Download V 1 +n($m)

Options on
Stock Indices, Currencies, and
Futures
Chapter 13
1
STOCK INDEX OPTIONS
ONE CONTRACT VALUE =
(INDEX VALUE)($MULTIPLIER)
One contract = (I)($m)
ACCOUNTS ARE SETTLED BY CASH
2
STOCK INDEX OPTIONS FOR
PORTFOLIO INSURANCE
Problems:
1. How many puts to buy?
2. Which exercise price will guarantee
the desired level of protection?
The answers are not easy because the
index underlying the puts is not the
portfolio to be protected.
3
The protective put with a single stock:
AT
STRATEGY
ICF
Hold the
stock
Buy put
-St
TOTAL
-St – p
-p
EKPIRATION
ST < K
ST
K - ST
K
ST ≥ K
ST
0
ST
4
The protective put consists of holding the
portfolio and purchasing n puts.
AT
STRATEGY
EXPIRATION (T = 1)
ICF (t = 0)
-V0
I1 < K
I1 ≥ K
V1
V1
Hold the
portfolio
Buy n puts
-n P($m)
TOTAL
-V0 –nP($m) V1+n($m)(K- I1)
n(K- I1)($m)
0
V1
5
WE USE
THE CAPITAL ASSET PRICING MODEL.
For any security i,the expected excess
return on the security and the expected
excess return on the market portfolio are
linearly related by their beta:
ER i  rF  βi (ER M  rF )
6
THE INDEX TO BE USED IN THE
STRATEGY, IS TAKEN TO BE A PROXY FOR
THE MARKET PORTFOLIO, M. FIRST,
REWRITE THE ABOVE EQUATION FOR THE
INDEX I AND ANY PORTFOLIO P :
ER p  rF  β p (ER I  rF ).
7
Second, rewrite the CAPM result, with
actual returns:
R p  rF  β p (R I  rF ).
In a more refined way, using V and I for the
portfolio and index market values,
respectively:
V1 - V0  D P
I1 - I 0  D I
 rF  β p [
 rF ].
V0
I0
8
NEXT, use the ratio Dp/V0 as the portfolio’s
annual dividend payout ratio qP and DI/I0
the index annual dividend payout ratio, qI.
V1 - V0
I1 - I0
 q P  rF  βp [
 q I  rF ]
V0
I0
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ]
V0
I0
The ratio V1/ V0 indicates the portfolio
required protection ratio.
9
For example:
V1
 .90,
V0
The manager wants V1, to be down to no
more than 90% of the initial portfolio
market value, V0:
V1 = V0(.9).
We denote this desired level by (V1/ V0)*.
This is the decision variable.
10
1.
The number of puts is:
V0
n  βp
.
($m)I 0
11
2. The exercise price, K, is determined by
substituting I1 = K and the required level,
(V1/ V0)* into the equation:
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ],
V0
I0
and solving for K:
V1
K
( ) * 1  q P  rF  β p [ - 1  q I - rF ].
V0
I0
I0 V1
K  [( ) * q p - (β p )q I  (1  rF )(β p -1)].
β p V0
12
We rewrite the Profit/Loss table for the protective put
strategy:
AT EXPIRATION
STRATEGY
INITIAL CASH
FLOW
Hold the portfolio
I1 < K
I1 ≥ K
-V0
V1
V1
-n P($m)
n(K - I1)($m)
0
V1+n($m)(K - I1)
V1
Buy n puts
TOTAL
-V0 - nP($m)
We are now ready to calculate the floor
level of the portfolio: V1+n($m)(K- I1)
13
We are now ready to calculate the floor
level of the portfolio:
Min portfolio value = V1+n($m)(K- I1)
This is the lowest level that the portfolio
value can attain. If the index falls below
the exercise price and the portfolio value
declines too, the protective puts will be
exercised and the money gained may be
invested in the portfolio and bring it to the
value of:
V1+n($m)K- n($m)I1
14
Substitute for n:
V0
n  βp
.
($m)I 0
V0
Min porfolio value  V1  β p
($m)K
($m)I 0
V0
 βp
($m)I 1
($m)I 0
V0
V0
Min portfolio value  V1  β p
K - βp
I1.
I0
I0
15
To substitute for V1 we solve the equation:
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ]
V0
I0
I1
V1  V0 (1  q P  rF  β p [ - 1  q I - rF ])
I0
V0
V1  β p
I1
I0
 V0 1  rF  q p  β p [q I  1  rF ]
16
3.
Substitution V1 into the equation for
the Min portfolio value
Min portfolio value 
V0
βp
K  V0 [β p q I  q p  (1  rF )(1  β p )].
I0
The desired level of protection is made at
time 0. This determines the exercise
price and management can also calculate
the minimum portfolio value.
17
EXAMPLE: A portfolio manager expects the
market to fall by 25% in the next six
months. The current portfolio value is
$25M. The manager decides on a 90%
hedge by purchasing 6-month puts on the
S&P500 index. The portfolio’s beta with the
S&P500 index is 2.4. The S&P500 index
stands at a level of 1,250 points and its
dollar multiplier is $100. The annual riskfree rate is 10%, while the portfolio and
the index annual dividend payout ratios are
5% and 6%, respectively. The data is
18
summarized below:
V1
V0  $25,000,000; ( )*  .9; I 0  1,250;
V0
$m  $100;
The annual rates are : rF  10%; q p  5%; q I  6%.
Finally, β  2.4.
Solution: Purchase
V0
n  βp
($m)I 0
$25,000,000
n  2.4
 480 puts.
($100)(1,2 50)
19
The exercise price of the puts is:
I 0 V1
K  [( ) * q p - (β p )q I  (1  rF )(β p - 1)].
β p V0
1,250
K
[.9  .025  (2.4).03  (1  .05)(2.4  1)
2.4
K  1,210.
Solution: Purchase n = 480 six-months
puts with exercise price K = 1,210.
20
Min portfolio value 
V0
βp
K  V0 [β p q I  q p  (1  rF )(1  β p )].
I0
$25,000,000
 2.4
1,210
1,250
 $25,000,000[2.4(.03) - .025  (1  .95)(1 - 2.4)]
 $22,505,000.
21
CONCLUSION:
Holding the portfolio and purchasing
480, 6-months protective puts on
the S&P500 index, with the exercise
price K = 1210, guarantees that the
portfolio value, currently $25M, will
not fall below $22,505,000 in six
months.
22
Example (page 274) protection for 3 months
V1
V0  $500,000; ( )*  .9; I 0  1,000;
V0
$m  $100;
The annual rates are : rF  12%; q p  4%; q I  4%.
Finally, β  2.0
Solution: Purchase
V0
n  βp
($m)I 0
$500,000
n  2.0
 10 puts.
($100)(1,0 00)
23
The exercise price of the puts is:
I 0 V1
K  [( ) *  q p - (β p )q I  (1  rF )(β p - 1)].
β p V0
1,000
K
[.9  .01  (2.0).01  (1  .03)(2.0  1)
2.0
K  960.
Solution: Purchase n = 10 six-months
puts with exercise price K = 960.
24
Min portfolio value 
V0
βp
K  V0 [β p q I  q p  (1  rF )(1  β p )].
I0
$500,000
 2.0
960
1,00
 $500,000[2.0(.01) - .01  (1  .03)(1 - 2.0)]
 $450,000.
25
CONCLUSION:
Holding the portfolio and purchasing
10, 3-months protective puts on the
S&P500 index, with the exercise
price K = 960, guarantees that the
portfolio value, currently $500,000
will not fall below $450,000 in three
months.
26
A SPECIAL CASE: In the case that
1. β = 1 and 2.
qP =qI, the portfolio is
statistically similar to the index.
In this case:
V0
n
($m)I 0
V1
K  I 0 ( ) * and
V0
Min portfolio value is  V0 (V1/V0 ) .
*
27
Assume that in the above example, βp =
1 and qP =qI, then:
V1
K  I 0 ( ) *  1,250(.9)  1,125.
V0
V0
$25,000,000
n

 200 puts and
($m)I 0
$100(1250)
Min portfolio value 
V  V0 (.9)  $25,000,000(.9)  $22,500,000.
*
1
*
28
Example: (page 273)
βp = 1 and qP =qI, then:
V1
K  I 0 ( ) *  1,000(.9)  900.
V0
V0
$500,000
n

 5 puts and
($m)I 0
$100(1,000)
Min portfolio value 
V  V0 (.9)  $500,000(.9)  $450,000.
*
1
*
29
FOREIGN CURRENCY FUTURES
Standardized Options Currencies Traded:
The PHLX lists six dollar-based
standardized currency option contracts,
which settle in the actual physical
currency. These are the most heavily
traded currencies in the global inter bank
market.
Contract Size: The amounts of currency
controlled by the various currency
options contracts are geared to the
needs of the widest possible range of
participants.
30
Currency options
Units
USD/AUD
50,000AUD
USD/GBP
31,250GBP
USD/CD
50,000CD
USD/EUR
62,500EUR
USD/JPY
6,250,000JPY
USD/CHF
62,500CHF
Exercise Style: American- or European
options available for physically settled
contracts; Long-term options are
31
European-style only.
Expiration/Last Trading Day
The PHLX offers a variety of expirations in
its physically settled currency options contracts,
including Mid-month, Month-end and Long-term
expirations. Expiration, which is also the last day
of trading, occurs on both a quarterly and
consecutive monthly cycle. That is, currency
options are available for trading with fixed
quarterly months of March, June, September and
December and two additional near-term months.
For example, after December expiration, trading
is available in options which expire in January,
February, March, June, September, and
December. Month-end option expirations are
available in the three nearest months.
32
With the Canadian dollar spot price currently
at a level of USD.6556/CD, strike prices would
Options
be listed inStandardized
half-cent intervals
ranging from 60
to 70. i.e., 60, 60.5, 61, …, 69, 69.5, 70. If the
Canadian dollar spot rate should move to say
USD.7060/CD, additional strikes would be
listed. E.G, 70, 70.5, 71, …, 75.
Exercise Prices
Exercise prices are expressed in terms of
U.S. cents per unit of foreign currency.
Thus, a call option on EUR with an
exercise price of 82 would give the
option buyer the right to buy Euros at 82
cents per EUR.
33
It is important that available exercise prices
relate closely to prevailing currency values.
Therefore, exercise prices are set at certain
intervals surrounding the current spot or
market price for a particular currency. When
significant price changes take place,
additional options with new exercise prices
are listed and commence trading.
Strike price intervals vary for the different
expiration time frames. They are narrower
for the near-term and wider for the longterm options.
34
Premium Quotation premiums for dollarbased options are quoted in U.S. cents
per unit of the underlying currency with
the exception of Japanese yen which are
quoted in hundredths of a cent.
Example:
A premium of 1.00 for a given EUR option
is one cent (USD.01) per EUR.
Since each option is for 62,500 EURs, the
total option premium would be
[62,500DM][USD.01/EUR] = USD625.
35
Customized Currency Options
Currencies Traded
Customized currency options can be traded on
any combination of the currencies currently
available for trading. Currently, AUD must be
denominated in U.S. dollars. AUD premiums must
be denominated in USD.
In the case of an option on the USD in CD, the
underlying currency is U.S. dollars. The strike
prices and premiums are quoted in Canadian
dollars. E.G, a call option on the USD with a
strike price of 1.542 gives the buyer the right
to purchase 50,000 USD at CD1.542/USD. 36
Underlying Currency
The underlying currency is that currency which is
purchased (in the case of a call) or sold (in the
case of a put) upon exercise of the contract.
Base Currency
The base currency is that currency in which terms
the underlying is being quoted, i.e. strike price.
Expiration/Last Trading Day
Expirations can be established for any business
day up to two years from the trade date.
Customized option contracts expire at 10:15 a.m.,
Eastern Time on the expiration day in contrast
with standardized options which expire at 11:59
p.m., Eastern Time on the expiration day.
37
In addition, the exercise and
assignment process for customized
options is more akin to the OTC market
in terms of expiration timeframe. Unlike
the process utilized for standardized
options, exercise notices must be
received by 10:00 a.m., Eastern Time
and the writer is then notified of the
number of contracts assigned. If
necessary, contact the PHLX for more
details.
The contract size for customized
currency options is:
38
Underlying currency
Contract size
Australian dollar
50,000AUD
British Pound
31,250GBP
Canadian dollar
50,000CD
Euro
62,500EUR
Japanese Yen
6,250,000JPY
Mexican Peso
250,000MXP
Spanish Peseta
5,000,000ESP
Swiss Frank
62,500CHF
USD
50,000USD.
39
Exercise Prices
Exercise or strike prices may be expressed
in any increment out to four characters. For
example, a USD/GBP option could have an
exercise price of 1.543.
Exercise-style European-style only.
Minimum Transaction Size
Since customized currency options were
designed for the institutional market, there
is a minimum opening transaction size
which equals or exceeds 50 contracts. 40
A call option on the EUR quoted in American
terms would have a strike price expressed in
USD. For example, USD.8484 per EUR.
A similar option expressed in European terms
would be a put option on USD with a strike
price expressed in EUR. For example, 1.1787
EUR per USD.
41
Contract Terms
An option may be expressed in
either American terms or European
terms (inverse terms). For example,
an option in American terms would
have exercise prices quoted in terms
of U.S. dollar per unit of foreign
currency. An option in European or
inverse terms would have exercise
prices quoted in terms of units of
foreign currency per U.S. dollar.
42
Trading Process
Trading is conducted in an open outcry
auction market, just as in standardized
option contracts. When initial interest in a
customized option series is expressed, a
floor member must first present a
Request For Quote (RFQ) to an Exchange
staff member for dissemination.
Subsequently, responsive quotes are
generated by competing market makers
and floor brokers representing off floor
interest.
43
Premiums
Premiums may be expressed either in
terms of the base currency per unit of
the underlying currency or in percent of
the underlying currency (based on
contract size). For example, the
premium for an option on the USD/EUR
(USD being the base currency and EURO
being the underlying currency) could be
expressed in U.S. cents per EUR or as a
percentage of 62,500 EUROS.
44
Price and Quote Dissemination
Request For Quotes (RFQs), responsive quotes and
trades will be disseminated to the Option Price
Reporting Authority (OPRA) for availability to quote
vendors
The premium for an option on the EUR with a strike
price in USD (EUR is the underlying currency and the
USD is the base currency) quoted in cents per
EUR(premium of 2.50) would be calculated as follows:
the aggregate premium for each contract =
USD1562.50(USD.025 x 62,500EUR per contract ).
Similarly, if this option were quoted in percentage of
underlying and the premium was 2.5%, the premium
amount for each contract = 1562.5 EUR (.025 x 62,500
45
EUR per contract).
Position Limits
Position limits are the maximum number of contracts
in an underlying currency which can be controlled by
a single entity or individual. Currently, position limits
are set at 200,000 contracts on each side of the
market (long calls and short puts or short calls and
long puts) for each currency except the Mexican
peso, and the Spanish peseta which are 100,000
contracts. For purposes of computing position limits,
all options involving the U.S. dollar against another
currency will be aggregated with each other for each
currency (i.e., USD/EUR and EUR/USD on the same
side of the market will be aggregated - USD/EUR
long calls and short puts with EUR/USD short calls
and long puts).
46
FX Options As Insurance
Options on spot represent insurance
bought or written on the spot rate. (options
on futures represent insurance bought or
written on the futures price.) An individual
with foreign currency to sell can use put
options on spot to establish a floor price on
the domestic currency value of the foreign
currency. For example, a put option on
EUR with an exercise price of USD.80/EUR
ensures that, if the value of the EUR falls
below USD.80/EUR, the EUR can be sold for
47
USD.80/EUR.
If the put option costs USD.01/EUR, the
floor price can be roughly approximated as:
USD.80/EUR - USD.O1/EUR = USD.79/EUR.
That is, if the option is used, the put holder
will be able to sell the EUR for the
USD.80/EUR strike price, but in the
meantime, have paid a premium of
USD.01/EUR. Deducting the cost of the
premium leaves USD.79/EUR as the floor
price established by the purchase of the
put. This calculation ignores fees and
interest rate adjustments.
48
Similarly, an individual who has to buy
foreign currency at some point in the
future can use call options on spot to
establish a ceiling price on the domestic
currency amount that will have to be
paid to purchase the foreign exchange.
49
For example, a call option on EUR with an
exercise price of USD.80/EUR will ensure
that, in the event the value of the EURO
rises above USD.80/EUR, the EUR can be
bought for USD.80/EUR anyway. If the call
option costs USD.02/EUR, this ceiling price
can be approximated:
USD.80/EUR + USD.02/EUR = USD.82/EUR
or the strike price plus the premium.
50
To summarize these two important points
involving FX puts and calls:
1- Foreign currency put options on spot can be
used as insurance to establish a floor price on
the domestic currency value of foreign
exchange. This floor price is approximately
Floor price = exercise price of put
- put premium
2- Foreign currency call options on spot can be
used as insurance to establish a ceiling price
on the domestic currency cost of foreign
exchange. This ceiling price is approximately
Ceiling price = exercise price of call
+ call premium.
51
These calculations are only approximate for
essentially two reasons. First, the exercise price and
the premium of the option on spot cannot be added
directly without an interest rate adjustment. The
premium will be paid now, up front, but the exercise
price (if the option is eventually exercised) will be
paid later. The time difference involved in the two
payment amounts implies that one of the two should
be adjusted by an interest rate factor. Second, there
may be brokerage or other expenses associated with
the purchase of an option, and there may be an
additional fee if the option is exercised. The
following two examples illustrate the insurance
feature of FX options on spot and show how to
calculate floor and ceiling values when some
additional transactions costs are included.
52
Example 1: A U.S. importer will have a net cash
out flow of £250,000 in payment for goods
bought in Great Britain. The payment date is not
known with certainty, but should occur in late
November. On September 16 the importer locks
in a ceiling purchase for pounds by buying 8 PHLX
calls [£250,000/£31,250 = 8] on the pound, a
strike price K = USD1.50/GBP and a December
expiration. The option premium on September 16
is USD.0220/GBP. With a brokerage commission
of $4/option, the total cost of the eight contracts
is:
8(£3l,250)($.0220/£) + 8($4) = $5,532.
53
Measured from today's viewpoint, the importer has
essentially assured that the purchase price for pounds
will not be greater than:
$5,532/£250,000 + $1.50/£ = $.02213/£ + $1.50/£ =
USD1.52213/GBP.
Notice here that the add factor USD.02213/GBP is
larger than the option premium of USD.0220/GBP by
USD.00013/GBP, which represents the dollar brokerage
cost per pound. The number USD1.52213/GBP is the
importer's ceiling price. The importer is assured he will
not pay more than this, but he could pay less. The price
the importer will actually pay will depend on the spot
price on the November payment date. To illustrate this,
we can consider two scenarios for the spot rate.
54
Case A. The spot rate on the November
payment date is USD1.46/GBP. The
importer would not exercise the call but
would buy pounds spot at the rate of
USD1.46/GBP. The importer then sell
the eight calls for whatever market
value they had remaining. Assuming, a
brokerage fee of $4 per contract for the
sale, the options would be sold as long
as their remaining market value was
greater than $4 per option. The total
cost will have turned out to be:
USD1.46/GBP+USD.02213/GBP
55
- (sale value of options- $32)/£250,000.
If the resale value is not greater than $32, then the total
cost per pound is $1.46 + $.02213 = $1.48213.
The USD.02213/GBP that was the original cost of the
premium and brokerage fee turned out in this case to be
an unnecessary expense.
[Now, to be strictly correct, a further adjustment to the
calculation should be made. Namely, the $1.46 and
$.02213 represent cash flows at two different times.
Thus, if R is the amount of interest paid per dollar over
the September 16 to November time period, the proper
calculation is
$1.46+$.02213(l+R)
- (sale value of options-$32)/250,000.]
56
Case B. The spot rate on the November payment
date is USD1.55/GBP. The importer can either
exercise his options or sell them for their market
value. Assume the importer sells them at a
current market value of $.055 and pays $32 total
in brokerage commissions on the sale of eight
option contracts. The importer then buys the
pounds in the spot market for USD1.55/GBP. The
total cost is, before adding the premium and
commission costs paid in September:
(USD1.55/GBP)(£250,000) –
(USD.055/GBP))( £250,000) + 8($4)
= $373,782.
This amount is:
$373,782/£250,000 = USD1.49513/GBP.57
Adding in the premium and commission
costs paid back in September, the total cost
is
USD1.49513/GBP + USD.02213(l + R)/GBP
If the importer chooses instead to exercise
the option, the calculations will be similar
except that the brokerage fee will be
replaced by an exercise fee.
This concludes Example 1.
58
Example 2 A Japanese company wants to
lock in a minimum yen value of USD50M,
this amount to be sold between July1 and
December 31. Since the company wishes to
sell USD and receive JPY, the company will
buy a put option on USD, with an exercise
price stated in terms of JPY.
Suppose the company buys from its bank
an American put on USD50M with a strike
price of JPY130/USD.
59
This call is purchased directly from the
bank thus, there is no resale value to the
option. The company pays a premium of
JPY4/USD. In this case, assume there are
no additional fees. Then, the Japanese
company has established a floor value for
its USD, approximately at
JPY130/USD - JPY4/USD = JPY126/USD.
Again, we can consider two scenarios, one
in which the yen falls in value to
JPY145/USD and the other in which the
yen rises in value to JPY115/USD.
60
Case A.
The yen falls to JPY145/USD. In this case
the company will not exercise the option
to sell dollars for yen at JPY13O/USD,
since the company can do better than this
in the exchange market. The company
will have obtained a net value of
JPY145/USD - JPY4/USD = JPY141/USD.
61
Case B.
The JPY rises to JPY115/USD. The
company will exercise the put and sell
each U.S. dollar for JPY130/USD. The
company will obtain, net,
JPY130/USD - JPY4/USD = JPY126/USD.
This is JPY11 better than would have
been available in the FX market and
reflects a case where the “insurance” paid
off. This concludes Example 2.
62
Writing Foreign Currency Options
General considerations. The writer of a foreign
currency option on spot or futures is in a different
position from the buyer of these options. The buyer
pays the premium up front and then can choose to
exercise the option or not. The buyer is not a source
of credit risk once the premium has been paid. The
writer is a source of credit risk, however, because the
writer has promised either to sell or to buy foreign
currency if the buyer exercises his option. The writer
could default on the promise to sell foreign currency if
the writer did not have sufficient foreign currency
available, or could default on the promise to buy
foreign currency if the writer did not have sufficient
domestic currency available.
63
If the option is written by a bank, this risk of default
may be small. But if the option is written by a
company, the bank may require the company to
post margin or other security as a hedge against
default risk. For exchange-traded options, as noted
previously, the relevant clearinghouse guarantees
fulfillment of both sides of the option contract. The
clearinghouse covers its own risk, however, by
requiring- the writer of an option to post margin. At
the PHLX, for example, the Options Clearing
Corporation will allow a writer to meet margin
requirements by having the actual foreign currency
or U.S. dollars on deposit, by obtaining an
irrevocable letter of credit from a suitable bank, or
by posting cash margin.
64
If cash margin is posted, the required deposit is the
current market value of the option plus 4 percent of
the value of the underlying foreign currency. This
requirement is reduced by any amount the option is
out of the money, to a minimum requirement of the
premium plus .75 percent of the value of the
underlying foreign currency. These percentages can
be changed by the exchanges based on currency
volatility. Thus, as the market value of the option
changes, the margin requirement will change. So an
option writer faces daily cash flows associated with
changing margin requirements.
65
Other exchanges have similar requirements for
option writers. The CME allows margins to be
calculated on a net basis for accounts holding both
CME FX futures options and IMM FX futures. That
is, the amount of margin is based on one's total
futures and futures options portfolio. The risk of an
option writer at the CME is the risk of being
exercised and consequently the risk of acquiring a
short position (for call writers) or a long position (for
put writers) in IMM futures. Hence the amount of
margin the writer is required to post is related to the
amount of margin required on an IMM FX futures
contract. The exact calculation of margins at the
CME relies on the concept of an option delta.
66
From the point of view of a company or individual,
writing options is a form of risk-exposure
management of importance equal to that of buying
options. It may make perfectly good sense for a
company to sell foreign currency insurance in the
form of writing FX calls or puts. The choice of strike
price on a written option reflects a straightforward
trade-off. FX call options with a lower strike price will
be more valuable than those with a higher strike
price. Hence the premiums the option writer will
receive are correspondingly larger. However, the
probability that the written calls will be exercised by
the buyer is also higher for calls with a lower strike
price than for those with a higher strike. Hence the
larger premiums received reflect greater risk taking
on the part of the insurance seller, ie., the option 67
writer.
Writing Foreign Currency
Options:
an example.
The following example will illustrate:
the risk/return trade-off for the case of
an oil company with an exchange rate
risk, that chooses to become an option
writer.
68
Example 3 Iris Oil Inc., a Houston-based
energy company, has a large foreign
currency exposure in the form of a CD
cash flow from its Canadian operations.
The exchange rate risk to Iris is that the
CD may depreciate against the USD. In
this case, Iris’ CD revenues, transferred
to its USD account will diminish and its
total USD revenues will fall. Iris chooses
to reduce its long position in CD by
writing CD calls with a USD strike price.
The strategy chosen is one of hedge ratio
1:1.
69
By writing options, Iris will receive an
immediate USD cash flow representing the
premiums. This cash flow will increase Iris'
total USD return in the event the CD
depreciates against the USD or, remains
unchanged against the USD, or appreciates
only slightly against the USD.
Clearly, the options might expire worthless
or they might be exercised. In either case,
however, Iris walks away with the full
amount of the options premium:
70
1. If the USD value of the CD remains
unchanged, the option premium
received is simply additional profit.
2. If the value of the CD falls, the premium
received on the written option will offset
part or all of the opportunity loss on the
underlying CD position.
3. If the value of the CD rises sharply, Iris
will only participate in this increased
value up to a ceiling level, where the
ceiling level is a function of the exercise
price of the option written.
71
In short, the payoff to Iris' strategy will
depend both on exchange rate
movements and on the selection of the
strike price of the written options.
To illustrate Iris' strategy, consider an
anticipated cash flow of CD300M over
the next 180 days. With hedge ratio of
1:1, Iris sells CD300,000,000/CD50,000
= 6,000 PHLX calls.
72
Assume: Iris writes 6,000 PHLX calls with a
6-month expiration; the current spot rate is
S = USD.75/CD and the 6-month forward
rate is:
F = USD.7447/CD.
For the current level of spot rate, logical
strike price choices for the calls might be X
= USD.74, or USD.75, or USD.76.
For the illustration, assume that Iris’
brokerage fee is USD4 per written call and
let the hypothetical market values of the
options be those listed in the following:
73
c(K = USD.74/CD) = USD.01379;
c(K = SUD.75/CD) = USD.00650;
c(K = USD.76/CD) = USD.00313.
n
K
Value
One call
.74
USD689.5
6,000
Total
Fees:
USD4/call
USD4,137,000 USD24,000
.75
USD325.0
6,000
USD1,950,000 USD24,000
.76
USD156.5
6,000
Value
Total Premium
USD939,000
USD24,000
74
The payoff to the total position depends on
the choice of exercise price, K, and the spot
exchange rate, S(USD/CD), at the calls’
expiration. We now introduce an additional
cost, that is associated with the exercise
fee, which exists in the real markets. If the
options are exercised, an additional
Options Clearing Corporation fee of USD35
per option is assumed. In our example
then, an exercise of the calls requires a
total OCC fee of: USD35(6,000) =
USD210,000 for the 6,000 written calls.
The total value of the long CD position of
Iris, plus short option position will be: 75
SPOT RATE
USD/CD
STRIKE PRICE
USD.74/CD
USD.75/CD
USD.76/CD
S<.74
S(CD300M)
+
USD4,113,000
S(CD300M)
+
USD1,926,000
S(CD300M)
+
USD915,000
.74<S<.75
USD225,903,000
S(CD300M)
+
USD1,926,000
S(CD300M)
+
USD915,000
.75<S.76
USD225,903,000
USD226,716,000
S(CD300M)
+
USD915,000
.76<S
USD225,903,000
USD226,716,000
USD228,705,000
76
Actually, the above table consolidates
three profit profile tables, each
corresponding to one of the three
strike prices under consideration.
The three tables are as follows:
77
Strategy
Initial
Cash Flow
Write 6,000,
.74 calls
USD4,113,000
Total
Total
Position:
Long
CD300M plus
short calls
Cash flow at Expiration
S< USD.74/CD
S>USD.74/CD
0
-(S-.74)300
USD210,000
USD4,113,000
-(S-.74)300
+
USD3,903,000
S(300M)
+
USD4.113,000
S(300M)
-(S-.74)300M
+
USD3,903,000
= USD225,903,000
78
Strategy
Initial
Cash Flow
Write 6,000,
.75 calls
USD1,926,000
Total
Total
Position:
Long
CD300M plus
short calls
Cash flow at Expiration
S< USD.75/CD
S>USD.75/CD
0
-(S-.75)300
USD210,000
USD1,926,000
-(S-.75)300
+
SUD1,716,000
S(300M)
+
USD1,926,000
S(300M)
-(S-.75)300M
+
USD1,716,000
=
USD226,716,000
79
Strategy
Write 6,000,
.76 calls
Total
Total
Position:
Long
CD300M plus
short calls
Initial
Cash Flow
USD915,000
Cash flow at Expiration
S< USD.76/CD
S>USD.76/CD
0
-(S-.76)300
USD210,000
USD915,000
-(S-.76)300
+
USD705,000
S(300M)
+
USD915,000
S(300M)
-(S-.76)300M
+
USD705,000
=
USD228,705,000
80
As illustrated by the consolidated table and
the three separate profit profile tables, the
lower the strike price chosen, the better
the protection against a depreciating CD.
On the other hand, a lower strike price
limits the corresponding profitability of the
strategy if the CD happens to appreciate
against the USD in six months. The optimal
decision of which strategy to take is a
function of the spot exchange rate at
expiration. One possible comparison is to
evaluate the options strategy vis-à-vis the
immediate forward exchange.
81
Recall that when Iris enters the
options strategy the forward
exchange rate is
F = USD.7447/CD.
Thus, a future break-even spot
rate can be calculated for every
corresponding exercise price
chosen:
82
F =.7447.
Iris may exchange today, CD300M
forward for:
CD300,000,000(USD.7447/CD)
= USD223,410,000.
IF: K =.74,
S(CD300M) + 4,113,000 = USD223,410,000
 SBE = USD.7310/CD.
IF
K= .75,
S(CD300M) + 1,926,000 = USD223,410,000
 SBE = USD.7383/CD.
IF
K= .76,
S(CD300M) + 915,000 = USD223,410,000
 SBE = USD.7416/CD.
83
CONCLUSION
Writing the calls will protect Iris’ flow
in USD six months from now better
than an immediate forward exchange,
for all spot rates (in six months) that
are above the corresponding breakeven exchange rates.
84
A second possible analysis of the optimal
decision depends on all possible values of
the spot exchange rate, given our
assumptions. Recall that the assumptions
are:
Iris either maintains an open long
position of CD300M un hedged.
Alternatively, Iris writes 6,000 PHLX calls
with 180-day expiration period. Possible
strike prices are USD.76/CD, USD.75/CD,
USD.74/CD. Current spot and forward
exchange rates are USD.75/CD and
USD.7447/CD, respectively.
85
The terminal spot rate is the market
exchange rate when the calls expire. It
is assumed Iris pays a brokerage-fee of
USD4 per option contract and an
additional fee of USD35 per option to
the Options Clearing Corporation if the
options are exercised.
86
Optimal Decision Under Iris' Strategy as a
Function Of The (Unknown) Terminal
Spot Rate
Terminal Spot rate Optimal Decision
S >.76235
Hold long
currency only
.75267 < S< .76235
Write options
with K = .76
.74477 < S< .75267
Write options
with K = .75
S < .74477
Write options
with K = .74
87
Final comments on Example 3.
Because of the large OCC fee of USD35 per
exercised call assumed in the example, it
might be less expensive for Iris to buy back
the calls and pay the brokerage fee of USD4
per call in the event the options were in
dancer of being exercised. In addition, it is
assumed that Iris will have the CD300M on
hand if the options are exercised. This
would not be the case if actual Canadian
dollar revenues were less than anticipated.
In that event, the options would need to be
88
repurchased prior to expiration.
Each of the three choices of strike price
will have a different payoff, depending on
the movement in the exchange rate. But
Iris' expectation regarding the exchange
rate is not the only relevant criterion for
choosing a risk-management strategy.
The possible variation in the underlying
position should also be considered.
89
Here are the maximal and minimal
payoffs for each of the call-writing
choices, compared to the un hedged
position and a forward market hedge:
90
Strategy
Max Value
Unhedged
Long
Position
None.
Sell:
forward: USD223,410,000
Min Value
.76 call:
USD228,705,000
.75 call
USD226,716,000
.74 call
USD225,903,000
Unhedged minimum
+ USD915,000.
Unhedged minimum
+ USD1,926,000.
Unhedged minimum
+ USD4,113,000.
Zero.
USD223,410,000
91
Mechanics of Call Futures Options
When a call futures option is
exercised the holder acquires
1. A long position in the futures
2. A cash amount equal to the excess of
the most recent settlement futures
price over the strike price
92
Mechanics of Put Futures Option
When a put futures option is
exercised the holder acquires
1. A short position in the futures
2. A cash amount equal to the excess of
the strike price over the most recent
settlement futures price
93
The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F0 – K
Payoff from put = K – F0
where F0 is futures price at time of
exercise
94
Put-Call Parity for Futures
Options (Equation 13.13, page 284)
Consider the following two portfolios:
1. European call plus Ke-rT of cash
2. European put plus long futures plus
cash equal to F0e-rT
They must be worth the same at time
T so that
c+Ke-rT=p+F0 e-rT
95
Valuing European Futures
Options
• We can use the formula for an option
on a stock paying a dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
• Setting q = r ensures that the
expected growth of F in a risk-neutral
world is zero
96
Black’s Formula
(Equations 13.17 and 13.18, page 287)
• The formulas for European options on
futures are known as Black’s formulas
F0 N(d1 )  K N(d 2 )
p  e  rT K N(d 2 )  F0 N(d1 )
ce
 rT
2
ln(F 0 /K)  σ T/2
where d1 
σ T
2
ln(F 0 /K)  σ T/2
d2 
 d1  σ T
σ T
97