(x)” Function
=NORMSDIST(1.02538) = 0.84741 = N(d1)
=NORMSDIST(0.90038) = 0.81604 = N(d2)
N(-d1) = 1 - N(d1):
N(-1.02538) = 1 – N(1.02538)
= 1 – 0.84741 = 0.15259 = N(-d1)
N(-0.90038) = 1 – N(0.90038)
= 1 – 0.81604 = 0.18396 = N(-d2)
16-26
Step 3a: The Call Price
C = SN(d1) – Ke–rTN(d2)
= \$50(0.84741) – 45(e-(.06)(.25))(0.81604)
= 50(0.84741) – 45(0.98511)(0.81604)
= \$6.195
16-27
Step 3b: The Put Price:
P =
–rT
Ke N(–d
2)
– SN(–d1)
= \$45(e-(.06)(.25))(0.18396) – 50(0.15259)
= 45(0.98511)(0.18396) – 50(0.15259)
= \$0.525
16-28
Verify Our Results Using Put-Call Parity
Note: Options must be European-style
C  P  S  Ke
 rT
\$5.985  \$0.565  50 - 45e  (0.060.25)
\$5.42  \$50.00  \$44.33  \$5.42
16-29
Valuing the Options Using Excel
Stock Price:
Strike Price:
Volatility (%):
Time (in years):
Riskless Rate (%):
50.00
45.00
25.00
0.2500
6.00
Stock:
Discounted Strike:
50.00
44.33
d(1):
N(d1):
1.02538
0.84741
N(-d1):
0.15259
d(2):
N(d2):
0.90038
0.81604
N(-d2):
0.18396
Call Price:
\$ 6.195
Put Price:
\$ 0.525
16-30
Using a Web-based Option
Calculator
• www.numa.com.
16-31
Varying the Option Price Input
Values
• An important goal of this chapter is to show how an
option price changes when only one of the five
inputs changes.
• The table below summarizes these effects.
16-32
Factors Influencing Option
Values
Effect on
Common
Option Value
Name
Call
Put
Table 16.3
Input Factor
Underlying stock price
S
Strike price of option contract
K
Time remaining to expiration
T
Volatility of the underlying stock price
σ
Risk-free interest rate
r
+
+
+
+
+
+
+
-
Delta
16-33
Varying the Underlying Stock
Price
• Changes in the stock price have a big effect
on option prices.
16-34
Varying the Time Remaining Until Option
Expiration
16-35
Varying the Volatility of the Stock
Price
16-36
Varying the Interest Rate
16-37
Calculating Delta
• Delta measures the dollar impact of a change in
the underlying stock price on the value of a stock
option.
Call option delta = N(d1) &gt; 0
Put option delta = –N(–d1) &lt; 0
• A \$1 change in the stock price causes an option
price to change by approximately delta dollars.
16-38
Example: Calculating Delta
Stock Price:
Strike Price:
Volatility (%):
Time (in years):
Riskless Rate (%):
50.00
45.00
25.00
0.2500
6.00
Stock:
Discounted Strike:
50.00
44.33
d(1):
N(d1):
1.02538
0.84741
N(-d1):
0.15259
d(2):
N(d2):
0.90038
0.81604
N(-d2):
0.18396
Call Price:
\$ 6.195
Put Price:
\$ 0.525
Call option delta =
Put option delta =
0.84741
-0.15259
16-39
The Call &quot;Delta&quot; Prediction:
• Call delta = 0.84741
• if the stock price increases by \$1, the call
option price will increase by about \$0.85
• From the previous example:
• Stock price = \$50
• Call option price = \$6.195
• If the stock price is \$51:
• Call option value = \$7.060
16-40
The Put &quot;Delta&quot; Prediction:
• Put delta value = -0.15259
• if the stock price increases by \$1, the put
option price will decrease by \$0.15.
• From the previous example:
• Stock price = \$50
• Put option price = \$0.525
• If the stock price is \$51:
• Put option value = \$0.390
16-41
Hedging with Stock Options
• You own 1,000 shares of XYZ stock and you want
protection from a price decline.
• Let’s use stock and option information from before—
in particular, the “delta prediction” to help us hedge.
• You want changes in the value of your XYZ shares
to be offset by the value of your options position.
That is:
Change in stock price  shares  Change in option price  # options
Change in stock price  shares  Option Delta  # options
16-42
Hedging Using Call Options—
The Prediction
• Delta = 0.8474; stock price declines by \$1:
Change in stock price  shares  Delta  # options
- 1  1,000  0.8474  # options
# options  - 1,000 / 0.8474  - 1,180.08
- 1,180.08 / 100  - 12.
Write 12 call options with a \$45 strike to hedge your
stock.
16-43
Hedging with Calls - Results
• Call option gain nearly offsets your loss of \$1,000.
• Why is it not exact?
• Call Delta falls when the stock price falls.
• Therefore, you did not sell quite enough call options.
Stock Price
Portfolio Value
Call Price
Call Position
Value
\$50
\$50,000
\$6.200
-\$7,440
\$49
\$49,000
\$5.370
-\$6,444
-\$1
(\$1,000)
-\$0.830
\$996
16-44
Hedging Using Put Options—The
Prediction
• Delta = -0.1526; stock price declines by \$1:
Change in stock price  shares  Delta  # options
- 1  1,000  - 0.1526  # options
# options  - 1,000 / - 0.1526  6,553.08
6,553.08 / 100  66.
Buy 66 put options with a strike of \$45 to hedge your
stock.
16-45
Hedging Using Put Options:
Results
• Put option gain more than offsets \$1,000 loss
• Why is it not exact?
• Put Delta also falls (gets more negative) when the stock
price falls.
• Therefore, you bought too many put options—this error
is more severe the lower the value of the put delta.
• To get closer: Use a put with a strike closer to at-themoney.
Stock Price
Portfolio Value
Put Price
\$50
\$50,000
\$0.530
Put Position
Value
\$3,498
\$49
\$49,000
\$0.700
\$4,620
-\$1
(\$1,000)
\$0.170
\$1,122
16-46
Hedging a Portfolio with Index
Options
• Many institutional money managers use stock index
options to hedge equity portfolios
Number of option contracts 
Portfolio beta  Portfolio value
Option delta  Option contract value
• Regular rebalancing needed to maintain an
effective hedge
• Underlying Value Changes
• Option Delta Changes
• Portfolio Value Changes
• Portfolio Beta Changes
16-47
Hedging with Stock Index
Options
You manage a \$10 million stock portfolio.
You attempt to maintain a portfolio beta of 1.00
index put options with a contract value of
\$125,000 and a delta of 0.579.
How many contracts do you need to buy?
Portfolio beta  Portfolio value
Number of option contracts 
Option delta  Option contract value
16-48
Hedging with Stock Index
Options
Portfolio value = \$10 million
Portfolio beta =
Index contract value = \$150,800
Number of option contracts 
Number of option contracts 
1.00
Option delta = 0.579
Portfolio beta  Portfolio value
Option delta  Option contract value
Portfolio Beta  Portfolio Value
Option Delta  Underlying Value  100
1.00  10,000,000

 115
0.579  1508  100
Sell 115 call options
16-49
Implied Standard Deviations
• Implied standard deviation (ISD)
• = Implied volatility (IVOL)
• Stock price volatility estimated from an option
price
• Of the six BSOPM input factors, only stock price
volatility is not directly observable
• Calculating an implied volatility requires:
• All 5 other input factors
• Either a call or put option price
16-50
CBOE Published Implied Volatilities for
Stock Indexes
• CBOE publishes 3 volatility indexes:
• S&amp;P 500 Index Option Volatility (VIX)
• S&amp;P 100 Index Option Volatility (VXO)
• Nasdaq 100 Index Option Volatility (VXN)
• Each volatility index calculated using ISDs from
eight options:
• 4 calls and 4 puts, each with two maturity dates:
• 2 slightly out of the money
• 2 slightly in the money
• Provides investors with information about market volatility in
the coming months.
16-51
Employee Stock Options, ESOs
• ESO = call option that a firm grants (gives)
to employees.
• ESO life generally 10 years
• Cannot be sold
• “Vesting” period of about 3 years
• If employee leaves the company before “vested,&quot;
the ESOs are lost
• Once vested, ESOs can be exercised any time
over its remaining life
16-52
52
Why are ESOs Granted?
• Motivates employees to make decisions
that help the stock price increase
• Powerful motivator
• Payoffs can be large.
• High stock prices: ESO holders gain and
shareholders gain.
• No upfront costs to the company
• Can act as a substitute for ordinary wages
16-53
53
ESO Repricing
• Generally issued exactly “at the money”
• Intrinsic value = zero.
• No value from immediate exercise.
• Still valuable
• “Underwater” Options
• Stock price falls after the ESO is granted
• “Restriking” or “Repricing.”
• Companies lower the strike prices
• Controversial practice
16-54
54
ESO Repricing Controversy
• PRO: Once an ESO is “underwater,” it loses
its ability to motivate employees.
• Small chance for payoff
• Employees may leave to get “fresh” options.
• CON: Lowering strike price = reward for
failing
• Decisions by employees made the stock price fall.
• If employees know that ESOs will be repriced, the
ESOs loose their ability to motivate employees.
16-55
55
ESOs Today
• Most companies award ESOs on a regular
basis.
• Quarterly
• Annually
• Therefore, employees will always have some
“at the money” options.
• Regular grants of ESOs mean that
employees always have some “unvested”
ESOs—giving them the added incentive to
remain with the company.
16-56
Valuing Employee Stock Options
• Companies must report estimates of ESO
values (FASB 123)
• BSOPM modified by Merton widely used for
this purpose
• For example:
• In December 2002, the Coca-Cola Company
granted ESOs with a stated life of 15 years.
• ESOs often exercised before maturity, so CocaCola also used a life of 6 years to value these
ESOs.
16-57
Employee Stock Option
Valuation
The Black-Scholes-Merton Option Pricing Model
incorporates the possibility that the underlying stock will
pay a dividend during the life of the option:
“y” = the annual dividend yield for the underlying stock
C  Se
 yT
N( d1 )  Ke
 rT
N( d 2 )
ln( S / K )  ( r  y   2 / 2 )T
d1 
 T
d 2  d1   T
“S” equals the stock’s price at the grant date
16-58
Example: Valuing Coca-Cola
ESOs Using Excel
Stock Price:
Discounted Stock:
44.55
35.10
Stock Price:
Discounted Stock:
44.55
40.23
Strike Price:
Discounted Strike:
44.66
19.13
Strike Price:
Discounted Strike:
44.66
36.41
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
25.53
15
5.65
1.59
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
30.20
6
3.40
1.70
d(1):
N(d1):
1.10792
0.86605
d(1):
N(d1):
0.50458
0.69307
d(2):
N(d2):
0.11915
0.54742
d(2):
N(d2):
-0.23517
0.40704
Call Price:
\$ 13.06
Call Price:
\$ 19.92
16-59
Summary: Coca-Cola
Employee Stock Options
16-60
Useful Websites
•
•
•
•
•
•
•
•
www.jeresearch.com (information on option formulas)
www.option-price.com (for a free option price calculator)