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Options on Stock Indices, Currencies, and Futures Chapter 13. STOCK INDEX OPTIONS ONE CONTRACT VALUE = (INDEX VALUE)($MULTIPLIER) One contract = (I)($m) ACCOUNTS ARE SETTLED BY CASH. STOCK INDEX OPTIONS FOR PORTFOLIO INSURANCE Problems: How many puts to buy?

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One contract = (I)($m)




  • Problems:
  • How many puts to buy?
  • 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.



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:




Second, rewrite the CAPM result, with actual returns:

In a more refined way, using V and I for the portfolio and index market values, respectively:


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.

The ratio V1/ V0 indicates the portfolio required protection ratio.


For example:

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.


2.The exercise price, K, is determined by substituting I1 = K and the required level, (V1/ V0)* into the equation:

and solving for K:


We rewrite the Profit/Loss table for the protective put strategy:

We are now ready to calculate the floor level of the portfolio: V1+n($m)(K- I1)


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


3. Substitution V1 into the equation for the Min portfolio value

The desired level of protection is made at time 0. This determines the exercise price and management can also calculate the minimum portfolio value.


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&P500index stands at a level of 1,250 points and its dollar multiplier is $100. The annual risk-free rate is 10%, while the portfolio and the index annual dividend payout ratios are 5% and 6%, respectively. The data is summarized below:


The exercise price of the puts is:

Solution: Purchase n = 480 six-months puts with exercise price K = 1,210.



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.


The exercise price of the puts is:

Solution: Purchase n = 10 six-months puts with exercise price K = 960.



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.


A SPECIAL CASE: In the case that

  • β = 1 and 2. qP =qI, the portfolio is
  • statistically similar to the index.
  • In this case:

Example: (page 273)

βp = 1 and qP =qI, then:



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.

Currency optionsUnits



USD/CD 50,000CD


USD/JPY 6,250,000JPY


Exercise Style: American- or European

options available for physically settled

contracts; Long-term options are

European-style only.

Expiration/Last Trading DayThe 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.
standardized options

With the Canadian dollar spot price currently at a level of USD.6556/CD, strike prices would be listed in 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.

Standardized Options

Exercise PricesExercise 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.

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 long-term options.

Premium Quotationpremiums for dollar-based 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.


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.

customized currency options
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.

Underlying CurrencyThe 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 CurrencyThe base currency is that currency in which terms the underlying is being quoted, i.e. strike price.

Expiration/Last Trading DayExpirations 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.

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:


Underlying currencyContract 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.

Exercise PricesExercise 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 SizeSince customized currency options were designed for the institutional market, there is a minimum opening transaction size which equals or exceeds 50 contracts.


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.

Contract TermsAn 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.
Trading ProcessTrading 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.
PremiumsPremiums 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.
Price and Quote DisseminationRequest 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 EUR per contract).

Position LimitsPosition 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).
FX Options As InsuranceOptions 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 USD.80/EUR.

If the put option costs USD.01/EUR, the floor price can be roughly approximated as:


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.

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.
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.

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 floorprice 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.

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.
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.

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.

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:


- (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


- (sale value of options-$32)/250,000.]

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.

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.

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.

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


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.

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


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,


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.

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 isa 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.

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.
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.
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.
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 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.

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.
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:

If the USD value of the CD remains unchanged, the option premium received is simply additional profit.
  • 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.
  • 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.
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.

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:


c(K = USD.74/CD) = USD.01379;

c(K = SUD.75/CD) = USD.00650;

c(K = USD.76/CD) = USD.00313.

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:

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:


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.

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


F =.7447. Iris may exchange today, CD300M forward for:


= 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.


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 break-

even exchange rates.

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.

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.
Optimal Decision Under Iris' Strategy as a Function Of The (Unknown) Terminal Spot Rate

Terminal Spot rateOptimal 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

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 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.

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:

StrategyMax ValueMin Value



Position None. Zero.


forward: USD223,410,000 USD223,410,000

.76 call: USD228,705,000 Unhedged minimum + USD915,000.

.75 call USD226,716,000 Unhedged minimum + USD1,926,000.

.74 call USD225,903,000 Unhedged minimum + USD4,113,000.

mechanics of call futures options
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

mechanics of put futures option
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

the payoffs
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

put call parity for futures options equation 13 13 page 284
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

valuing european futures options
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 = rensures that the expected growth of F in a risk-neutral world is zero
black s formula equations 13 17 and 13 18 page 287
Black’s Formula (Equations 13.17 and 13.18, page 287)
  • The formulas for European options on futures are known as Black’s formulas