Financial options and applications in corporate finance chapter

Financial Options and Applications in Corporate Finance
CHAPTER 8
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classroom use.
Topics in Chapter
Financial Options Terminology
Option Price Relationships
Black-Scholes Option Pricing Model
Put-Call Parity
classroom use.
The Big Picture: The Value of a Stock Option
classroom use.
What is a financial option?
An option is a contract which gives its holder the right, but not
the obligation, to buy (or sell) an asset at some predetermined

price within a specified period of time.
classroom use.
What is the single most important
characteristic of an option?
It does not obligate its owner to take any action. It merely
gives the owner the right to buy or sell an asset.
classroom use.
Option Terminology (1 of 8)
Call option: An option to buy a specified number of shares of a
security within some future period.
Put option: An option to sell a specified number of shares of a
security within some future period.
classroom use.
Strike (or exercise) price: The price stated in the option
contract at which the security can be bought or sold.
Expiration date: The last date the option can be exercised.

classroom use.
Exercise value: The value of a call option if it were exercised
today =
Max[0, Current stock price - Strike price]
Note: The exercise value is zero if the stock price is less than
the strike price.
Option price: The market price of the option contract.
classroom use.
Time value: Option price minus the exercise value. It is the
additional value because the option has remaining time until it
expires.
classroom use.
Writing a call option: For every new option, there is an investor
who “writes” the option.
A writer creates the contract, sells it to another investor, and
must fulfill the option contract if it is exercised.

For example, the writer of a call must be prepared to sell a
share of stock to the investor who owns the call.
classroom use.
Covered option: A call option written against stock held in an
investor’s portfolio.
Naked (uncovered) option: An option written without the stock
to back it up.
classroom use.
In-the-money call: A call whose strike price is less than the
current price of the underlying stock.
Out-of-the-money call: A call option whose strike price
exceeds the current stock price.
classroom use.
LEAPS: Long-term Equity AnticiPation Securities that are
similar to conventional options except that they are long-term

options with maturities of up to 2 ½ years.
classroom use.
Consider the following data:
Strike price = $25.Stock PriceCall Option Price$25$3.00 30
7.50 3512.00 4016.50 4521.00 5025.50
classroom use.
Exercise Value of OptionPrice of
stock (a)Strike
price (b)Exercise value
of option (a)–(b)$25.00$25.00$0.0030.0025.00
5.0035.0025.0010.0040.0025.0015.0045.0025.0020.0050.0025.0
025.00
classroom use.
Market Price of OptionPrice of
stock (a)Strike
price (b)Exer.
val. (c)Mkt. Price
of opt. (d)$25.00$25.00$0.00 $3.0030.0025.00 5.00

7.5035.0025.0010.0012.0040.0025.0015.0016.5045.0025.0020.0
021.0050.0025.0025.0025.50
classroom use.
Time Value of OptionPrice of
stock (a)Strike
price (b)Exer.
Val. (c)Mkt. P of
opt. (d)Time value
(d) – (c)$25.00$25.00$0.00 $3.00$3.0030.0025.00 5.00 7.50
2.5035.0025.0010.0012.00 2.0040.0025.0015.0016.50
1.5045.0025.0020.0021.00 1.0050.0025.0025.0025.50 0.50
classroom use.
Call Time Value Diagram
classroom use.
Option Time Value Versus Exercise Value
The time value, which is the option price less its exercise value,
declines as the stock price increases.

This is due to the declining degree of leverage provided by
options as the underlying stock price increases, and the greater
loss potential of options at higher option prices.
classroom use.
The Binomial Model
Stock assumptions:
Current price: P = $27
In next 6 months, stock can either
Go up by factor of 1.41
Go down by factor of 0.71
Call option assumptions
Expires in t = 6 months = 0.5 years
Exercise price: X = $25
Risk-free rate: rRF = 6%
classroom use.
Binomial Payoffs at Call’s Expiration
classroom use.

Create portfolio by writing 1 option and buying Ns shares of
stock.
Portfolio payoffs:
Stock is up: Ns(P)(u) − Cu
Stock is down: Ns(P)(d) − Cd
classroom use.
The Hedge Portfolio with a Riskless Payoff
Set payoffs for up and down equal, solve for number of shares:
Ns= (Cu − Cd) / P(u − d)
In our example:
Ns= ($13.07 − $0) / $27(1.41 − 0.71)
Ns=0.6915
classroom use.
Riskless Portfolio’s Payoffs at Call’s Expiration: $13.26
classroom use.
Riskless payoffs earn the risk-free rate of return.
Find PV of riskless payoff. Discount at risk-free rate

compounded daily.
N = 0.5(365)
I/YR = 6/365
PMT = 0
FV = −$13.26 (because we want to know how much we would
want now to give up the FV)
PV = $12.87
classroom use.
Alternatively, use the PV formula
(daily compounding).
PV = $13.26 / (1 + 0.06/365)365*0.5
= $12.87
classroom use.
The Value of the Call Option
Because the portfolio is riskless:
VPortfolio = PV of riskless payoff
By definition, the value of the portfolio is:
VPortfolio = Ns(P) − VC
Equating these and rearranging, we get the value of the call:
VC = Ns(P) − PV of riskless payoff

classroom use.
Value of Call
VC = Ns(P) − Payoff / (1 + rRF/365)365*t
VC = 0.6915($27)
− $13.26 / (1 + 0.06/365)365*0.5
= $18.67 − $12.87
= $5.80
(VC = $5.81 if no rounding in any intermediate steps.)
classroom use.
Portfolio Replicating the Call Option
From the previous slide we have:
VC = Ns(P) − Payoff / (1 + rRF/365)365*t
The right side of the equation is the same as creating a portfolio
by buying Ns shares of stock and borrowing an amount equal to
the present value of the hedge portfolio’s riskless payoff (which
must be repaid).
The payoffs of the replicating portfolio are the same as the
option’s payoffs.
classroom use.
Replicating Portfolio Payoffs: Amount Borrowed and Repaid
Amount borrowed:
PV of payoff = $12.87

Repayment due to borrowing this amount:
Repayment = $12.87 (1 + rRF/365)365*t
Repayment = $13.26
Notice that this is the same as the payoff of the hedge portfolio.
classroom use.
Replicating Portfolio Net Payoffs
Stock up:
Value of stock = 0.6915($38.07) =$26.33
Repayment of borrowing = $13.26
Net portfolio payoff = $13.07
Stock down:
Value of stock = 0.6915($19.17) =$13.26
Repayment of borrowing = $13.26
Net portfolio payoff = $0
Notice that the replicating portfolio’s payoffs exactly equal
those of the option.
classroom use.
Replicating Portfolios and Arbitrage
The payoffs of the replicating portfolio exactly equal those of
the call option.
Cost of replicating portfolio
= Ns(P) − Amount borrowed
= 0.6915($27) − $12.87
= $18.67 − $12.87

= $5.80
If the call option’s price is not the same as the cost of the
replicating portfolio, then there will be an opportunity for
arbitrage.
classroom use.
Arbitrage Example
Suppose the option sells for $6.
You can write option, receiving $6.
Create replicating portfolio for $5.80, netting $6.00 −$5.80 =
$0.20.
Arbitrage:
You invested none of your own money.
You have no risk (the replicating portfolio’s payoffs exactly
equal the payoffs you will owe because you wrote the option.
You have cash ($0.20) in your pocket.
classroom use.
Arbitrage and Equilibrium Prices
If you could make a sure arbitrage profit, you would want to
repeat it (and so would other investors).
With so many trying to write (sell) options, the extra “supply”
would drive the option’s price down until it reached $5.80 and
there were no more arbitrage profits available.
The opposite would occur if the option sold for less than $5.80.

classroom use.
Multi-Period Binomial Pricing
If you divided time into smaller periods and allowed the stock
price to go up or down each period, you would have a more
reasonable outcome of possible stock prices when the option
expires.
This type of problem can be solved with a binomial lattice.
As time periods get smaller, the binomial option price
converges to the Black-Scholes price, which we discuss in later
slides.
classroom use.
Assumptions of the
Black-Scholes Option Pricing Model
The stock underlying the call option provides no dividends
during the call option’s life.
There are no transactions costs for the sale/purchase of either
the stock or the option.
Risk-free rate, rRF, is known and constant during the option’s
life.
classroom use.

Assumptions
Security buyers may borrow any fraction of the purchase price
at the short-term risk-free rate.
No penalty for short selling and sellers receive immediately full
cash proceeds at today’s price.
Call option can be exercised only on its expiration date.
Security trading takes place in continuous time, and stock prices
move randomly in continuous time.
classroom use.
What are the three equations that make up the OPM?
classroom use.
What is the value of the following call option according to the
OPM?
Assume:
P = $27
X = $25
rRF = 6%
t = 0.5 years
σ = 0.49

classroom use.
First, find d1.
classroom use.
Second, find d2.
d2 = d1
d2 = 0.4819
d2 = 0.1355
classroom use.
Third, find N(d1) and N(d2)
N(d1) = N(0.4819) = 0.6851
N(d2) = N(0.1355) = 0.5539
Note: Values obtained from Excel using NORMSDIST function.
For example:
N(d1) = NORMSDIST(0.4819)

classroom use.
Fourth, find value of option.
VC = $27(0.6851) - $25 e-(0.06)(0.5) (0.5539)
= $18.4977 - $13.4383
= $5.06
classroom use.
What impact do the following parameters have on a call
option’s value?
Current stock price: Call option value increases as the current
stock price increases.
Strike price: As the exercise price increases, a call option’s
value decreases.
classroom use.
Impact on Call Value (1 of 2)
Option period: As the expiration date is lengthened, a call
option’s value increases.
Longer time to expiration increases probability of very high
stock price, which has big payoff.
Also increases the probability of a very low stock price, but
payoff is zero for any price below the strike price.
Risk-free rate: Call option’s value tends to increase as rRF
increases (reduces the PV of the exercise price).

classroom use.
Impact on Call Value (2 of 2)
Stock return variance: Option value increases with variance of
the underlying stock.
Higher variance increases probability of very high stock price,
which has big payoff.
Also increases the probability of a very low stock price, but
payoff is zero for any price below the strike price.
classroom use.
Put Options
A put option gives its holder the right to sell a share of stock at
a specified stock on or before a particular date.
classroom use.
Put-Call Parity
Portfolio 1:
Put option,
Share of stock, P
Portfolio 2:

Call option, VC
PV of exercise price, X
classroom use.
Portfolio Payoffs at Expiration Date T for PT<X and PT≥X
PT<X PT≥XPort. 1Port.
2Port. 1Port. 2Stock PTPTPutX-PT0Call0PT-XCashX
XTotalXXPTPT
classroom use.
Put-Call Parity Relationship
Portfolio payoffs are equal, so portfolio values also must be
equal.
Put + Stock = Call + PV of Exercise Price
Put + P = VC + Xe-rRFt
Put = VC – P + Xe-rRFt
classroom use.
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Ethics in Research Presentation – Module 4
Purpose:
The purpose of this team presentation, like all assignments in
this course, is to help prepare you to be a successful nurse. The
assignments in this course are to challenge you, work on
collaborating as a team, empower you to research, and present
your findings to your peers.
Skills:
The purpose of this presentation is to help you practice the
following skills that are essential to your success in the course:
· Module Outcome #1: Examine the historical background that
led to the development of ethical guidelines for the use of
human subjects in research
Knowledge:
This assignment will also help you become familiar with the
following important content knowledge in this discipline:
· Communicate together as a team on ethical guidelines in the
use of human subjects in research
· Explore and research about the role ethics plays in research
Task:
1. Examine the historical background that led to the
development of ethical guidelines for the use of human subjects
in research. Analyze the assigned experiment/study to assess the
purpose of the study and methods/intervention employed on
subjects to identify ethical violations.
2. Your faculty will assign you to one of the following research
studies with another individual, you will work on this in a team.

a. Tudor Study aka Monster Study;
b. Little Albert;
c. NYSPI/Mount Sinai/CUNY Fen-Phen Study;
d. Tuskegee Syphilis Study
e. Stanford Prison Experiment;
f. Tearoom Trade/Sex Study;
g. Willowbrook Hepatitis Experiments/Study;
h. Sloan-Kettering Study;
i. Milgram Study;
j. Study of Response of Human Beings exposed to Significant
Beta and Gamma Radiation due to Fall-out from High-Yield
Weapons aka Project 4.1
k. Project MKULTRA, or MK-ULTRA;
l. Aversion Study
m. Unit 731;
n. Nazi Human Experimentations
o. Guatemala Syphilis Experiment
p. San Antonio Contraceptive Study
q. Henrietta Lacks
r. Gene Therapy Research: Case of Jesse Gelsinger
s. Human Plutonium Inject Experiments
t. Project Bluebird
u. Stateville Penitentiary Malaria Study
v. The Oregon and Washington Experiments: Effect of
Radiation on Testicles
3. Develop a presentation with voice overlay (create using
slideshow technology such as PowerPoint with voice overlay;
Adobe Spark, or VoiceThread, for example. Pro TIP: If you’re
interested in using Adobe Spark or VoiceThread, sign up with
your student email address and select the Free version for either
one).
4. Submit the presentation to the discussion forum by the end of
Week 7.
5. In Week 8 you will host, with your team the discussion by
reviewing other teams to your team presentation/concluding
questions and respond accordingly.

6. In turn, you will respond to a minimum of two team’s
presentations. For specifics review the criteria outlines below.
Required Presentation Components
1. A 8-12 slide presentation with narration.
2. The content of the presentation should include:
a. Title Slide (name and presentation title)
b. Introduction slide with an attention-getter introduction the
focus of the presentation.
c. A description of the research study:
i. Purpose
ii. Subjects/participants
iii. Research methods employed and intervention
iv. Credentials of researchers and affiliation (institution
supporting the research)
d. Implications on study participants (how were they affected
during and after study/experiment)
e. An explanation of 2 ethical principles that were violated.
f. Conclude presentation with the following questions for your
audience/classmates to address in Week 8 discussion forum:
i. Identification of two-three takeaways. In other words, what
did you learn about the ethical concept that you can apply to
your own decision making in the future?
ii. Do you think that such a violation could occur today – why
or why not (justify response).
Criteria for Success (Rubric)
Criteria
Points
Attention-Getter
/5
Description of research study
/20

Implications on Study Participants
/15
Discussion of Ethical Violations
/15
Conclusion Questions
/5
Scholarly Presentation & Posting to Discussion Forum on
Sunday of Week 7
/10
Hosting of Discussion During Week 8
/20
Response to two peer presentations.
/10
Total
/100
Clinical Inquiry – NURS 3270
Ethics in Research Presentation
1. Define each of the following terms:
Call option
Put option
Strike price or exercise price
Expiration date
Exercise value
Option price
Time value

Writing an option
Covered option
Naked option
In-the-money call
Out-of-the-money call
LEAPS
2. The current price of a stock is $50. In 1 year, the price will
be either $65 or $35. The annual risk-free rate is 10%. Find the
price of a call option on the stock that has an exercise price of
$55 and that expires in 1 year. (Hint: Use daily compounding.)
3. The exercise price on one of Chrisardan Company’s call
options is $20, its exercise value is $27, and its time value is
$8. What are the option’s market value and the price of the
stock?
Submit your answers in a Word document.
ChapterTool KitChapter 811/20/18 Financial Options and
Applications in Corporate Finance8-1 Overview of Financial
OptionsAn option is a contract which gives its holder the right
to buy (or sell) an asset at a predetermined price within a
specified period of time. Option contracts, though often quoted
in terms of single shares, usually are contracts for a 100 shares.
A call option describes a situation in which one investor may
sell to someone the right to buy his/her shares of a stock over

some interval of time. In this scenario, the writer of the call
option (the party that surrenders the right to exercise) is said to
hold a short position on the option. Meanwhile, the party that
has purchased this right to buy is said to hold a long position on
the option. The predetermined price that the stock may be
purchased for is called the strike, or exercise, price. When an
investor "writes" call options against stock held in his/her
portfolio, this is called a "covered call". When the call options
are written without the stock to back them up, they are they are
called "naked calls". When the strike price is below the current
market price, the call option is said to be "in-the-money".
Likewise, when the strike price exceeds the current market
price, the call option is said to be "out-of-the-money". For
instance, if you believed that the price of stock was primed to
rise, a call option would allow you to capture a profit off of the
rise in price.A put option allows you to buy the right to sell a
stock at a specified price within some future period. If you
happened to believe that the price of a stock was ready to fall, a
put option would allow you to turn a profit out of that decline.
In the cases of both call and put options, the profit or loss
made on an options transaction is determined by the value of the
underlying asset, the strike price of the option, and the price of
the option.FOR A CALL, AT EXPIRATIONIf the value of the
underlying asset exceeds the strike price, the profit/loss from
the call transaction would be equal to the difference between
the value of the asset and the strike price less the price of the
call. In this case there could be either a net profit or loss
depending upon the exercise value and the price of the call.If
the value of the underlying asset equals the strike price, the
profit/loss from the call transaction would be equal to the price
of the call, because whether exercised or unexercised the call
value would be zero. In this case there is a loss equal to the
price of the call.If the value of the underlying asset is less than
that of the strike price, the profit/loss from the call transaction
would be equal to the price of the call, because the option
would not be exercised if the strike price was greater than the

market price. In this case there is a loss equal to the price of
the call.FOR A PUT, AT EXPIRATIONIf the value of the
underlying asset is less than the strike price, the profit/loss
from the put transaction would be equal to the difference
between the strike price and value of the asset less the price of
the put. In this case there could be either a profit or loss
depending upon the exercise value and the price of the put.If the
value of the underlying asset equals that of the strike price, the
profit/loss from the put transaction would be equal to the price
of the call, because whether exercised or unexercised the put
value would be zero. In this case there is a loss equal to the
price of the put.If the value of the underlying asset exceeds that
of the strike price, the profit/loss from the put transaction
would be equal to the price of the put, because the option would
not be exercised if the market price was greater than the strike
price. In this case there is a loss equal to the price of the
put.Table 8-1January 8, 2019, Listed Options
QuotationsCALLS—LAST QUOTEPUTS—LAST
QUOTEClosing PriceStrike
PriceFebruaryMarchMayFebruaryMarchMayGeneral Computer
Corporation
(GCC)53.50504.254.755.500.651.402.2053.50551.302.053.152.
65r4.5053.50600.300.701.506.65r8.00U.S.
Biotec56.65555.256.108.002.253.75rFood
World56.65553.504.10r0.70rrNote: r means not traded on this
date.Suppose you purchase GCC's May call option with a strike
price of $50 and the stock price goes to $60. What is the rate of
return on the stock? What is the rate of return on the
option?Stock ReturnIntital stock price$53.50Final stock
price$60.00Rate of return on stock12.1%Call Option
ReturnIntital cost of option$5.50Market price of
stock$60.00Strike price$50.00Profit from exercise$10.00Rate of
Return81.8%Suppose you purchase GCC's May put option with
a strike price of $50 and the stock price goes to $45. What is
the rate of return on the stock? What is the rate of return on the
option?Stock ReturnIntital stock price$53.50Final stock

price$45.00Rate of return on stock-15.9%Put Option
ReturnIntital cost of option$2.20Market price of
stock$45.00Strike price$50.00Profit from exercise$5.00Rate of
Return127.3%What is the exercise value of GCC's May call
option with a strike price of $50? What is the exercise value of
GCC's May call option with a strike price of $55?Exercise
ValueStock price$53.50Strike price$50.00Exercise
value$3.50Stock price$53.50Strike price$55.00Exercise
value$0.008-2 The Single-Period Binomial Option Pricing
ApproachConsider a call option on a stock. The stock's current
price, denoted by P, is $40 and the strike price, denoted by X, i s
$35. The option expires in 6 months. The nominal annual risk-
free rate is 8%.PAYOFFS IN A SINGLE-PERIOD BINOMIAL
MODELAt expiration, the stock can take on only one of two
possible values. It can either go up in price by a factor of 1.25,
or down in price by a factor of 0.80.Inputs:Key output:Current
stock price, P =$40.00VC =$7.71Nominal annual risk-free rate,
rRF =8%Strike price, X =$35.00Up factor for stock price, u
=1.25Down factor for stock price, d =0.80Years to expiration, t
=0.50Number of periods until expiration, n =1
Mike Ehrhardt: Do not change this input.Consider the value of
the stock and the payoff of the option.Figure 8-1Binomial
PayoffsStrike price: X =$35.00Current stock price: P
=$40.00Up factor for stock price: u =1.25Down factor for stock
price: d =0.80Cu,Ending up ending upstock priceoption payoffP
(u) =MAX[P(u) − X, 0] ==A146*D135 =$50.00=MAX[D141−
D133,0] =$15.00P,VC,currentcurrentstock priceoption
price$40?Cd,Ending down ending downstock priceoption
payoffP (d) =MAX[P(d) − X, 0] ==A146*D136
=$32.00=MAX[D151− D133,0] =$0.00THE HEDGE
PORTFOLIO APPROACHWe can form a portfolio by writing 1
call option and purchasing Ns shares of stock. We want to
choose Ns such that the payoff of the portfolio if the stock price
goes up is the same as if the stock price goes down. This is a
hedge portfolio because it has a riskless payoff. Step 1. Find the

number of shares of stock in the hedge portfolio.Ns =Cu -
Cd=0.83333P(u - d)Step 2. Find the hedge portfolio’s payoff.If
the stock price goes up:Portoflio payoff =Ns (P)(u) -
Cu=$26.6667If the stock price goes down:Portoflio payoff =Ns
(P)(d) - Cd=$26.6667Figure 8-2The Hedge Portfolio with
Riskless PayoffsStrike price: X =$35.00Current stock price: P
=$40.00Up factor for stock price: u =1.25Down factor for stock
price: d =0.80Up option payoff: Cu = MAX[0,P(u)-X]
=$15.00Down option payoff: Cd =MAX[0,P(d)-X]
=$0.00Number of shares of stock in portfolio: Ns = (Cu - Cd) /
P(u-d) =0.83333Stock price = P (u) =$50.00P,Portfolio's stock
payoff: = P(u)(Ns) =$41.67currentSubtract option's payoff: Cu
=$15.00stock pricePortfolio's net payoff = P(u)Ns - Cu
=$26.67$40Stock price = P (d) =$32.00Portfolio's stock payoff:
= P(d)(Ns) =$26.67Subtract option's payoff: Cd
=$0.00Portfolio's net payoff = P(d)Ns - Cd =$26.67Step 3. Find
the present value of the hedge portfolio's riskless payoff.The
present value of the riskless payoff disounted at the risk-free
rate (we assume daily compounding) is:Frequency compounded
in year:365Nominal risk-free rate, rRF:8%Years to end of
binomial period:0.5Payoff of hedged portfolio:$26.6667Inputs
to PV function:N =182.5 = t(Frequency)I/YR =0.02192% = Rrf
/ FrequencyPMT =0FV =-$26.6667 Negative because the PV is
the amount we want in exchange for payoff of hedged
portfolioPV of payoff =$25.6212 Use the PV function:
=PV(I/YR,N,PMT,FV)Alternatively, you can use the present
value equation and adjust for the frequency of compounding:Pv
of payoff =Payoff=$26.6667=$25.6212(1 +
rRF/365)365*(t/n)1.04081Step 4. Find the option's current
value.The current value of the hedge portolio is the the stock
value (Ns x P) less the call value (VC). But the hedge portfolio
has a riskless payoff, so the hedge portfolio's value must also be
equal to the present value of the riskless payoff disounted at the
risk-free rate (we assume daily compounding). With a little
algebra, we get: VC =Ns (P) - Present value of riskless
payoffVC =$7.71THE REPLICATING PORTFOLIOIf a

portfolio can be formed such that is has the same cash flows as
an option, the the option value must equal the value of this
replicating portfolio. It is possible to replicate an option's cash
flows with a portfolio of stock and risk-free bonds, as we show
in the next section.Suppose we form a portfolio with Ns shares
of stock (as determined by the formula for the number of shares
of stock in the hedge portfolio). How much could we borrow so
that the net payoff from the stock and the repayment of the loan
(and its interest) has the same payoff as the
option?Inputs:Current stock price, P =$40.00Risk-free rate, rRF
=8%Strike price, X =$35.00Up factor for stock price, u
=1.25Down factor for stock price, d =0.80Years to expiration, t
Mike Ehrhardt: Do not change this input.Intermediate
calculations:Up payoff for stock, Pu =$50.00Down payoff for
stock, Pd =$32.00Cu =$15.00Cd =$0.00Ns =Cu - Cd=0.8333P(u
- d)If we form a portfolio with Ns shares of stock, how much
can we afford to borrow so that the portfolio's net payoff is
equal to the option's payoff?Value of stock in portfolio if up
=Ns P u=$41.67Cu =$15.00Amount of borrowing (plus interest)
that can be repaid =$26.6667Value of stock in portfolio if down
=Ns P d=$26.67Cd =$0.00Amount of borrowing (plus interest)
that can be repaid =$26.6667Notice that the amount of
borrowing (plus interest) that we can afford to repay is the same
whether the stock goes up or down. To find the amount we can
borrow, we find the present value fo the amount we can repay.
Option pricing assumes that interest rates are compounded very
frequently. We will assume daily compounding (which is a good
approximation for continuous compounding).Amount borrowed
=Amount repaid=25.6212(1 + rRF/365)365*(t/n)A summary of
the replicating portfolio value and payoff's is shown
below:Replicating Portfolio PayoffsNumber of shares of stock:
Ns =0.8333Current stock price: P =$40.00Up factor for stock
price: u =1.2500Up stock price: P(u) =$50.00Down factor for
stock price: d =0.8000Down stock price: P(d) =$32.00Risk-free

rate: rRF =8.00%Years to expiration: t =0.50Number of periods
until expiration: n =1Amount of principal and interest repaid
=$26.67Amount borrowed =$25.62(Ns) x (Pu) =$41.67Loan
repayment =$26.67Net portfolio payoff =$15.00Current value of
portfolio:(Ns) x (P) =$33.33Amount borrowed =$25.62Total
portfolio net cost =$7.71(Ns) x (Pd) =$26.67Loan repayment
=$26.67Net portfolio payoff =$0.00The call option has the same
cash flows as the replicating portfolio, so the call's price must
be equal to the value of the replicating portfolio:VC = Total
portfolio value =$7.718.3 The Single-Period Binomial Option
Pricing FormulaThe step-by-step hedge portfolio approach
works fine, but for problems in which you want to change the
inputs, it is easier to use the binomial option pricing formula
shown below. Inputs:P =$40.00X =$35.00u =1.25d =0.80Cu
=$15.00Cd =$0.00Risk-free rate, rRF =8%Years to expiration, t
Mike Ehrhardt: Do not change this input.VC =$7.7123The
Simplified Binomial Option Pricing FormulaP =$40.00X
=$35.00u =1.25d =0.80Cu =$15.00Cd =$0.00Risk-free rate, rRF
=8%Years to expiration, t =0.50Number of periods until
expiration, n =1
Mike Ehrhardt: Do not change this input.We can simplify the
model by defining pu and pd as:The binomial option pricing
model then simplifies to:VC = Cu pu + Cd pd For Western's 6-
month options, we have: pu =0.5142 pd =0.4466We can find the
value of Western's 6-month call with a $35 strike price:VC =Cu
x pu +Cd x pd VC =$15.00x0.5142+$0.00x0.4466VC
=$7.71Find the value of a 6-month call option with a $30 strike
price:x =$30.00Cu =MAX[0,Pu-X] =$20.00Cd =MAX[0,Pd-X]
=$2.00VC =Cu x pu +Cd x pd VC
=$20.00x0.5142+$2.00x0.4466VC =$11.18In fact, we can use
the p's to find the value of any security with payoffs that depend
on Western's 6-month stock price. 8-4 The Multi-Period
Binomial Option Pricing ModelSuppose we divide the year into

two 6-month periods. We will allow the stock to only go up or
down each period, but because there are more periods there will
be more possible stock prices. The key is to keep the standard
deviation of the stock's return the same as we divide the year
into smaller periods. If we know the standard deviation of the
stock's return and the number of periods, there is a formula that
will show us what u and d must be.s is the standard deviation of
stock return. Here are the formulas relating s to to u and d:The
standard deviation of Western's stock return is shown below.
Notice that this provides the values for u and d that we used in
the single-period model.Multi-periodSingle-periodAnnual
standard deviation of stock return, σ =0.3155730.315573Years
to expiration, t =0.50.5Number of periods prior to expiration, n
=2
Mike Ehrhardt: The number of periods per year may not be
changed by the user.
Mike Ehrhardt: Do not change this input.1
Mike Ehrhardt: The number of periods per year may not be
changed by the user.
Mike Ehrhardt: Do not change this input.u =1.17091.250d
=0.85400.8000Here are the other data for Western, taken from
the original problem:Current stock price, P =$40.00Risk-free
rate, rRF =8%Strike price, X =$35.00Because we are going to
solve a binomial problem repeatedly, it will be easier if we go
ahead and calculate the p's now. pu =0.51400 pd
=0.46620Applying these values of u and d to the intital stock
price gives the possible stock prices after 3 months. We can
then apply u and d to these 3-month values to get the stock
values at the end of 6 months, as shown below. Notice that
because d = 1/u, the "middle" stock value at the end of the year
is the same whether the stock initially went up and then went
down, or whether it went down and then went up.Notice that the

range of final outcomes at 6 months is wider than the previous
problem. However, the standard deviation of stock returns is
the same as before, because most of the time the stock price will
end up at the middle outcome rather than at the top or bottom
outcomes.Figure 8-3The 2-Period Binomial Lattice and Option
ValuationStandard deviation of stock return: σ
=0.315573Current stock price: P =$40.00Up factor for stock
price: u =1.1709Down factor for stock price: d =0.8540Strike
price: X =$35.00Risk-free rate: rRF =8.00%Years to expiration:
t =0.50Number of periods until expiration: n =2Price of $1
payoff if stock goes up: πu =0.51400Price of $1 payoff if stock
goes down: πd =0.46620Now3 months6 monthsStock = P (u) (u)
=$54.84Cuu = Max[P(u)(u) − X, 0]Cuu = $19.84Stock = P (u)
=$46.84Cu = Cuuπu + CudπdCu = $12.53Stock = P (u) (d) = P
(d) (u) P =$40.00Stock =$40.00VC = Cuπu + CdπdCud = Cdu =
Max[P(u)(d) − X, 0]VC =$7.64Cud = $5.00Stock = P (d)
=$34.16Cd = Cduπu + CddπdCd = $2.57Stock = P (d) (d)
=$29.17Cdd = Max[P(d)(d) − X, 0]Cdd = $0.00To find the
current value of the option, we can break the binomial lattice
into three problems. Problem #1 is to find the option value at
the end of six months, given that the stock moved upward from
its initial value. Problem #2 is to find the option value at the
end of six months, given that the stock moved downward from
its intitial value. Finally, problem #3 is to find the current
value of the option, given its two possible values at the end of
six months.In this example, we divided time into two periods.
If we were to divide time into more periods, we would get a
distribution of stock prices in the last period that would be very
realistic, which would give a very accurate option price. It is
true that dividing time into more periods would create more
binominal problems to solve, but each problem is very easy and
computers can solve them very quickly.8-5 The Black-Scholes
Option Pricing Model (OPM) In deriving this option pricing
model, Black and Scholes made the following assumptions:1.
The stock underlying the call option provides no dividends or
other distributions during the life of the option.2. There

are no transaction costs for buying or selling either the stock or
the option.3. The short-term, risk-free interest rate is known
and is constant during the life of the option.4. Any purchaser
of a security may borrow any fraction of the purchase price at
the short-term, risk-free interest rate.5. Short selling is
permitted, and the short seller will receive immediately the full
cash proceeds of today's price for a security sold short.6.
The call option can be exercised only on its expiration date.7.
Trading in all securities takes place continuously, and the stock
price moves randomly.The derivation of the Black-Scholes
model rests on the concept of a riskless hedge. By buying
shares of a stock and simultaneously selling call options on that
stock, an investor can create a risk-free investment position,
where gains on the stock are exactly offset by losses on the
option. Ultimately, the Black-Scholes model utilizes these
three formulas:VC =P[ N (d1) ] - X e-r t [ N (d2) ]Note: r is
the risk free rate, rRF.d1 ={ ln (P/X) + [rRF + σ2 /2) ] t } / (σ
t1/2)d2 =d1 - σ (t 1 / 2)In these equations, V is the value of
the option. P is the current price of the stock. N(d1) is the area
beneath the standard normal distribution corresponding to the
value of d1. X is the strike price. rRF is the risk-free rate. t is
the time to maturity. N(d2) is the area beneath the standard
normal distribution corresponding to the value of d2. σ is the
volatility of the stock price, as measured by the standard
deviation. Looking at these equations we see that you must first
solve d1 and d2 before you can proceed to value the
option.First, we will lay out the input data given earlier for
Western Cellular's call option.Inputs:Key Output:P =$40VC
=$7.39X =$35rRF =8.00%t =0.5s =0.31557Now, we will use
the formula from above to solve for
d1.d1=0.8892=(LN(B527/B528)+(B529+((B531^2)/2))*B530)/((
B531)*(B530^0.5))Having solved for d1, we will now use this
value to find d2.d2=0.6661=D535-(B531*(B530^0.5))At this
point, we have all of the necessary inputs for solving for the
value of the call option. We will use the formula for V from
above to find the value. The only complication arises when

entering N(d1) and N(d2). Remember, these are the areas under
the normal distribution. Luckily, Excel is equipped with a
function that can determine cumulative probabilities of the
standard normal distribution. This function is located in the list
of statistical functions, as "NORMSDIST". For both N(d1) and
N(d2), we will follow the same procedure of using this function
in the value formula. The data entries for N(d1) are shown
below.N(d1)=0.8131N(d2)=0.7473=NORMSDIST(D539)By
applying this method for cumulative distributions, we can solve
for the option value using the formula
above.VC=$7.39=(B527*D563)-(B528*EXP(-
B529*B530))*D566EFFECTS OF OPM FACTORS ON THE
VALUE OF A CALL OPTIONThe figure below shows 3 of
Westerns's call options, each with a $35 strike price. One option
has 1 year until expiration, 1 has 6 months (0.5 years), and 1
has 3 months (0.25 years).Figure 5-5Data for the figure.Time
until expirationWestern Cellular’s Call Options with a Strike
Price of $3510.50.25Stock
Price$5$9.37$7.39$6.20$0.00$0.00$0.00$0.00$0.00$2.50$0.00$
0.00$0.00$0.00$5.00$0.00$0.00$0.00$0.00$7.50$0.00$0.00$0.0
0$0.00$10.00$0.00$0.00$0.00$0.00$12.50$0.00$0.00$0.00$0.0
0$15.00$0.00$0.02$0.00$0.00$17.50$0.00$0.07$0.00$0.00$20.
00$0.00$0.22$0.02$0.00$22.50$0.00$0.53$0.09$0.01$25.00$0.
00$1.05$0.28$0.04$27.50$0.00$1.82$0.68$0.18$30.00$0.00$2.
86$1.37$0.56$32.50$0.00$4.16$2.41$1.32$35.00$0.00$5.70$3.
78$2.54$37.50$2.50$7.45$5.46$4.20$40.00$5.00$9.37$7.39$6.
20$42.50$7.50$11.42$9.51$8.44$45.00$10.00$13.59$11.76$10.
80$47.50$12.50$15.85$14.11$13.24$50.00$15.00$18.17$16.51$
15.71$52.50$17.50$20.54$18.95$18.20$55.00$20.00$22.94$21.
42$20.70$57.50$22.50$25.37$23.90$23.19The figure shows
that:1.Option prices increase as the stock price increases
relative to the strike price.2.Option prices increase as time to
expiration increases.3.Obviousy, an increase in the strike price
will cause the option price to fall.The impact of changes in
σ.We keep all inputs constant except the standard
deviation:Standard deviationCall option

price0.00001$6.370.10000$6.380.31557$7.390.40000$8.070.60
000$9.870.90000$12.70The impact of changes in the risk-free
rate.We keep all inputs constant except the risk-free rate:Risk-
free rate (rRF)Call option
price0%$6.414%$6.898%$7.3912%$7.9020%$8.938-6 The
Valuation of Put OptionsConsider two portfolios. The first has
a put option and a share of stock. The second has a call option
and cash equal to the present value of the strike price
(discounted with continuous compounding; see Chapter 04 Web
Extension 04C). What are the payoffs at expiration date T of
the two portfolios if the stock price is less than the strike price
at expiration? If it is above the strike price at
expiration?PT<XPT>=XPutX-PT0StockPTPTPortfolio
1:XPTCall0PT - XCashXXPortfolio 2:XPTAs the table shows,
the two porfolios have the same payoffs. Therefore, they must
have the same value today. This is called put-call parity.Put +
Stock = Call + PV of strike priceVP = VC - P + X exp(-rRF
t)Suppose you have the following information. What is the
value of the put?P =$40X =$35rRF =8%t =0.50V (call price)
=$7.39Put =VC-P+X exp(-rRF t)=$7.39-$40+33.63=$1.02If you
do not already have the value of the call option, you can use the
following formula to directly calculate the value fo the put.Put
=P[ N (d1) - 1 ] - X e-r t [ N (d2) -1 ]Note: r is the risk free
rate, rRF.The formulas for d1 and d2 are the same as for the
Black-Scholes call option model. In fact, the only differences
between the two models is that the formula for puts subtracts 1
from N(d1) and N(d2).Put =$1.02
4.8999999999999998E-3 2.5 5 7.5 10 12.5 15 17.5 20
22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50
52.5 55 57.5 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 2.5 5 7.5 10 12.5 15
17.5 20 22.5 4.8999999999999998E-3 2.5 5 7.5 10
12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40
42.5 45 47.5 50 52.5 55 57.5
1.5150622246796405E-173 8.551037223004708E-17
1.0707391850392869E-9 1.7888330135090295E-6

1.3585249859497206E-4 2.3209063311913428E-3
1.7007818860363233E-2 7.3417460290248315E-2
0.22265267412135037 0.52767094353637489
1.0464025940164552 1.8180793522198648
2.8578489451071842 4.1588837090380917
5.6985569916309835 7.4454485292264891
9.3652391884590784 11.424793940210201
13.594503070659179 15.849283904615337
18.16870054198052 20.536584798365151
22.940430900902154 25.370735065231614
4.8999999999999998E-3 2.5 5 7.5 10 12.5 15
17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45
47.5 50 52.5 55 57.5 0 2.034371665111521E-32
2.1793899956117436E-18 4.4422907445989101E-12
1.9320356049049042E-8 4.3342499037726459E-6
1.8125098221685731E-4 2.6564218893350283E-3
1.93802973823062 05E-2 8.7036692966296569E-2
0.27589396880531591 0.67705362154287307
1.3723892350350759 2.4075803490956478
3.7831612533390917 5.4631471584671552
7.3917425251585094 9.5090391158147796
11.761291275124691 14.105483202163523
16.509863251030701 18.952374964508827
21.418401108009341 23.898608547159135
4.8999999999999998E-3 2.5 5 7.5 10 12.5 15
17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45
47.5 50 52.5 55 57.5 0 3.1006815734593928E-63
2.4111167130696169E-35 7.2193048440695481E-23
1.0108472623198529E-15 3.8160640339364244E-11
5.0384429453763521E-8 8.1939514959356247E-6
3.3051895956191443E-4 5.0541777747451638E-3
3.8545487259123323E-2 0.17667576564411558
0.55543116117534286 1.3190314214951382
2.5430641729822128 4.2030784897650015
6.2045576034450711 8.435082930148436
10.800892256507687 13.238645143905515

15.711472657439714 18.200209025287549
20.695740198428446 23.194031602069494
Stock Price ($)
$
Exercise Value
VC if t = 1
VC if t = 0.5
VC if t = 0.25
8-1SECTION 8-1SOLUTIONS TO SELF-TEST Brighton
Memory's stock is currently trading at $50 a share. A call option
on the stock with a $35 strike price currently sells for $21.
What is the exercise value of the call option? What is the time
value?Stock price$50Strike price$35Market price of
option$21Exercise value of option$15.00Time value of
option$6.00
8-2SECTION 8-2SOLUTIONS TO SELF-TEST Lett
Incorporated's stock price is now $50 but it is expected to either
go up by a factor of 1.5 or down by a factor of 0.7 by the end of
the year. There is a call option on Lett's stock with a strike
price of $55 and an expiration date one year from now. What
are the stock's possible prices at the end of the year? What are
the call option's payoffs if the stock price goes up? If the stock
price goes down? If we sell one call option, how many shares of
Lett's stock must we buy to create a riskless hedged portfolio
consisting of the option position and the stock? What is the
payoff of this portfolio? If the annual risk free rate is 6
percent, how much is the riskless portfolio worth today
(assuming daily compounding)? What is the current value of
the call option?Inputs:Current stock price$50Strike
price$55u1.50d0.70Risk-free rate6%Time to exercise1.00Stock
price if u$75.00Stock price if d$35.00Option payoff if
u$20.00Option payoff if d$0.00N0.50Hedge portfolio payoff if
u$17.50Hedge portfolio payoff if d$17.50Portfolio value today

(PV of payoff)$16.48Current option value$8.52
8-3SECTION 8-3SOLUTIONS TO SELF-TEST Yegi's Fine
Phones has a current stock price of $30. You need to find the
value of a call option with a strike price of $32 that expires in 3
months. The annual risk-free rate is 6%. Use the binomial model
with 1 period until expiration. The factor for an increase in
stock price is u = 1.15; the factor for a downward movement is
d = 0.85. What are the possible stock prices at expiration? What
are the option's possible payoffs at expiration? What are pu and
pd? What is the current value of the option (assume each month
is 1/12 of a year)? Inputs for ProblemCurrent stock
price$30.00Strike price$32.00u =1.15d =0.85Time in years to
expiration0.25Number of periods until expiration

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