Session 6
Options and Corporate Finance
FIN 625: Corporate Finance
Learning Objectives
LO 1: Explain the basic characteristics and
terminology of options.
LO 2: Determine the intrinsic value of
options at expiration date.
LO 3: Use the Black-Scholes Option Pricing
Model to determine the fair value of an
option before expiration.
LO 4: Explain how option pricing can be
applied to corporate finance.
Outline
1.
2.
3.
Option Basics and Value at Expiration
The Black-Scholes Option Pricing Model
Applications of Options in Corporate
Finance
1. Option Basics and Value at
Expiration


An option gives the holder the right, but not the
obligation, to buy or sell a given quantity of an asset on (or
before) a given date, at prices agreed upon today.
Exercising the Option


Strike or Exercise Price


The act of buying or selling the underlying asset
The fixed price in the option contract at which the holder can buy or
sell the underlying asset
Expiration Date

The maturity date of the option
Option Basics

European versus American options



In-the-Money


Exercising the option would result in a positive payoff.
At-the-Money


European options can be exercised only at expiration date.
American options can be exercised at any time up to maturity date.
Exercising the option would result in a zero payoff (i.e., stock price
equals to exercise price).
Out-of-the-Money

Exercising the option would result in a negative payoff.
Call Options


Call options give the holder the
right, but not the obligation, to
buy a given quantity of some asset
on or before some time in the
future, at prices agreed upon
today.
When exercising a call option, you
“call in” the asset.
Call Option Pricing at Expiration

At expiration, an American call option is
worth the same as a European option with
the same characteristics.


If the call is in-the-money, it is worth ST – E.
If the call is out-of-the-money, it is worthless.
Hence,
C = Max[ST – E, 0]
Where
ST is the stock price at expiration (time T)
E is the exercise/strike price.
C is the (intrinsic) value of the call option at expiration
Value of Call Option at Expiration
Option value ($)
60
40
20
20
40
50
60
80
100
–20
Exercise price = $50
120
Stock price ($)
Put Options


Put options gives the holder the
right, but not the obligation, to
sell a given quantity of an asset on
or before some time in the future,
at prices agreed upon today.
When exercising a put, you “put”
the asset to someone.
Put Option Pricing at Expiration



At expiration, an American put option
is worth the same as a European
option with the same characteristics.
If the put is in-the-money, it is worth E
– ST.
If the put is out-of-the-money, it is
worthless. Hence,
P = Max[E – ST, 0]
where P is the (intrinsic) value of the put option at
expiration
Value of Put Option at Expiration
Option value ($)
60
50
40
20
0
–20
0
20
40
50
60
80
100
Exercise price = $50
Stock price ($)
2. The Black-Scholes Option Pricing Model

The last section
concerned itself
with the value of
an option at
expiration.

This section
considers the
value of an option
prior to the
expiration date.

A much more
interesting
question.
Option Value Determinants
1.
2.
3.
4.
5.
Stock price
Exercise price
Interest rate
Volatility in the stock price
Expiration date
Call
+
–
+
+
+
Put
–
+
–
+
+
The Black-Scholes Model
C0  S  N( d1 )  Ee
σ2
ln( S / E )  (r  )T
2
d1 
 T
d 2  d1   T
 rT
 N( d 2 )
N(d) = Probability that a
standardized, normally
distributed, random variable
will be less than or equal to
d.
Where,
C0 = the value of a European option at time t = 0
S = current stock price
E = exercise price of call
r = annual risk-free rate of return, continuously compounded
σ2 = variance (per year) of the stock return
T = time (in years) to expiration date
e = the base of natural logarithm, equals 2.7182818…
Put-Call Parity
If we know the intrinsic value of a call
option, the intrinsic value of a put option
with identical features (same maturity,
exercise price, stock price, risk-free
interest rate, and equity volatility) can be
determined by the put-call parity:
P + S = C + PV(E) = C + e(-rT)*E
So P = C + e(-rT)*E - S

Example
Find the value of a six-month call and put option
on Microsoft with an exercise price of $150.
The current price of Microsoft stock is $160.
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying stock is 30% per
annum.
Example
We solve the value of the call option first. We then
use the put-call parity to get the put option value.
First calculate d1 and d2
ln( S / E )  (r  .5σ 2 )T
d1 
 T
ln(160 / 150)  (.05  .5(0.30) 2 ).5
d1 
 0.53
0.30 .5
Then,
d 2  d1   T  0.52815  0.30 .5  0.32
Example
C0  S  N( d1 )  Ee
d1  0.53
d 2  0.32
 rT
 N( d 2 )
N(d1) = N(0.53) = 0.70194
N(d2) = N(0.32) = 0.62552
C0  $160  0.70194  150e .05.5  0.62552
C0  $20.80
P0  C0  e
P0  $7.10
 rT
* E  S  $20.80  e
.05.5
* $150  $160
3. Applications of Options in Corporate
Finance


We can view bondholders, rather than
shareholders as the owner of the firm.
Bondholders write a call option to
shareholders.



The underlying asset is the asset of the firm.
The strike price is the payoff of the bond.
The expiration date of the call option is the
debt maturity.
Stock As a Call Option


If at the maturity of the debt, the assets of
the firm are greater in value than the debt
(the strike price), the shareholders have an
in-the-money call. They will pay the
bondholders and “call in” the assets of the
firm.
If at the maturity of the debt the
shareholders have an out-of-the-money
call, they will not pay the bondholders (i.e.
the shareholders will declare bankruptcy)
and let the call expire.
Example
Consider a company with the following
characteristics:





MV assets = $40 million
Face value debt = $25 million
Debt maturity = 5 years
Asset return standard deviation = 40%
Risk-free rate = 4%
What is the market value of equity and debt?
Example
S = $40 million, E = $25 million, T = 5,
σ = 0.4, r = 0.04
First calculate d1 and d2
ln( S / E )  (r  .5σ 2 )T
d1 
 T
ln(40 / 25)  (.04  .5(0.4) 2 ) * 5
d1 
 1.1963
0.40 5
Then,
d 2  d1   T  1.1963  0.4 5  0.3019
Example
C0  S  N( d1 )  Ee
d1  1.20
d 2  0.30
 rT
 N( d 2 )
N(d1) = N(1.20) = 0.8849
N(d2) = N(0.30) = 0.6179
C0  $40  0.8849  $25  e .045  0.6179  $22.75
So the market value of equity is $22.75 million.
The market value of debt = market value of firm
– market value of equity = $40 million - $22.75
million = $17.25 million.
Readings

Chapter 17