Applications of Option Methods in Corporate Finance

Applications of Option
Methods in Corporate
Finance
Timothy A. Thompson
Financial Decisions
Goals of option valuation
Purpose is not to be derivatives traders

We want to understand what options are
present in financial contracts
We want to understand what the economic
function these options have in financial
contracting
We are going to talk about simple option
pricing models (Black Scholes/binomial) to get
a ballpark of the value of imbedded options
We want to be able to understand when the
ballpark is big or small
What are options?
A call/put option represents a right, not an

obligation, for the holder of the option to
buy/sell an underlying assets for a fixed
price (exercise price or strike price) on or
before a specified future date (expiration
date)
American vs. European
Examples
Calls and puts on CBOE

Warrants
Caps and Floors
Options, options, everywhere
Warrants, convertibles, callables

Embedded options in PERCS, LYONS,
etc.
Real asset options
Option to wait
Option for follow up investments
Flexibility options
Abandonment options
Determinants of option prices
Parameters of call and put prices, C t

and Pt
Price of the underlying asset (stock, etc.),
St
Time to maturity, , T - t
Strike price (exercise price), X
Risk free interest rate, r
Volatility (std dev of ror on underlying),
Dividend yield on underlying asset
Interesting what parameters are not

there
Expected return on the underlying

Expected risk premium on stocks over risk
frees
Risk aversion of investors
Why arent these there?
Because they are there: they are in the stock
price
Options are derivative assets: they derive their
value from the value of the underlying asset
Black Scholes model
The mathematics behind the Black Scholes model is difficult

But for our purposes the model is like a black (no pun
intended) box
We put in parameters
We get an answer
We want to know how the answer depends on the

parameters
We want to know whether the model will get us in the
ballpark
If the model really is not appropriate for an application, we
would go to a model that could be modified for the
application
Like the binomial method or numerical estimation methods
Assumptions of Black Scholes

model
Perfect markets
Option is European
No taxes/transactions costs, information costs

This is crucial. Next slide.
Stock follows a diffusion process

People can borrow or lend at r
r, and are known constants
X and T are known constants
Black Scholes Equation

Ct S t e N (d1 ) Xe r N (d 2 )
where
d1
Se
ln
r
Xe
and
d 2 d1
When will a European model work

when pricing American options?
Generally, it wont
An American option is always worth at least as much as

its European counterpart
Because you can do anything with an American option that

you can do with an European option and
You can exercise it prior to maturity. This right cant have
negative value.
Important no-arbitrage result from options
An American call option on a non-dividend paying

underlying asset will never be optimally exercised prior to
maturity
If the option we need to value can be characterized as a
call option on a non-dividend paying stock, then BS will be
reasonable
As a practical matter, as long as dividends arent large
enough to induce early exercise, then BS will be reasonable
Luckily the computer does the math

for us!
Call option value
Call value and volatility
Call value and maturity
Estimating parameters for traded

call options
Time to expiration
Calendar time to expiration
Risk free interest rate
Nearest Treasury strip to maturity of option

Annualized and restated to be continuously compounded
Exercise price (strike price)

Stock price
Current market price of underlying asset
Dividends
Annualized dividend to price ratio and cont. comp.

Or subtract present value of dividends from stock price
Volatility
Standard deviation of the rate of return on the underlying

asset
Volatility estimation
Historical sample standard deviation
Implied volatility
Estimate all the B/S parameters except
for volatility
Using the market price of an option,
back into the value of volatility
parameter that equates the B/S value of
the option to its market price
Assumptions behind historical and

implied volatility
Historical volatility
Assuming that historical volatility is a

reasonable forecast of future volatility
Same as many other issues we face (betas,
etc.)
Implied volatility
Assuming that the option is priced correctly by

the Black Scholes model
Assuming that the option price and underlying
asset price are efficiently priced and available
at the same time
Warrants
What is a warrant?
Security giving the holder the right to purchase the

underlying stock for a fixed price and given duration of
time.
Sounds just like an American call option
Differences between warrants and calls
Warrant is a primary market instrument for firm

Issued for cash or consideration, which is cash inflow to the
firm when issued
If warrants exercised, the exercise funds are cash inflow to the
firm and there are more shares outstanding (dilution)
Executive stock options are warrants in this sense.
Warrants typically have longer maturities than calls

Can have much more flexible terms than exchange traded
options
Applying Black Scholes model to

value warrants
Addiitonal notation:
W = Warrant value
N = Number of shares of stock outstanding
before exercise of warrants
M = number of warrant shares outstanding
Assumptions
The warrants being valued are the only
securities convertible into common stock
Assume all warrants would be exercised only
at maturity
Warrants and common are

options on total firm equity value
The value of a European warrant is

equivalent to the value of a European call
option on the stock of on an otherwise
identical firm with no warrants outstanding
Same number of shares outstanding, N

Multiplied by dilution factor M/(N+M)
The value of the total equity of the

identical firm is NS*, equal to
The value of the total equity of this firm =
NS + MW, so S* = S + (M/N)W
Firm with equity and warrants
Black Scholes Warrant Model
N M
Wt S *t e N (d1 ) Xe r N (d 2 )
where
d1
S *e
ln
r
Xe

, d 2 d1 *
2
and
M
S S
W
N
*
Debt and equity as options
Assumptions
Company has only one debt issue (Face value = F, Zero
coupon, Maturing in T years) and equity outstanding
Company pays no dividends on common
Bankruptcy costs are zero and absolute priority will be
observed
At maturity (date T), the value of the equity is

given by ET = max[0, VT F]
Value of the debt at maturity (date T) is given by
DT = min[VT, F]
Equity payoff is identical to the payoff on a call

option written on the assets (value) of the firm with
a strike price equal to the face value of the debt and
maturity equal to the maturity of the debt.
Risky debt is riskless debt minus a

put option
From above we have Et = Callt
The options are European here and there
are no dividends (or coupons on debt) so
we can use put-call-parity formula (PCP):
Et = Vt PV(F) + Putt
Using the balance sheet constraint, D =
V-E
Dt = Vt Vt + PV(F) Putt, or
Dt = PV(F) - Putt
Value of loan guarantee as a put

option
Suppose the government were to

guarantee a firms debt
If the firm were to default, the
government pays the bondholders their
promised payments
Bonds become like riskless debt
Put option from the last slide is
contingent liability that the government
assumes.

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Applications of Option

Goals of option valuation

Purpose is not to be derivatives traders

What are options?

A call/put option represents a right, not an

Calls and puts on CBOE

Options, options, everywhere

Warrants, convertibles, callables

Determinants of option prices

Parameters of call and put prices, C t

Interesting what parameters are not

Expected return on the underlying

Black Scholes model

The mathematics behind the Black Scholes model is difficult

We want to know how the answer depends on the

Assumptions of Black Scholes

No taxes/transactions costs, information costs

Stock follows a diffusion process

Black Scholes Equation

When will a European model work

An American option is always worth at least as much as

Because you can do anything with an American option that

Important no-arbitrage result from options

An American call option on a non-dividend paying

Luckily the computer does the math

Call option value

Call value and volatility

Call value and maturity

Estimating parameters for traded

Calendar time to expiration

Risk free interest rate

Nearest Treasury strip to maturity of option

Exercise price (strike price)

Current market price of underlying asset

Annualized dividend to price ratio and cont. comp.

Standard deviation of the rate of return on the underlying

Historical sample standard deviation

Assumptions behind historical and

Assuming that historical volatility is a

Assuming that the option is priced correctly by

Security giving the holder the right to purchase the

Differences between warrants and calls

Warrant is a primary market instrument for firm

Warrants typically have longer maturities than calls

Applying Black Scholes model to

Warrants and common are

The value of a European warrant is

Same number of shares outstanding, N

The value of the total equity of the

Firm with equity and warrants

Black Scholes Warrant Model

Debt and equity as options

At maturity (date T), the value of the equity is

Equity payoff is identical to the payoff on a call

Risky debt is riskless debt minus a

Value of loan guarantee as a put

Suppose the government were to

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