16: Lecture Options and
Corporate Finance: II
Topics
Option Pricing: Black-Scholes Formula
Application: Equity Value as Option
Black-Scholes Model
The binomial approach provide good intuition but
they do not work effectively in the real world.
There are many more than two possibilities for future
stock prices.
Black and Scholes Model
Nobel-Price winning theory developed by Fisher Black
and Myron Scholes.
Based on replicating portfolio that duplicates a call
over an infinitesimal horizon.
Major tool used by the finance industry to price
options
Black-Scholes Model
C0 = S N(d1 ) Ee rT N(d 2 )
Where
C0 = the value of a European option at time t = 0
r = the risk-free interest rate. T=the maturity of option
N(d) = Probability that a
2
ln(S / E ) + (r + )T standardized, normally
2
d1 =
distributed, random variable
T
d 2 = d1 T
will be less than or equal to
d.
The Black-Scholes Model allows us to value options in the real
world.
4
Black-Scholes Model: Example
Find the value of a six-month call option on
Microsoft with an exercise price of $150.
The current value of a share of Microsoft 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 Microsoft stock return is 30% per
annum.
Before we start, note that the intrinsic value of the
option is ____our answer must be at least that
amount.
Black-Scholes Model: Example
First calculate d1 and d2
ln(S / E ) + (r + .5 2 )T
d1 =
T
ln(160 / 150) + (.05 + .5(0.30) 2 ).5
d1 =
= 0.52815
0.30 .5
Then,
d 2 = d1 T = 0.52815 0.30 .5 = 0.31602
Black-Scholes Model: Example
C0 = S N(d1 ) Ee rT N(d 2 )
d1 = 0.52815
d 2 = 0.31602
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
C0 = $160 0.7013 150e .05.5 0.62401
C0 = $20.92
In-Class Practice
Find the value of a six-month call option on
Ashley Co. with an exercise price of $80.
The current value of a share of Ashley is $85.
The interest rate available in the U.S. is r = 6%.
The option is maturity 18 months (one and half
years).
The volatility of Ashley stock is 15% per annum.
Equity Value as Call Option
Recall: call option payoff
Call options gives holder the right to purchase
underlying asset at exercise price E.
On the maturity date:
Zero payoff if S, price of the underlying asset, is lower
than E.
Payoff equals S-E if S is higher than E.
Equity Value as Call Option
Payoff of equity:
If at the maturity of the debt, the asset value is lower
than the debt level, then payoff is zero (bankruptcy).
If at the maturity of their debt, the assets of the firm
(A) are greater in value than the debt (D), then payoff
is A-D.
Levered equity is actually a call option.
The underlying asset comprises the assets of the
firm.
The strike price is the payoff of the bond.
Equity Value as Call Option: Example
Popov Company has been awarded the concessions at
the Olympic Games in Antarctica. Popov has $800 debt
due after the games. The companys assets will be worth
$1,000 if the games are successful and $500 if the
games are unsuccessful.
Shareholders payoffs
If the games are successful, the shareholders will pay the
bondholders $800. Their payoff is $200.
If the games are unsuccessful, the shareholders will declare
bankruptcy. Their payoff is $0.
The equity has the same payoff as a call option on the
firms assets with exercise price $800.
Practice: Mapping Equity to Option
Ashley Company has $20 million debt due in
one year. The companys assets will be worth
$25million if they are doing well and $15million
they are doing poorly.
Map Ashleys equity to an option
Exercise price is __________
Underlying asset is ____________________
It is a ____ option
In Class Practice
Tony Corporations market value of assets are $40
million. It has debt with face value of $18 million due in 4
years. Tony Corporations asset return standard
deviation is 40%. Risk free rate is 4%.
Calculate the market values of Tonys Corporations
equity and debt using Black-Scholes Formula.