1/6/2015
SITUATION
LOOKING AT EXERCISE AND MARKET VALUE OF AN OPTION
Strike) price = $25
Price of Strike Exercise
the stock Price Value
$0 $20.00 $0.00
$5 $20.00 $0.00
$25
Chapter 5. Mini Case for Financial Options
(2.) What happens to the option’s time value (the difference between the option price and its exercise value)
as the stock price rises? The time value falls as the stock price increases; see the graph below. Why?
Answer: See Chapter 05 Mini Case Show
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To begin, you gathered some outside materials on the subject and used these materials to draft a list of
pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a
question-and-answer format. Now that the questions have been drafted, you have to develop the answers.
Assume that you have just been hired as a financial analyst by Triple Play Inc., a mid-sized California
company that specializes in creating high-fashion clothing. Since no one at Triple Play is familiar with the
basics of financial options, you have been asked to prepare a brief report that the firm’s executives could
use to gain at least a cursory understanding of the topics.
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d. Consider a stock with a current price of P = $27. Suppose that over the next 6 months the stock price will
(1.) Using the binomial model, what are the ending values of the stock price? What are the payoffs of the
(1.) What are the corresponding exercise values and option time values?
Exercise Values Option Time Values
Strike Price=
a. What is a financial option? What is the single most important characteristic of an option? Answer: See
b. Options have a unique set of terminology. Define the following terms: (1) call option; (2) put option; (3)
exercise price; (4) striking, or strike price; (5) option price; (6) expiration date; (7) exercise value; (8) writing
c. Consider Triple Play’s call option with a $25 strike price. The following table contains historical values
for this option at different stock prices:
Suppose a stock has the strike price shown below. The Exercise Value is the profit if you choose to exercise
Stock Price Option Price
$25.00
$50.00
Exercise Value vs. Stock Price
$25.00
$30.00
Exercise Values and Option Time Values vs. Stock Price
Current stock price, P = $27.00
Binomial Payoffs
Strike price: X = $25.00
Ending “up” stock price = P (u) = $38.07
Option payoff: C
u
= MAX[0,P(u)-X] = $13.07
Current
stock price
d
N
s
=
C
u
– C
d=0.69153
P(u – d)
The Hedge Portfolio with Riskless Payoffs
Strike price: X = $25.00
Current stock price: P = $27.00
Up factor for stock price: u = 1.41
Down factor for stock price: d = 0.71
Up option payoff: C
u
= MAX[0,P(u)-X] = $13.07
Down option payoff: C
d
=MAX[0,P(d)-X] = $0.00
Number of shares of stock in portfolio: N
s
= (C
u
– C
d
) / P(u-d) = 0.69153
N =
182.5
I/YR =
0.0164%
PMT =
0
call option?
(2.) Suppose you write 1 call option and buy Ns shares of stock. How many shares must you buy to create a
portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio?
(3.) What is the present value of the hedge portfolio’s riskless payoff? What is the value of the call option?
The present value of the riskless payoff disounted at the risk-free rate (we assume daily
compounding) is:
N
s
= 0.69153
Amount borrowed = PV of riskless payoff = $12.86
Repayment of riskless payoff = $13.26
Payoff if stock is up:
(4.) What is a replicating portfolio? What is arbitrage?
BLACK-SCHOLES OPTION PRICING MODEL
Key Inputs:
Key output:
V
Now, we will use the formula from above to solve for d
)
)
.
e. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes Option Pricing Model (OPM).
In deriving this option pricing model, Black and Scholes made the following assumptions:
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:
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
e. (2.) Write out the three equations that constitute the model.
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.
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 standard normal distribution. Luckily, Excel is equipped with a
function that can determine cumulative probabilities of the 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.
e. (3.) What is the value of the following call option according to the OPM?
Looking at these equations we see that you must first solve d
1
and d
2
before you can proceed to value the
option.
This model is widely used by options traders and is generally considered to be the standard for option
pricing. Many hand-held calculators and computer programs have this formula permanently stored in. We
now use Excel to write a “program”, if you will, for the Black-Scholes pricing model in Excel.
Using the NORMSDIST function:
(d
1
) = 0.6851
(d
2
) = 0.5539
(1.) Current stock price
(2.) Strike price
(3.) Option’s term to maturity
(4.) Risk-free rate
t = 0.5 r
rF
= 6%
s
2
= 0.11
Price of Strike Exercise Option
the stock
Price
Value
Price
$0 $25 $0.00 0.0000
Using the Black-Scholes formula and the cumulative distributions, we can solve for the option value.
EFFECTS OF OPM FACTORS ON THE VALUE OF A CALL OPTION
f. What impact does each of the following call option parameters have on the value of a call option?
Let us now turn our attention to determining how sensitive the call option value is to the five factors of the
Black-Scholes OPM. We will set up data tables for each factor determining the call value if the specified
input is changed plus or minus 15% and 30%.
Data
Change the inputs below to see the impact on the option’s price (X=25 for all cases).
$25.00
Option Pricing: Sensitivity Analysis