Chapter 12
DERIVATIVES: ANALYSIS AND
VALUATION
Chapter 12 Questions
How are spot and futures prices related?
What is basis risk?
What is program trading and stock index
arbitrage? How can futures be used to hedge
or speculate on changes in yield curve
spreads and credit quality spreads?
Why would investors want to invest in an
option on a futures contract?
Chapter 12 Questions
What factors influence the price of an option?
How does one use the Black-Scholes optionpricing model?
Why are the terms delta,theta, vega, rho, and
gamma important to option investors?
How do option-like features affect the price of
bonds?
Futures Valuation Issues
Cost of Carry Model
Suppose that you needed some commodity in
three months. You have at least the following
two options:


Purchase the commodity now at the current spot
market price (S0) and “carry” the commodity for 3
months
Buy a futures contract for delivery of the
commodity in 3 months for the current futures
price (F0,3)
Futures Valuation Issues
Cost of Carry Model
The futures prices and spot prices must be
related to one another in order for there to be
no arbitrage opportunities for investors.
If the carrying cost only amounts to forgone
interest at a risk-free rate (rf) for T time
periods, then the following relationship must
hold:
F0,T = S0 (1+rf)T
Futures Valuation Issues
Cost of Carry Model Example: Suppose that
you can buy gold in the spot market for $300.
The monthly risk-free is .25%. You need the
gold in three months.
What should be the current futures price?
F0,T = 300 (1+.0025)3 = 302.26
What if the futures price is $305?

You have a risk-less profit opportunity. Buy gold at
$300, sell futures at $305. In three months,
delivery the gold, pay the known interest, pocket
the difference.
Futures Valuations
Issues
Similar futures-spot price relationships can be
derived when there are “market
imperfections” involved with carrying the
commodity or financial asset
Incorporating storage and insurance costs as
a percentage of contract value (SI):
F0,T = S0 (1+rf +SI)T
Incorporating ownership benefits lost with a
futures position, especially dividends(d):
F0,T = S0 (1+rf +SI -d)T
Futures Valuation Issues
Basis


Basis is the difference between the spot and
futures prices.
For a contract expiring at time T, the basis at time t
is:
Bt,T = St – Ft,T


Over time, the spot and futures prices converge,
and basis becomes zero at expiration
Between time t and expiration, basis can change
as the difference between spot and futures prices
vary (known as basis risk)
Advanced Applications
of Financial Futures
Stock Index Arbitrage
An example of a program trading strategy
designed to take advantage of temporarily
“mis-pricing” of securities
 Monitor the parity condition:
F0,T = S0 (1+rf +-d)T
 If it does not hold, construct a risk-free
position to take advantage of the situation.

Advanced Applications
of Financial Futures
T-Bond/T-Note Futures Spread
“Note over bond” (NOB) spread
 Strategies based on speculating the
changing slope of the yield curve

Options on Futures
Also known as Futures Options
Options on Stock Index Futures

Gives the owner the right to buy (call) or
sell (put) a stock futures contract
Options on Treasury Bond Futures

Gives the owner the right to buy (call) or
sell (put) a Treasury bond futures contract
Options on Futures
Why would they be attractive?


If exercised, it would seem to have been better to
simply buy a futures contract instead (no option
premium to pay)
One primary advantage can be found when
looking at all the potential price movements


Futures contracts used for hedging offset portfolio value
changes; thus, advantageous price movements for a
portfolio are offset by the futures position
Options give the right (but not the obligation) to purchase
the futures contract; thus, favorable price movements will
be offset only by the option premium rather than by a
corresponding loss on the futures position
Valuation of Options
Factors influencing the value of a call option:

Stock price (+)


Exercise price (-)


For a given exercise price, the higher the stock price, the
greater the intrinsic value of the option (or at least the
closer to being in-the-money)
The lower the price at which you can buy, the more value
Time to expiration (+)

The longer the time to expiration, the more likely the
option will be valuable
Valuation of Options
Factors influencing the value of a call
option:

Interest rate (+)


Options involve less money to invest, lower
opportunity costs
Volatility of underlying stock price (+)

The greater the volatility of the underlying
stock, the more likely that the option position
will be valuable
Valuation of Options
Factors influencing the value of a put option:






The same listed, but different directions for several
items.
Stock price (-)
Exercise price (+)
Time to expiration (+)
Interest rate (-)
Volatility of underlying stock price (+)
Black-Scholes Option
Pricing Model
Model for determining the value of
American call options
This work warranted the awarding of the
1997 Nobel Prize in Economics!
Black-Scholes Option
Pricing Formula
P0 = PS[N(d1)] - X[e-rt][N(d2)]
where:
P0 = market value of call option
PS = current market price of underlying stock
N(d1) = cumulative density function of d1 as defined later
X = exercise price of call option
r = current annualized market interest rate for prime
commercial paper
t = time remaining before expiration (in years)
N(d2) = cumulative density function of d2 as defined later
Black-Scholes Option
Pricing Formula
P0 = PS[N(d1)] - X[e-rt][N(d2)]
The cumulative density functions are defined as:
 ln( P / X  (r  0.5 2 )t 
S

d1  
1


 (t ) 2
d 2  d1   (t ) 2 


1
Where:
ln(PS/X) = natural logarithm of (Ps/X)
S = standard deviation of annual rate of
return on underlying stock
Using the Black-Scholes
Formula
Besides mathematical values, there are five
inputs needed to use this model:





Current stock price (Ps)
Exercise price (X)
Market interest rate (r)
Time to expiration (t)
Standard deviation of annual returns ()

Of these, only the last in not observable
Also, using the put/call parity, we can value
put options as well after calculating call value
Option Valuation
Terminology
Delta
The sensitivity of an option’s price to the
price of the underlying security
 Positive for calls, negative for puts

Theta

Measures how the option premium
changes as expiration approaches
Option Valuation
Terminology
Vega

The sensitivity of the option premium to the price
volatility () of the underlying security
Rho

Measures the sensitivity of the option premium to
changes in interest rates
Gamma

Measures the sensitivity of delta to changes in the
underlying security price
Option-like Securities
Several types of securities contain
embedded options:
Callable and Putable Bonds
 Warrants
 Convertible Securities

Callable and Putable
Bonds
Callable Bonds contain a “call provision”



The issuer has the option of buying the bonds
back at the call (exercise) price rather than having
to wait until maturity
Attractive option for issuers if interest rates fall,
since they can purchase back old bonds and
refinance (refunding) with new, lower interest
bonds
Typically will trade at no more than the call price,
since call becomes likely at that point
Callable and Putable
Bonds
Putable Bonds contain a “put provision”
Investors may resell the bonds back to the
issuer prior to maturity at the put (exercise)
price, often par value
 Puts can generally be exercised only when
designated events take place

Warrants
Warrant is an option to buy a stated
number of shares of common stock at a
specified price at any time during the life
of the warrant
Similar to a call option, but usually with
a much longer life
Issued by the company whose stock the
warrant is for
Warrants
Intrinsic value is the difference between the
market price of the common stock and the
warrant exercise price
Intrinsic Value = (Stock Price – Exercise Price)
x Number of Share
Speculative value is the value of the warrant
above its intrinsic value

Like other options, the value is higher than
intrinsic value, except at maturity
Convertible Securities
Allows the holder to convert one type of
security into a stipulated amount of another
type (usually common stock) at the investor’s
discretion
With convertible securities, value depends
both on the value of the original asset and the
value if conversion takes place

Value cannot fall below the greater of the two
values
Convertible Securities
Convertible Bonds
Advantages to issuing firms
Lower interest rate on debt
 Debt represents potential common stock

Advantages to investors
Upside potential of common stock
 Downside protection of a bond

Convertible Securities
Convertible bonds
Conversion ratio = number of shares
obtained if converted
 Conversion price = Face Value/Number of
shares

Valuation of convertible bonds
Combination value of stock and bond
 Two step process to determine minimum
value

Convertible Securities
Convertible Bonds
Value of a convertible as a bond


Determine the bond’s value as if it had no
conversion feature
This is the convertible’s investment value or floor
value
Value of a convertible as stock


Compute the value of the common stock received
on conversion
This is the conversion value
Convertible Securities
Convertible Bonds
Minimum Value = Max (Bond Value,
Conversion Value)
Like other options, including embedded
options, they typically only sell at their
minimum, intrinsic value only at maturity.

Conversion Premium = (Market Price – Minimum
Value)/Minimum Value
Convertible Securities
Convertible Bonds
Conversion Parity Price = Market
Price/Conversion Ratio

An risk-free profit opportunity would exist if the
price of the convertible below this price, since
immediate conversion of the bond and then selling
the stock would yield a profit
Payback

How long it takes the higher-interest income from
the convertible bond (compared to the stock
dividend) to make up for the conversion premium
Convertible Securities
Convertible Preferred Stock
Combination of preferred stock and common
stock
Common characteristics:





Cumulative but not participating dividends
No sinking fund or purchase fund
Fixed conversion rate
Waiting period not required before conversion
Conversion privilege does not expire
Usually issued in connection with mergers
Convertible Securities
Convertible Preferred Stock
Value as preferred stock
Value as common stock, given the
conversion rate
Parity relationships imply that the value
has to be higher than the maximum of
the two values