Options
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Option Terminology
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Call Option – the right to buy an asset at some
point in the future for a designated price.
Put Option – the right to sell an asset at some
point in the future at a given price
Review of Option Terminology
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Expiration Date The last day the option can be
exercised (American Option)
also
called the strike date,
maturity,
and exercise
date
Exercise Price
The price specified in the
contract
American Option Can be exercised at any time up
to the expiration date
European Option Can be exercised only on the
expiration date
Review of Option Terminology
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Long position: Buying an option
Long Call: Bought the right to buy the asset
Long Put: Bought the right to sell the asset
Short Position: Writing or Selling the option
Short Call Agreed to sell the other party the
right to buy the underlying asset, if the other
party exercises the option you deliver the
asset.
Short Put - Agreed to buy the underlying
asset from the other party if they decide to
exercise the option.
Review of Terminology
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In - the - money options
when the spot price of the underlying asset for a call
(put) is greater (less) than the exercise price
Out - of - the - money options
when the spot price of the underlying asset for a call
(put) is less (greater) than the exercise price
At the money options
when the exercise price and spot price are equal.
Interest Rate Options
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Traded on Chicago Board of Options Exchange
(CBOE)
Interest rate Options are traded on 13 Week,
5 year, 10 year and 30 year treasury securities
Options on Futures
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Options on futures are as popular or even more
popular than on the actual asset.
Options on futures do not require payments for
accrued interest.
The likelihood of delivery squeezes is less.
Current prices for futures are readily available, they
are more difficult to find for bonds.
Futures Options
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Call option holder will own a long futures position
if the option is exercised.
The writer of the call option accepts the
corresponding short position at the exercise
price.
Mechanics of Options on
Futures
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Call Option Example
Exercise price = $85 Current Futures price = $95
Upon exercise both the long position and the short
owned by the writer of the short option is set to $85.
When marked to market the holder of the long makes
$10, the holder of the short looses $10.
The holders of the short and long position then face the
same risks as any other holder of the futures
contract.
Buyer Margin Requirements
on Futures Options
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The buyer of the call option is not required to
place any margin deposits. The most that could
be lost is the cost of the option.
Call Option Writer
Margin Requirements
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The writer of the call option accepts all of the
risk since the buyer will not exercise if there
would be a loss.
The writer is required to deposit the original
margin that would be required on the futures
contract and the option price that is received for
writing the option. The writer is also required to
deposit variation margin as the contract is
marked to market.
Call Option Profit
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Call option – as the price of the asset increases
the option is more profitable.
Once the price is above the exercise price (strike
price) the option will be exercised
If the price of the underlying asset is below the
exercise price it won’t be exercised – you only
loose the cost of the option.
The Profit earned is equal to the gain or loss on
the option minus the initial cost.
Profit Diagram Call Option
(Long Call Position)
Profit
S-X-C
S
Cost
X
Spot Price
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Call Option Intrinsic Value
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The intrinsic value of a call option is equal to
the current value of the underlying asset
minus the exercise price if exercised or 0 if
not exercised.
In other words, it is the payoff to the investor
at that point in time (ignoring the initial cost)
the intrinsic value is equal to
max(0, S-X)
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Payoff Diagram Call Option
Payoff
S-X
X
S
Spot
Price
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Example: Naked Call Option
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Assume that you can purchase a call option on
an 8% coupon bond with a par value of $100
and 20 years to maturity. The option expires in
one month and has an exercise price of $100.
Assume that the option is currently at the money
(the bond is selling at par) and selling for $3.
What are the possible payoffs if you bought the
bond and held it until maturity of the option?
Five possible results
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The price of the bond at maturity of the option is
$100. The buyer looses the entire purchase
price, no reason to exercise.
The price of the bond at maturity is less than
$100 (the YTM is > 8%). The buyer looses the
$3 option price and does not exercise the option.
Five Possible Results continued
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The price of the bond at maturity is greater than
$100, but less than $103. The buyer will
exercise the option and recover a portion of the
option cost.
The price of the bond is equal to $103. The
buyer will exercise the option and recover the
cost of the option.
The price of the bond is greater than $103. The
buyer will make a profit of S-$100-$3.
Profit Diagram Call Option
(Long Call Position)
Profit
S-100-3
103
-3
100
S
Spot Price
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Price vs. Rate
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Note buying a call on the price of the bond is
equivalent to buying a put on the interest rate
paid by the bond.
As the rate decreases, the price increases
because of the time value of money.
Profit Diagram Call Option
(Short Call Position)
Profit
X
C+X-S
S Spot Price
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Put option payoffs
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The writer of the put option will profit if the
option is not exercised or if it is exercised and
the spot price is less than the exercise price plus
cost of the option.
In the previous example the writer will profit as
long as the spot price is less than $103.
What if the spot price is equal to $103?
Put Option Profits
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Put option – as the price of the asset decreases
the option is more profitable.
Once the price is below the exercise price (strike
price) the option will be exercised
If the price of the underlying asset is above the
exercise price it won’t be exercised – you only
loose the cost of the option.
Profit Diagram Put Option
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Profit
X-S-C
Spot Price
S
Cost
X
Put Option Intrinsic Value
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The intrinsic value of a put option is equal to
exercise price minus the current value of the
underlying asset if exercised or 0 if not
exercised.
In other words, it is the payoff to the investor
at that point in time (ignoring the initial cost)
the intrinsic value is equal to
max(X-S, 0)
Payoff Diagram Put Option
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Profit
X-S
S
Cost
X
Spot Price
Profit Diagram Put Option
Short Put
Profit
S
S-X+C
X
Spot Price
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Pricing an Option
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Arbitrage arguments
Black Scholes
Binomial Tree Models
PV and FV in continuous time
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e = 2.71828 y = lnx x = ey
FV = PV (1+k)n for yearly compounding
FV = PV(1+k/m)nm for m compounding periods per
year
As m increases this becomes
FV = PVern =PVert
let t =n
rearranging for PV
PV = FVe-rt
Lower Bound of Call Option
Price
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Assume that you have an asset that does not pay
a cash income (A non dividend paying stock for
example)
Consider the case of an option as it expires.
In this case, regardless of whether it is an
American or European option it will be worth its
intrinsic value (max(S-X,0)).
Assuming a positive value the lower bound is
given by:
S-X
Formal Argument
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Consider two portfolios
A: One European call option on the stock of Widget
Inc. plus cash equal to Xe-rT
B: One share of stock in Widget Inc.
Note: If the cash in portfolio A is invested at r, it
will grow to be worth X at time T.
Portfolio A
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There are two possible outcomes at time T
depending upon the value of S at time T
ST > X Exercise the option and purchase the asset
with a current value of ST (The value of portfolio A at
time T is ST ).
ST < X Do not exercise the option, The portfolio is
then worth the value of the cash, X.
Therefore the portfolio is worth:
max(ST,X)
Portfolio B
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The value of portfolio B is simply the value of the
stock at time T, ST.
Comparing A to B
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Combining the two results it is easy to see that
portfolio A (the option and the cash) is always
worth at least as much as portfolio B (owning the
stock), and sometimes it is worth more than B.
Without arbitrage, the same relationship should
be true today as well as at time T in the future.
Equal value of the portfolios
today
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Let c be the call price (value of option) today.
Then the value of portfolio A is: c + Xe-rT
The value of portfolio B is: S
Since the value of A is always worth as much as B
and sometimes it is worth more:
c + Xe-rT >S
or rearranging
c > S - Xe-rT
Final result
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The worst outcome to buying a call option is that it
expires worthless, so the option is worth either
nothing or S-Xe-rT
Therefore:
c > max(S - Xe-rT,0)
Put Option
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Similar to a call option the put option should
always have a positive value.
Considering the case of an option as it expires
(either an American or European Option), the
value of the option should be equal to its intrinsic
value. The lower bound is therefore:
X-S
European Put Option
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Again, in the case of a European option prior to
maturity this equation will not hold and it is
necessary to account for the time value of
money. In this case the lower bound for the
option is given by:
Xe-rT - S
Formal Argument:
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Consider two Portfolios
C: One European put option plus one share
and
D: An amount of cash equal to Xe-rT
Portfolio C:
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There are two possibilities:
ST < X Exercise the option at time T and the
portfolio is worth X.
ST > X The option expires and the portfolio is
worth ST
Portfolio C is therefore worth max(ST,X)
Portfolio D
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Investing the amount at a rate equal to r, the
portfolio will be worth X at Time T.
Combining the two arguments it is easy to see
that portfolio C is always worth at least the same
amount as portfolio D and sometimes it is worth
more.
Comparing the two
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Let p = the value (price) of the put option
Without arbitrage opportunities:
p + S > Xe-rT
or rearranging
p > Xe-rT - S
the value of the put option is then given as
p > max(Xe-rT-S,0)
Put Call Parity
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Consider portfolio A and C above
A: One European call option plus an amount of
cash equal to Xe-rT
C: One European put option plus one share
Both portfolios are have a value of max(X,ST) at
the expiration of the options. If no arbitrage
opportunities exist, they should also have the
same value today which implies:
c + Xe-rT = p + S
Put Call Parity
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In other words, the value of a European call with
a given exercise date can be deduced from the
value of a European put with the same exercise
date and exercise price.
Put call Parity
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Without this condition arbitrage opportunities
exist:
Put-Call Parity specifies that:
c + Xe-rT = p + S
which rearranges to
p = c + Xe-rT - S
A Fixed Income Example
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Previously we discussed had a call option on an
8% coupon bond with a par value of $100 and
20 years to maturity. The option expires in one
month and has an exercise price of $100. The
option is currently at the money (the bond is
selling at par) and selling for $3.
Assume we also have a put option on the same
bond and the put option is selling for $2.
A Fixed Income Example
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For now, ignore the coupon payments on the
bond.
Consider three possible strategies
simultaneously
Buy the bond in the spot market for $100
Enter into a short call position (sell the call) for $3
Buy a put at a price of $2
Possible Outcomes at expiration
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Bond Price > $100
The call option is exercised so you are forced to sell
the bond at a price equal to $100. The put option
expires. You make $1 profit from the difference in the
call and put prices
Bond Price <$100
You exercise the Put option and sell the Bond for
$100, which is the same price you paid. The Call
expires worthless . You make a $1 from the difference
in the price.
Arbitrage
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Regardless of the price of the bond at expiration,
there was a $1 profit.
Three possible things would eliminate arbitrage.
An increase in the price of the bond today.
A decrease in the call option price.
An increase in the put option price.
Arbitrage continued
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Assume that the price of the bond doesn’t
change:
There would be an increase in market
participants attempting to short call options. To
compete with each other they lower the price
and the call price will decrease.
There will be an increase in the number of
market participants wanting to purchase long put
options. To compete with each other they will
offer higher prices increasing the price.
Put Call Parity Revisited
p = c + Xe-rT - S
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We have ignored the Time value of money so the
relationship becomes:
p=c+X-S
In our example X=S which implies that p should
equal c.
If both the put and call price equaled each other
there would be no arbitrage profits regardless of
what happened to the bond price at maturity.
Put Call Extensions
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We ignored the time value of money and the
coupon payments paid by the bond.
The coupon payment can be treated similar to
the price. If you own the bond you will receive
the cash payment in the future.
The put call parity relationship for a coupon bond
is simply:
p=c+Xe-rT+CPe-rT-S
where CP is the coupon payment received at time T
Put Call Parity and
American Options
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Put-Call Parity holds only for European Options
but it is possible to use the relationship to specify
some generalizations concerning the relationship
between American Puts and Calls.
American call Option vs.
European Call Option
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Should an American call option on a non dividend
paying stock be exercised prior to maturity?
NO (assuming that the investor plans to hold the
stock past the maturity date of the option.)
Should an American call Option
be Exercised Early?
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Assume that the option currently is deep in the
money, The following possibilities exist
1) S > X The investor can earn interest amount
of cash equal to X and then still pay X for the
stock upon expiration of the option.
2) S < X The investor can then purchase the
stock at the spot price and let the option expire.
3) S=X Again there is no reason to exercise the
option, and the investor will let the option expire.
Exercising Call Options
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Since it is never optimal to exercise the call early,
the value of the American Call (C) should be
equal to the value of the European Call (c).
Exercising Put Options
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Should an American put option on a non dividend
paying stock be exercised prior to maturity?
Yes (if it is sufficiently in the money)
The general argument is that the Put option
serves as insurance and that early exercise is a
good idea if the investor realizes a significant
gain from the exercise of the option
Exercising Put Options
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The price of an American put option should be
above that of an Equivalent European option
(P>p)
The value of an American Call should equal the
value of an European Call.
Using the put call parity relationship and
substituting generalizations can be made about
American Options:
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P > p = c + Xe-rT - S
P > C + Xe-rT - S
Which rearranges to
C - P < S - Xe-rT
It can also be shown that C - P > S-X
Which combines with the above equation to
prove:
S - X < C - P < S - Xe-rT
Black Scholes
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The basic starting point for the actual pricing of
an European option is the model developed by
Fisher Black, Myron Scholes, and Robert Merton.
Black Scholes Assumptions
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1)
2)
3)
4)
5)
6)
7)
Stock prices follow a lognormal distribution
with m and s constant.
There are no transaction costs or taxes and all
securities are perfectly divisible
There are no dividend on the asset during the
life of the option
There are no riskless arbitrage opportunities
Security trading is continuous
Investors can borrow and lend at the same risk
free rate
The short term risk free rate is constant
Inputs you will need
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S = Current value of underlying asset
X = Exercise price
t = life until expiration of option
r = riskless rate
s2 = variance
Black Scholes
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Value of Call Option = SN(d1)-Xe-rtN(d2)
S = Current value of underlying asset
X = Exercise price
t = life until expiration of option
r = riskless rate
s2 = variance
N(d ) = the cumulative normal distribution
(the probability that a variable with a
standard normal distribution will be less than
d)
Black Scholes (Intuition)
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Value of Call Option
SN(d1)
-
Xe-rt
N(d2)
The expected PV of cost
Risk Neutral
Value of S
of investment Probability of
if S > X
S>X
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Value of Call Option = SN(d1)-Xe-rtN(d2)
Where:
S
s
ln(
)  (r 
)t
X
2
d1 
s t
2
d 2  d1  s
t
Extending Black Scholes to
Futures Options
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Black extended the original model to price
options on futures.
c  e  rt ( F0 N (d1 )  XN (d 2 ))
pe
 rt
( XN (d 2 )  F0 N ( d1 ))
F0 
2

ln 
s T /2

X

d1 
s T
d 2  d1  s T
Time Value of an Option
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The time value of an option is the difference in
the theoretical price of the option and the
intrinsic value.
It represents the the possibility that the value of
the option will increase over the time it is owned.
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Time Value of Call Option
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Payoff
S-X
Time value
of option
X
S
Spot
Price
Delta of an option
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The delta of the option shows how the
theoretical price of the option will change with a
small change in the underlying asset.
change in price of call option
delta 
change in price of underlying bond
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Time Value of Call Option
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Payoff
S-X
Time value
of option
X
S
Spot
Price
Delta is the slope of the tangent line
at the given stock price
Delta of an option
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Intuitively a higher stock price should lead to a
higher call price. The relationship between the
call price and the stock price is expressed by a
single variable, delta.
The delta is the change in the call price for a
very small change it the price of the underlying
asset.
Delta
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Delta can be found from the call price equation as:
c

 N (d1 )
S
Using delta hedging for a short position in a European
call option would require keeping a long position of N(d1)
shares at any given time. (and vice versa).
Delta explanation
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Delta will be between 0 and 1.
A 1 cent change in the price of the underlying
asset leads to a change of delta cents in the
price of the option.
Delta
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For deep in the money call options the delta will
be close to 1.
For deep out of the money call options the delta
will be close to zero.
Gamma
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Gamma measures the curvature of the
theoretical call option price line.
change in price of option
gamma 
Change in price of underlying bond
Gamma of an Option
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The change in delta for a small change in the
stock price is called the options gamma:
Call gamma =
e
 d 12 / 2
Ss 2T
Other Measures
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change in th e price of option
theta 
decrease in time to expiration
Change in option Price
kappa 
Change in expected
Hedging
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If a firm is worried about an increase in
borrowing costs it could buy a call option on the
relevant interest rate. Any gains on the call
options will offset the increased borrowing.
Similarly if the firm is worried about a decline in
rates decreasing income it could buy a put option
on the interest rate. Any decline in income
would be offset by the change in rates.
A Short Hedge
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Agree to sell 10 Eurodollar future contracts (each
with an underlying value of $1 Million).
We want to look at two results the spot market
and the futures market. Assume you close out
the futures position and that the futures price will
converge to the spot at the end of the three
months.
Rates increase to 8%
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Spot position:
Need to pay 8% + 1% = 9% on $10 Million $10
Million(.09/4) = $225,000
Futures Position:
Fut Price = $92 interest rates increased by .9%
Close out futures position:
profit = ($10 million)(.009/4) = $22,500
Rates Increase to 8%
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Net interest paid
$225,000 - $22,500 = $202,500
$10 million(.0810/4) = $202,500
Rates decrease to 6%
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Spot position:
Need to pay 6% + 1% = 7% on $10 Million
$10 Million(.07/4) = $175,000
Futures Position:
Fut Price = $94 interest rates decreased by 1.1%
Close out futures position:
loss = ($10 million)(.011/4) = $27,500
Rates Decrease to 8%
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Net interest paid
$175,000 + $27,500 = $202,500
$10 million(.0810/4) = $202,500
Results of Hedge
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Either way the final interest rate expense was
equal to 8.10 % or 100 basis points above the
initial futures rate of 7.10%
Should the position be hedged?
It locks in the interest rate, but if rates had
declined you were better off without the hedge.
Hedging with options
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Assume that you can purchase a put option on
the futures contract with a strike price of $93
(7%) and a cost of .40
If interest rates rates increase to 8% the put
guarantees that the worst case would be a rte of
7% +1% +.4% = 8.4% which implies a total
interest cost of $210,000
If rates decrease to 7% you only have the added
cost of the option resulting in a total interest
expense of 7.4% or $185,000
Hedging Interest Rate Risk
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Previously we purchased a put option on a
eurodollars futures contract to hedge against a
change in interest rates.
The result was that it limited the upper rate that
might be paid and allowed a decrease in rates to
decease the actual rate paid (the option wasn’t
exercised).
There is a (sometimes substantial) cost to
entering into the option contracts to accomplish
this.
Option position
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In the example the option profited as the level of
interest rates increased (the price of bonds
decreased)
Profit Diagram Put Option
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Profit
X-S-C
Spot Price
S
Cost
X
Spot Position
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The investor was attempting to hedge against an
increase in the level of interest rates, in other
words, they paid a higher borrowing cost as rates
increased.
This is the same idea as saying the they lost
money as the price of the bonds declined.
This was offset by the profit from the option, but
you incur the cost of the option.
Diagramming the spot
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The spot position could be represented by a
straight line that represents the corresponding
savings in interest rates.
The line will also slope up to the right. As the
price increases (rate decreases) there is a
relative improvement since the rate decrease
saves the investor money.
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Profit Diagram Spot
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Profit
Spot Price
Cost
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Assume that the current interest rate is just
below the rate implied by the strike price.
The two positions could be represented on a
single graph which explains the results of the
hedge.
At rates above the strike price the profit on the
option cancels out the loss from the increased
rates. At rates below the strike price you gain,
but the gain is reduced by the cost of the option.
Profit Diagram Put Option
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X
Combined
Position
“Selling off” benefits
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It is possible to decrease the impact of the option’s
cost
One approach would be buying a cap as before, but
also selling at cap at a higher rate (lower price).
The money received from selling the option offsets
the initial cost of the other option position.
The downside is that if rates increased above the
second level, you are exposed to the interest rate
change.
Selling off benefits –
options position
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However if the level of interest rates increased
too high the option that was sold would
experience a loss offsetting the gains from the
original position.
Therefore the original position is no longer
hedged.
Assume that the two options are for the same
expiration date and are both European
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Profit Diagram
Put Options
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Short Put
Spot Price
Long Put
Profit Diagram
Bear Spread
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Spot price
Bear Spread
Bear Spread
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The previous example was essentially buying an
interest rate cap (buying put on price of bonds)
and selling an interest rate cap (selling a put on
the price of bonds). The position could also be
thought of as buying and selling call options on
the level of interest rates.
Below the lower price (above the higher yield)
the two options cancel each other out so the
increased cost associated with the spot position
is unhedged.
Adding the spot position
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Again assume that the current level of interest
rates is slightly below the lower of the two strike
prices.
Profit Diagram
Bear Spread
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Spot price
Hedged
Position
Bear Spread
Another Strategy
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To avoid the downside of the previous example
you could buy the same interest rate cap, but sell
an interest rate floor at a higher price (lower
yield).
Interest Rate Floor
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A call option on the price of the bond can be
represented as a floor on the level of interest
rates.
The option will be profitable if the price of the
bond increases above the strike price (the
interest rate decreases below the strike).
This would offset a loss on an asset that is rate
sensitive and effectively limit the loss.
Profit Diagram Call Option
(Long Call Position)
Profit
S-X-C
S
Cost
X
Spot Price
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Spot position
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Since the option profits as the rates decrease
(the price of a bond increases) this would offset
lost income on an asset that is rate sensitive.
In our new position we want to sell the option.
Profit Diagram Call Option
(Short Call Position)
Profit
X
C+X-S
S Spot Price
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Another Strategy
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The new strategy is to avoid the downside of the
previous example you could buy the same
interest rate cap, while selling an interest rate
floor at a higher price (lower yield).
Profit Diagram Call Option
(Short Call Position)
Profit
Long Put
Short Call
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Profit Diagram
Costless Collar
Profit
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Long Put
Combined
Profit
Short
Call
Combined with Spot
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By using options selling at the same price the net cost is
zero.
At prices above the higher strike price (below the lower
yield) the gain is offset by a loss in the option position.
At prices below the lower strike price (above the higher
yield) the loss is offset by gains in the option.
You have limited both the gain and loss.
Assume that the current interest rate is exactly between
the two strike prices
Profit Diagram
Costless Collar (Fence)
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Profit
Costless
Collar
Complications
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In the previous examples we ignored many real
world complications.
This is especially true if you are buying options
on futures (treasury bond futures for example).
Many of these complications arise from the ideas
of basis risk presented earlier.
New Example: Protective Put
(From Fabozzi)
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Assume that you own a corporate bond and you
are afraid that an increase in interest rates will
decrease the value of the bond.
It would be possible to use futures or futures
options to lock in a future sale price for the
bonds.
Assume that the coupon on the bond is 11.75%
and they mature on April 19, 2023. Today is
April 19, 1985 and you plan to sell the bond in
June 1985.
The Hedge
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To hedge against the possible increase in interest
rates you decide to buy a put option on the
treasury bond futures contract.
If interest rates increase, the price of the
underlying bonds will decrease allowing you to
own a short futures position with the higher
futures price (equal to the strike price).
Determining the Strike Price
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The strike price will effectively set a cap on the
level of interest rates since it rates increase
above the rate corresponding to the strike price
you profit from the option offsetting the loss in
bond value.
Assume that you do not want the price of the
bond to drop below $87.668.
Target Strike Price = 87.668
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The problem is that the futures option is not for
the bond which you own, it is for a treasury
bond.
You need to set a strike price for the Treasury
bond that corresponds with a price of 87.688 for
the corporate bond.
Price Vs. Yield
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Choosing the minimum price is equivalent to
selecting the maximum yield on the corporate
bond.
A price of 87.668 implies that the corporate bond
will be paying a yield of 13.41% (since it is
selling at a large discount the yield will be above
the coupon rate)
Yield on CTD treasury
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The futures contract underlying the option has a
large set of acceptable treasuries that can be
delivered. You can find the cheapest to deliver
at the current date.
Assume that after finding the cheapest to deliver
bond, you find that is has a current yield 90 basis
points less than current yield on the corporate.
Assume that the yield spread stays fixed.
Price of CTD Treasury
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Given that the yield spread stays fixed at 90
basis points and that the maximum acceptable
yield on the corporate bond is 13.41% it implies
a yield of 12.51% on the treasury.
This implies a price of $63.756 for the treasury
that is currently CTD.
The price used in the futures option will not be
this price, it must be found using the conversion
factor
Finding the Strike Price
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Given the treasuries price of $63.756 and the
conversion factor for the treasury of .9660, a
futures price is then found to be:
63.756/.9660 = 66
Therefore a strike price of 66 on the treasury
futures option contract would be used.
Hedge ratio
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Hedging the position with just futures contract
would have required finding the hedge ratio, this
still applies.
Assuming that you found the hedge ratio to be
1.24, you will need 1.24 put futures options for
each spot position.
Another approach:
A Covered Call
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The assumption is that the change in interest
rates will be small. To hedge against a possible
decline in rates, the holder of a bond (or
portfolio) sells out of the money calls.
The income from the sale of the option provides
income to offset a possible increase in rates that
lowers the bond value.
The Maximum
Effective Call Price
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Assume that the maximum effective call price
you set is 102.66 plus the premium from the call
option.
The price of 102.66 corresponds to a yield of
11.436% on the corporate bonds.
Keeping the 90 basis point spread the yield on
the CTD treasury should be 10.536% which
implies a 75.348 price for the treasury
The strike price
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Using the conversion factor, this implies a futures
price of 75.348/.9960 = 78 which is also the
strike price on the call option you sell.
Comparing Strategies
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Comparing a basic futures position to the
covered call and protective put in the previous
examples shows that each has its own
advantages and disadvantages.
The basic futures position sets the price (and
yield) regardless of what happens to the level of
interest rates in the economy. However the
other two provide scenarios where you are better
off than this ( and scenarios where you would
have been worse off)
Comparing Strategies
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The protective put does better if rates decrease
and the call option in the covered call option is
exercised. The protective put also outperforms
the basic futures option if as rates decline, but it
is outperformed by the covered call.
For extreme rate increases, the option strategies
are both outperformed by the basic futures
position.