1. The exercise price on one of Flanagan Company's options is $14, its exercise value is $23, and its time value is $6. What are the option's market value and the price of the stock? Market value $ Price of the stock $ 2. Suppose you believe that Delva Corporation's stock price is going to decline from its current level of $82.50 sometime during the next 5 months. For $510.25 you could buy a 5-month put option giving you the right to sell 100 shares at a price of $85 per share. If you bought this option for $510.25 and Delva's stock price actually dropped to $60, what would your pre-tax net profit be? a. $2,193.70 b. $2,089.24 c. −$510.25 d. $2,303.38 e. $1,989.75 3. The current price of a stock is $50, the annual risk-free rate is 6%, and a 1-year call option with a strike price of $55 sells for $7.20. What is the value of a put option, assuming the same strike price and expiration date as for the call option? a. $7.71 b. $7.33 c. $8.55 d. $9.00 e. $8.12 4. A call option on the stock of Bedrock Boulders has a market price of $7. The stock sells for $29 a share, and the option has an exercise price of $25 a share.What is the exercise value of the call option? $ What is the option's time value? $ 5. Deeble Construction Co.'s stock is trading at $30 a share. Call options on the company's stock are also available, some with a strike price of $25 and some with a strike price of $35. Both options expire in three months. Which of the following best describes the value of these options? a. The options with the $25 strike price will sell for $5. b. The options with the $25 strike price have an exercise value greater than $5. c. If Deeble's stock price rose by $5, the exercise value of the options with the $25 strike price would also increase by $5. d. The options with the $35 strike price have an exercise value greater than $0. e. The options with the $25 strike price will sell for less than the options with the $35 strike price. Ch08 P08 Build a Model Spring 1, 20137/22/12Chapter 8. Ch 08 P08 Build a ModelExcept for charts and answers that must be written, only Excel formulas that use cell references or functions will be accepted for credit. Numeric answers in cells will not be accepted.You have been given the following information on a call option on the stock of Puckett Industries:P =$65X =$70t =0.5rRF =4%s =50.00%a. Using the Black-Scholes Option Pricing Model, what is the value of the call option?First, we will use formulas from the text to solve for d1 and d2.Hint: use the NORMSDIST function.(d1)=N(d1) =(d2)=N(d2) =Using the formula for option value and the values of N(d) from above, we can find the call option value.VC=b. Suppose there is a put option on Puckett's stock with exactly the same inputs as the call option. What is the value of the put?Put option using Black-Scholes modified form ...

1. 1. The exercise price on one of Flanagan Company's options is
$14, its exercise value is
$23, and its time value is $6. What are the option's market value
and the price of the
stock?
Market value $
Price of the stock $
2. Suppose you believe that Delva Corporation's stock price is
going to decline from its
current level of $82.50 sometime during the next 5 months. For
$510.25 you could buy a
5-month put option giving you the right to sell 100 shares at a
price of $85 per share. If
you bought this option for $510.25 and Delva's stock price
actually dropped to $60, what
would your pre-tax net profit be?
a. $2,193.70
b. $2,089.24
c. −$510.25
d. $2,303.38
e. $1,989.75

2. 3. The current price of a stock is $50, the annual risk-free rate
is 6%, and a 1-year call
option with a strike price of $55 sells for $7.20. What is the
value of a put option,
assuming the same strike price and expiration date as for the
call option?
a. $7.71
b. $7.33
c. $8.55
d. $9.00
e. $8.12
4. A call option on the stock of Bedrock Boulders has a market
price of $7. The stock
sells for $29 a share, and the option has an exercise price of $25
a share.What is the
exercise value of the call option?
$
What is the option's time value?
$

3. 5. Deeble Construction Co.'s stock is trading at $30 a share.
Call options on the
company's stock are also available, some with a strike price of
$25 and some with a strike
price of $35. Both options expire in three months. Which of the
following best describes
the value of these options?
a. The options with the $25 strike price will sell for $5.
b. The options with the $25 strike price have an exercise value
greater than $5.
c. If Deeble's stock price rose by $5, the exercise value of the
options with the $25
strike price would also increase by $5.
d. The options with the $35 strike price have an exercise value
greater than $0.
e. The options with the $25 strike price will sell for less than
the options with the $35
strike price.
Ch08 P08 Build a Model Spring 1, 20137/22/12Chapter 8. Ch
08 P08 Build a ModelExcept for charts and answers that must
be written, only Excel formulas that use cell references or
functions will be accepted for credit. Numeric answers in cells
will not be accepted.You have been given the following
information on a call option on the stock of Puckett Industries:P

4. =$65X =$70t =0.5rRF =4%s =50.00%a. Using the Black-
Scholes Option Pricing Model, what is the value of the call
option?First, we will use formulas from the text to solve for d1
and d2.Hint: use the NORMSDIST function.(d1)=N(d1)
=(d2)=N(d2) =Using the formula for option value and the values
of N(d) from above, we can find the call option value.VC=b.
Suppose there is a put option on Puckett's stock with exactly the
same inputs as the call option. What is the value of the put?Put
option using Black-Scholes modified formula =Put option using
put-call parity =
Sheet27/22/12
Chapter8/13/10Chapter 8. Tool Kit for Financial Options and
Applications in Corporate FinanceFINANCIAL OPTIONS
(Section 8.1)An 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

5. 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 the opportunity 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 that of 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 that of 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 that of 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

6. put.Table 8-1January 8, 2010, 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.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
opiotn$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$0.00THE SINGLE-PERIOD BINOMIAL OPTION
PRICING MODEL (Section 8.2)Consider a call option on a
stock. The stock's current price, denoted by P, is $40 and the
strike price, denoted by X, is $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. =$7.71Risk-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 8-1: Binomial
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 8-2: The 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,Portoflio's stock
payoff: = P(u)(Ns) =$41.67currentSubtract option's payoff: Cu
=$15.00stock pricePortoflio's net payoff = P(u)Ns - Cu
=$26.67$40Stock price = P (d) =$32.00Portoflio's stock payoff:
= P(d)(Ns) =$26.67Subtract option's payoff: Cd
=$0.00Portoflio's net payoff = P(d)Ns - Cd =$26.67Step 3. Find

8. 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: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 =0.50Number of periods until expiration, n =1
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

9. 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 =$7.71THE SINGLE-PERIOD BINOMIAL
OPTION PRICING FORMULA (Section 8.3)The 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 =0.50Number of periods until
expiration, n =1
Mike Ehrhardt: Do not change this input.VC =$7.71The
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

10. model by define 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 =0.5141 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 =$15.00x0.5141+$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
=$20.00x0.5141+$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. THE MULTI-PERIOD
BINOMIAL OPTION PRICING MODEL (Section 8.4)Suppose
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, s =31.5573%31.5573%Years 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

11. =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.(1 + rRF/365)365(T/n) =1.0202
pu =0.5140 pd =0.4662Applying 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 8-3: The 2-Period Binomial Lattice and
Option ValuationStandard deviation of stock return: s
=31.557%Current 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: pu =0.51400Price of $1 payoff if stock
goes down: pd =0.46621Now3 months6 monthsStock = P (u) (u)
=$54.84Cuu = Max[P(u)(u) − X, 0] =$19.84Stock = P (u)
=$46.84Cu = Cuupu + Cudpd =$12.53 P =$40.00Stock = P (u)
(d) = P (d) (u) =$40.00VC=Cupu+Cdp =$7.64Cud = Cdu =
Max[P(u)(d) − X, 0] =$5.00Stock = P (d) =$34.16Cd = Cudpu +
Cddpd =$2.57Stock = P (d) (d) =$29.17Cdd = Max[P(d)(d) −
X, 0] =$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

12. 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.BLACK-SCHOLES OPTION PRICING MODEL
(Section 8.5)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 + s2 /2) ] t } / (s
t1/2)d2 =d1 - s (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 (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 (d2). s is the volatility of the
stock price, as measured by the standard deviation. Looking at

13. 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 =31.557%Now, we will use the formula from
above to solve for d1.d1=0.8892Having solved for d1, we will
now use this value to find d2.d2=0.6661At 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)=0.7473By applying this method for
cumulative distributions, we can solve for the option value
using the formula above.VC=$7.39EFFECTS 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 yearsuntil expiration, 1 has 6 months
(0.5 years), and 1 has 3 months (0.25 years).Figure 8-5. Western
Cellular’s Call Options with a Strike Price of $35Data for the
figure.Time until expiration10.50.25Stock
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$16.51$

14. 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 sWe
keep all inputs constant except the standard deviation:Standard
deviationCall option
price0.001%$6.3710.000%$6.3831.557%$7.3940.000%$8.0760.
000%$9.8790.000%$12.70The impact of changes in the risk-
free rateWe keep all inputs constant except the risk-free
rate:Risk-free rate (rRF)Call option
price0%$6.414%$6.898%$7.3912%$7.9020%$8.93THE
VALUATION OF PUT OPTIONS (Section 8.6)Consider 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 4 Web Extension 4C). 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
0.0049 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0

15. 27.5 30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5
55.0 57.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 2.5 5.0 7.5 10.0 12.5
15.0 17.5 20.0 22.5 0.0049 2.5 5.0 7.5 10.0 12.5
15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0
42.5 45.0 47.5 50.0 52.5 55.0 57.5 1.51506222467964E-
173 8.551037404862E-17 1.0707398540399E-9
1.78882936371088E-6 0.000135852498602879
0.00232090633119555 0.0170078188603651
0.0734174602902471 0.222652674121347
0.527670943536376 1.046402594016459
1.818079352219861 2.857848945107179
4.158883709038092 5.698556991630983
7.445448529226486 9.365239188459078
11.4247939402102 13.59450307065918
15.84928390461534 18.16870054198051
20.53658479836515 22.94043090090215
25.37073506523161 0.0049 2.5 5.0 7.5 10.0 12.5
15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0
42.5 45.0 47.5 50.0 52.5 55.0 57.5 0.0
2.03437166585549E-32 2.1793900216619E-18
4.44229145562589E-12 1.93203845722943E-8
4.33424982656166E-6 0.000181250982217191
0.00265642188932642 0.0193802973823096
0.0870366929662917 0.275893968805316
0.677053621542877 1.372389235035076
2.407580349095648 3.783161253339091
5.463147158467155 7.391742525158513
9.509039115814772 11.76129127512468
14.10548320216353 16.50986325103071
18.95237496450883 21.41840110800933
23.89860854715915 0.0049 2.5 5.0 7.5 10.0 12.5
15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0
42.5 45.0 47.5 50.0 52.5 55.0 57.5 0.0
3.10068157347839E-63 2.41111671358491E-35
7.21930486657584E-23 1.01084729435005E-15

16. 3.81606506301323E-11 5.03845361193844E-8
8.19395127732203E-6 0.000330518959587531
0.0050541777747499 0.0385454872591215
0.17667576564412 0.555431161175346
1.319031421495138 2.54306417 2982212
4.203078489765005 6.204557603445071
8.43508293014844 10.8008922565077
13.23864514390552 15.71147265743971
18.20020902528756 20.69574019842843
23.19403160206953
Stock Price ($)
$
Exercise Value
T = 1
T = 0.5
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 0.7by 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

17. 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. 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 expiration1
Mike Ehrhardt: The number of periods per year may not be
changed by the user.Risk-free rate, rRF =6%Binomial lattice of
stock prices:P(u) =$34.50P =$30.00P(d) =$25.50Cu =$2.50Cd
=$0.00 pu =0.5422 pd =0.4429Current value of option:VC
=$1.36
8.4SECTION 8.4SOLUTIONS TO SELF-TEST Ringling Cycle’s
stock price is now $20. You need to find the value of a call
option with an strike price of $22 that expires in 2 months. You
want to use the binomial model with 2 periods (each period is a
month). Your assistant has calculated that u = 1.1553, d =
0.8656, pu = 0.4838, and pd = 0.5095. Draw the binomial lattice
for stock prices. What are the possible prices after 1 month?
After 2 months? What are the option's possible payoffs at
expiration? What will the option's value be in 1 month if the

18. stock goes up? What will the option's value be in 1 month if the
stock price goes down? What is the current value of the option
(assume each month is 1/12 of a year)? Previous work done by
your assistant:Annual standard deviation of stock return, s
=50.000%Years to expiration, T =0.1667Number of periods per
year, n =2
Mike Ehrhardt: The number of periods per year may not be
changed by the user.u =1.1553d =0.8656Risk-free rate, rRF
=8%(1 + rRF/365)365(T/n) =1.0067 pu =0.4838 pd
=0.5095Inputs for ProblemCurrent stock price$20.00Strike
price$22.00u =1.1553d =0.8656Time in years to
expiration0.1667Number of periods until expiration2pu
=0.4838pd =0.5095Binomial lattice of stock prices:P(u)(u)
=$26.69P(u) =$23.11P =$20.00P(u)(d) = P(d)(u) =$20.00P(d)
=$17.31P(d)(d) =$14.99Option payoffs at expirationCuu
=$4.69Cud =$0.00Cdd =$0.00Value of option in 1 month if
stock goes up:Cu =$2.27Value of option in 1 month if stock
goes up:Cd =$0.00Current value of option:VC =$1.10
8.5SECTION 8.5SOLUTIONS TO SELF-TEST What is the
value of a call option with these data: P = $35, X = $25, rRF =
6%, t = 0.5 (6 months), and s = 0.6?
P$35X$25rRF6.0%t0.50s0.6(d1)1.076(d2)0.652N(d1)0.8590N(d
2)0.7427V =$12.05
8.6SECTION 8.6SOLUTIONS TO SELF-TEST A put option
written on the stock of Taylor Enterprises (TE) has an exercise
price of $25 and six months remaining until expiration. The
risk-free rate is 6 percent. A call option written on TE has the
same exercise price and expiration date as the put option. TE's
stock price is $35. If the call option has a price of $12.05, what
is the price (i.e., value) of the put option? P =$35.00X
=$25.00rRF =6.00%t =0.50V (call price) =$12.05Put =$1.31
n
/
t
e

19. u
s
=
n/t
eu
u
1
d
=
u
1
)
n
/
t
(
365
RF
)
n
/
t
(
365
RF
u
)
365
/
r
1
(
d

20. u
d
)
365
/
r
1
(
+
ú
ú
û
ù
ê
ê
ë
é
-
-
+
=
p
)
n
/
t
(
365
RF
)
n
/
t
(
365
RF

21. d
)
365
/
r
1
(
d
u
)
365
/
r
1
(
u
+
ú
ú
û
ù
ê
ê
ë
é
-
+
-
=
p
)
n
/
t
(
365

22. RF
)
n
/
t
(
365
RF
d
)
n
/
t
(
365
RF
u
C
)
365
/
r
1
(
d
u
)
365
/
r
1
(
u
C
d
u

23. d
)
365
/
r
1
(
C
V
+
ú
ú
û
ù
ê
ê
ë
é
-
+
-
+
ú
ú
û
ù
ê
ê
ë
é
-
-
+
=