Slide 22.35 American Call
Lecture Tip: You may want to discuss the importance of arbitrage
in the valuation of options. The classic definition of arbitrage is
trading in more than one market simultaneously to earn a riskless
profit. It is designed to exploit price discrepancies between
markets. Risk arbitrage, on the other hand, is used to exploit the
apparent mispricing of stocks involved in a takeover. The “risk
arbitrageur” buys the stock of the firm being acquired and shorts
the stock of the acquiring firm. The goal is to profit from the
tendency of target firm prices to increase and acquiring firm
prices to decrease. The difference here is that there is risk involved
because there is no guarantee that the prices will move
“normally.”
Lecture Tip: The phrase “intrinsic value” is important in the field
of finance, but it has more than one meaning. In this context, it
refers to the lower bound on options. In the investments area,
however, it is used by fundamental analysts to refer to the “true”
value of a financial asset.
.A The Factors Determining Call Option Values
We discuss five key determinants of the value of an option and the
sensitivity of an option’s price to each determinant.
Determinant Relation to Call Relation to Put
Stock price Positive Negative
Strike price Negative Positive
Risk-free rate Positive Negative
Volatility of the stock Positive Positive
Time to expiration Positive Usually positive
The relationship between the value of an option and the stock price
and exercise price is intuitive based on the intrinsic values.
Risk-free Rate of Interest
The present value of the strike price discounted at the risk-free rate
of interest will affect the value of the options. For a call option, a
higher risk-free rate implies a lower PV of the strike price. Since
the holder of the call option must pay the strike price to exercise
the option, higher risk-free rate increases the value of a call option.
The argument for the negative relationship between the risk-free
rate and the value of a put option follows the same logic.
Volatility of the Underlying Stock
An option is worthless and will not be exercised if it expires out-
of-the-money. If the variability of the underlying stock is greater
(i.e. a more dispersed probability distribution), the probability of
an option expiring in-the-money increases. Therefore, the value of
both put and call options increases with the volatility of the
underlying stock.
Time to Expiration Date
Increasing the time to expiration has two effects:
1. It decreases the present value of the strike price. Note that
the buyer of a call option pays the strike price and the buyer of a
put option receives the strike price. Consequently, call option
values increase and put option values decrease when the time to
expiration increases.
2. It increases the likelihood of favorable outcomes for both
call and put options, so that both call and put values increase.
Therefore, the relationship between time to expiration and the
value of a call option is always positive. The value of a put option
may increase or decrease depending on the values of other
parameters. Deep out-of-the-money put options tend to decrease in
value with increasing time to expiration. Decrease in the present
value of the strike price (which the put buyer will receive if
exercised) dominates the change in the probability of the put
expiring in-the-money. Deep in-the-money puts tend to increase in
value with increasing time to expiration as the increase in the
likelihood of favorable outcomes dominates the decrease in the
present value of the strike price.
Options and Dividends
CBOE options do not receive dividends. This means that an
increase in a dividend (and a corresponding decrease in future
stock price) causes call options to fall and put options to rise in
value.
An American call option on a non-dividend paying stock should
never be exercised early. Consequently, such options may be
valued in the same way as European call options.
.B A Quick Discussion of Factors Determining Put Option Values
See discussion above.
Slide 22.36 Option Value Determinants
22.2. An Option Pricing Formula
Slide 22.37 An Option Pricing Formula
.A A Two-State Option Model
A Simple Model: Part I
One way to illustrate option pricing is to show equivalent cash
flows with different sets of securities.
Suppose a stock currently sells for $62, and its price will be either
$70 or $90 in one period. Assume there is a call option with a
strike price of $65. The risk-free rate for one period is 10%.
Portfolio 1: Buy the stock
Portfolio 2: Buy the call and lend $59.09 for one period (PV(E))
In one period the stock (portfolio 1) will be worth either $70 or
$90. The value of portfolio 2 will equal the value of the call + $65
(proceeds from the loan).
Stock = $70; Portfolio 2 = $70 – $65 + $65 = $70
Stock = $90; Portfolio 2 = $90 – $65 + $65 = $90
Since portfolio 1 and portfolio 2 will have equal values at the end,
they must have equal values today. Otherwise, you would buy the
“cheap” one and sell the “expensive” one and make a riskless
profit.
Therefore, S0 = C0 + PV(E)
C0 = S0 – PV(E) = $62 – $59.09 = $2.91
This can be extended to any stock price where the option finishes
in-the-money.
A Simple Model: Part II
The option may, however, finish out-of-the money. So, adjust your
second portfolio as follows: instead of loaning the PV(E), loan the
PV(lowest possible S1) at the risk-free rate and buy enough calls to
have the difference between the lowest and highest possible prices.
Example: Suppose a stock is currently selling for $67 and can have
a price of $60 or $80 in one period. There is a call option with a
strike price of $70. The risk-free rate is 9%.
Loan PV(60) = $55.05 at the risk-free rate. The loan proceeds will
be $60 at expiration, by design.
Difference between $80 and $60 = $20. If the stock price is $80,
then the payoff on one call is $80 – $70 = $10. Therefore, it takes 2
calls to make up the difference between the two possible prices.
So, your two possible portfolios are:
1. Buy the stock
2. Buy 2 calls and loan $55.05 at 9%
At expiration:
Stock = $60; Portfolio 2 = $0 + $60 = $60
Stock = $80; Portfolio 2 = 2($80 – $70) + $60 = $80
Since the two portfolios have the same ending value, they must sell
for the same amount today.
S0 = 2C0 + $55.05
C0 = ($67 – $55.05)/2 = $5.98
A Closer Look
Let S equal the difference in the possible stock prices, and let C
be the difference in option values at expiration. Then the number
of options needed to replicate the possible stock prices is S/C,
and this is referred to as delta.
Slide 22.38 –
Slide 22.44 Binomial Option Pricing Model
Slide 22.45 –
Slide 22.46 Delta
Since we can replicate outcomes, if we desired, we could replicate
a risk free asset. (This can be illustrated by rearranging the put-call
parity equation.) Thus, we could value options in a “risk-free”
world.
Slide 22.47 –
Slide 22.49 The Risk-Neutral Approach
Slide 22.50 –
Slide 22.53 Example of Risk-Neutral Valuation
Note that the two approaches (replication and risk neutral) provide
the same values for the option.
Slide 22.54 Risk-Neutral Valuation and the Replicating Portfolio
.B The Black-Scholes Model
The binomial model reflected a discrete world, but the B&S Model
reflects a continuous world, which is more realistic.
The Formula:
C = SN(d1) – Ee-RtN(d2)
where N(d1) and N(d2) are probabilities that we compute using the
following formulas and then look the numbers up in the standard
normal tables.
tσdd
tσ
t
2
σ
R
E
S
ln
d
12
2
1
where is the standard deviation (or volatility) of the underlying asset
returns.
Slide 22.55 The Black-Scholes Model
Example: Consider a stock that is currently selling for $35. You are
looking at a call option that has an exercise price of $30 and
expires in 6 months. The risk-free rate is 4%, compounded
continuously. The volatility of stock returns is .25. What is the call
price?
89676.
12
6
25.07353.1d
07353.1
12
6
25.
12
6
2
25.
04.
30
35
ln
d
2
2
1
N(d1) = N(1.07353) = .8585
N(d2) = N(.89676) = .8151
C = 35(.8585) – 30e-.04(6/12)(.8151) = $6.08
Slide 22.56 –
Slide 22.58 The Black-Scholes Model
22.3. Stocks and Bonds as Options
.A The Firm Expressed in Terms of Call Options
The underlying asset is the value of the firm (the value of its
assets). The stockholders have a call on this value with a strike
price equal to the face value of the firm’s debt. If the firm’s assets
are worth more than the debt, the option is in-the-money and
stockholders exercise the option by paying off the debt. If,
however, the face value of the debt is greater than the value of the
firm’s assets, the option expires unexercised (i.e., the company
defaults on its debt). Thus, the bondholders can be viewed as
owning the firm’s assets and having written a call against them.
Slide 22.59 Stocks and Bonds as Options
.B The Firm Expressed in Terms of Put Options
Alternatively, we could view owners as having a put option that
gives them the right to put the firm to the creditors in the case of
bankruptcy.
Slide 22.60 Stocks and Bonds as Options
.C A Resolution of the Two Views
These two views are just opposite sides of the parity relationship.
Slide 22.61 Stocks and Bonds as Options
.D A Note about Loan Guarantees
Default free debt = risky debt + put option
22.4. Options and Corporate Decisions: Some Applications
A. Mergers and Diversification
Slide 22.62 Mergers and Diversification
Use option valuation to investigate whether diversification is a good
reason for a merger – from a stockholder’s viewpoint.
If synergies do not exist, then a merger will reduce volatility without
increasing cash flow.
Decreasing volatility decreases the value of the call option (equity) and
the put option. Decreasing the value of the put increases the value
of the debt.
So, a merger for diversification reasons transfers value from the
stockholders to the bondholders.
Slide 22.63 –
Slide 22.65 Example
Slide 22.66 M&A – Conclusions
B. Options and Capital Budgeting
Slide 22.67 Options and Capital Budgeting
If a firm has a substantial amount of debt, stockholders may prefer
riskier projects, even if they have a lower NPV.
The riskier project increases the volatility of the asset returns. The
increased volatility increases the value of the call (equity) and the
put. The increased put value decreases the value of the debt. This
transfers wealth from the bondholders to the stockholders.
Slide 22.68 –
Slide 22.69 Example: Low NPV
Stockholders may even prefer a negative NPV project if it increases
volatility enough.
The wealth transfer from bondholders to stockholders may outweigh
the negative NPV.
Slide 22.70 –
Slide 22.72 Example: Negative NPV
Lecture Tip: Bondholders recognize the desire of stockholders to take
on riskier projects. Consequently, provisions are typically put into
the bond indentures to try to prevent this wealth transfer. These
provisions add to the firm’s cost either directly through a higher
interest rate or through additional monitoring costs. These costs
are all considered agency costs.
Lecture Tip: Option valuation can explain how a company that
has filed for Chapter 11 could still have a positive equity value,
even though it is unlikely to be able to pay off its creditors.
Consider the following example:
Market value of assets = 1,000
Bond value = 1500
Equity value = ?
The company has the following two potential investments:
1. Project A has an expected payoff of $1,000, is extremely risky,
and has a NPV of $50.
2. Project B has an expected payoff of $400, is very safe, and has a
NPV of $200.
Project B, with the higher NPV, would normally be preferred,
but if accepted, its payoff when combined with the current $1000
value of assets would still fall short of the $1500 required to payoff
the creditors. The only project which may save the company is the
high risk project, with the low NPV, but possibly high payoff. If
project A fails, stockholders are not any worse off.
Slide 22.73 Options and Capital Budgeting
22.5. Investment in Real Projects and Options
Real options provide the right to buy or sell real assets. These
options often apply in capital budgeting situations and can be very
valuable.
Explicit options – contracts giving the holder the right to buy or
sell the asset
Implicit options – options that exist in many capital budgeting
situations, but are often “hidden”
Timing – if we take a project today, we cannot take it later.
Consequently, even though a project has a positive NPV, it does
not mean we should take it now. It may be worth more to us if we
wait one year, or two, or three …
The option to wait is particularly valuable when the economy or
market is expected to be bigger in the future. It is not valuable
when trying to capitalize on current fads.
The option to wait may actually turn a bad project into a good
project – waiting a year or two may allow the firm to capture
higher cash flows.
Managerial options are options to modify a project once it has been
implemented. Examples include:
1. Option to expand – ability to make the project bigger if it is
a successful. We underestimate the NPV if we ignore this
option.
2. Option to abandon – ability to shut down the project if things
do not go as planned. We underestimate the NPV if we ignore
this option.
3. Option to suspend or contract – ability to downscale when
the market is weaker than expected.
4. Strategic options – using a project to explore possible new
ventures or strategies. These projects open up a wide number of
future opportunities but are more difficult to analyze with
traditional DCF analysis.
Slide 22.74 Investment in Real Projects an Options
Slide 22.75 Quick Quiz