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12/10/2012
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
$10 $20.00 $0.00
$25
a. What is a financial option? What is the single most important characteristic of an option? Answer: See Chapter 08
Mini Case Show
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 the stock.
If the current price of the stock is greater than the strike price, then the Exercise Value is the current stock price minus
the strike price; otherwise, it is zero (you would never exercise the option if the stock price is less than the strike price.)
(1.) What are the corresponding exercise values and option time values?
Strike Price=
Chapter 8. Mini Case for Financial Options and Applications in Corporate Finance
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.
$45.00
$50.00
Exercise Value vs. Stock Price
Current stock price, P = $27.00
Risk-free rate, rRF = 6%
Strike price, X = $25.00
Binomial Payoffs
Strike price: X = $25.00
Current stock price: P = $27.00
Up factor for stock price: u = 1.41
Stock Price
Option Price
$25.00
Exercise Values
Option Time Values
d. Consider a stock with a current price of P = $27. Suppose that over the next 6 months the stock price will either go
up by a factor of 1.41 or down by a factor of 0.71. Consider a call option on the stock with a strike price of $25 which
expires in 6 months. The risk-free rate is 6%.
(1.) Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?
(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
08 Mini Case Show
$3.00
$0.00
$3.00
Exercise Values and Option Time Values vs. Stock Price
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: Cu = MAX[0,P(u)-X] = $13.07
Down option payoff: Cd =MAX[0,P(d)-X] = $0.00
Number of shares of stock in portfolio: Ns = (Cu - Cd) / P(u-d) = 0.69153
N = 182.5
I/YR = 0.0164%
PMT = 0
FV = ($13.26)
PV = $12.865 Using the PV function.
(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:
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 payoff
VC = $5.81
Ns = 0.69153
Amount borrowed = PV of riskless payoff = $12.86
Repayment of riskless payoff = $13.26
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:
If you borrow an amount equal to the present value of the riskless payoff and buy Ns shares of
stock, the payoffs of this portfolio replicate the payoffs of the call option.
(4.) What is a replicating portfolio? What is arbitrage?
BLACK-SCHOLES OPTION PRICING MODEL
Now, we will use the formula from above to solve for d1.
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 d1 and d2 before you can proceed to value the option.
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
e. (1.) What assumptions underlie the OPM?
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, or sigma, is the volatility
of the stock price, as measured by the standard deviation.
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.
3. The short-term, risk-free interest rate is known and is constant during the life of the option.
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:
(d1)=0.4819
Having solved for d1, we will now use this value to find d2.
(d2)=0.1355
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
Using the NORMSDIST function:
(d1) = 0.6851
(d2) = 0.5539
(1.) Current stock price
(2.) Strike price
t = 0.5
rrF = 6% s2 = 0.11
Price of Strike Exercise Option
the stock Price Value Price
$0 $25 $0.00 0.0000
$5 $25 $0.00 0.0000
$10 $25 $0.00 0.0001
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? Answer: See the
sensitivity analysis below; also, see Chapter 08 Mini Case Show
Data
$20.00
$25.00
Option Pricing: Sensitivity Analysis
$40 $25 $15.00 15.7819
$45 $25 $20.00 20.7490
($5.00)
$0.00
$5.00
$0 $20 $40 $60
Option Price
Stock Price
Exercise value Option price
