Strike

Price

360
380
400
420
440
460
480
500
530

Strike °

Price

360
380
400
420
440
460
480
500
530

Table
Prices of Gold Futures and Options on Gold Futures
(December 10 closing prices, WSJ)

Gold Future Prices
Spot 436.00
Feb. 441.50

April 448.30
August 462.50

Gold Future Options

Calls Puts
Feb. April August Feb. April August
88.50 2.00
62.00 71.00 88.00 1.50 4.50 9.50
44.50 55.50 74.50 3.50 7.50 14.00 Increasing
27.50 41.50 62.00 7.50 12.00 21.50
Decreasing 16.00 31.50 50.50 14.00 20.50 28.50
9.00 21.00 42.50 26.50 31.00 37.50
4.50 15.00 33.00 43.00 49.50
3.00 10.00 27.00 57.00
6.00 18.00 78.00
Increasing ——py Increasing ~—fy
Gold Future Options Less Intrinsic Value
Calls Puts
Feb. April August Feb. April August
20 “2.00 Decreasing
50 2.70 3.50 1.50 4.50 9.50
3.00 7.20 12.00 3.50 7.50 14.00 ?
6.00 13.20 19.50 7.50 12.00 21.50
15.50 23.20 28.00 14.00 20.50 28.50 (Peak at futures price)
9.00 21.00 40.00 8.00 19.30 37.50
4.50 15.00 33.00 11.30 32.00 t
3.00 10.00 27.00 5.30
6.00 18.00 10.50 Decreasing

Increasing ——> Increasing ——{>


-4-

method of hedging in this case would be for the importer to purchase 1
million marks spot and buy a put (an option to sell) for 1 million marks one year
hence. The costs of these alternative means of hedging must be equivalent or
arbitrage will occur. If, for instance, the cost of a call option plus the cost if
exercised exceeds the cost of the put option (for the same strike price) plus the
interest opportunity cost of purchasing and holding mark denominated assets,
arbitragers could make profits by writing calls (selling options to purchase) and
purchasing puts, and would bid up the put price and bid down the call price until
arbitrage opportunities disappeared.

That is, for a particular strike price (the Price at which the
option to purchase or sell the currency is set), the following relationship

will hold, evaluated at the expiration date of the option:

(2) (14rg) Pg + S+X(l4rg)/(ltr,_) =
(1trg)PA g + GX

where PS

price G and PG = the price of a call option to buy X units of currency

.G = the price of a put option to sell X units of currency at

at price G.

A third method of hedging in this case is to purchase 1 million
marks forward rather than spot while buying a put. By substituting (1) into
(2) this yields the following relationship between the put and call prices as

a function of the forward rather than the spot exchange rate:
(3) (14rg)PE ig + FX = (14rg)PX Gg + Gx

If the strike price equals the forward rate*, the price of the put

and call options must be identical. If the call price exceeds the put price,

*In market jargon, an at-the-money option.


B-2

McKinnon, R. Money in International Finance. 1979.

Merton, R. C. Theory of Rational Option Pricing. Bell Journal of Economics.
Spring 1973.

Philadelphia Stock Exchange. Foreign Currency Options: The Third Dimension
to Foreign Exchange. 1982

Schmalensee, Richard and Trippi, Robert. Common Stock Volatility
Expectations Implied by Option Premia. Journal of Finance. March 1978.

Smith, Cifford, Jr. Option Pricing: A Review. Journal of Financial
Economics. 1976

Wolf, Avner. Fundamentals of Commodity Options on Futures. Working Paper
Series #CSFM-35. Columbia University. July 1982.


-9-

d) A speculator expects a currency to depreciate (appreciate) more
than the forward rate.

The speculator can sell (buy) futures and buy call (put) options.
The option provides insurance as to the investor's maximum loss and also
Provides 'staying power'. With only a futures contract, an adverse rate
movement can wipe out the initial margin requirement. A call option limits
the maximum loss and can prevent a margin call if the speculator wants to
maintain his position.

e) A speculator expects a currency to depreciate (appreciate) more than
the current foward rate.

The speculator can write call (put) options. This strategy has
unlimited loss potential (of course, the sSpeculator can close out the
position by buying a call (put) option at a subsequent time). The speculator
will gain the amount of the option price if the exhange rate moves as
expected, as the option will be allowed to expire. By writing calls (puts)
with higher (lower) strike Prices, the investor has less likelihood of loss,
but will receive smaller fees from writing the options.

£) An institution holding foreign currency wishes to hedge against a
decline in its value.

This might be considered pure hedging, but could also be considered
hedging a speculative open position. The institution could purchase put
options.

g) An institution holds several currencies for transactions purposes,

and expects the future spot rate to equal the forward rate,


-6-

However, while an option to purchase a futures contract might be
better suited to the importer's needs than an option to purchase spot, note
that the importer could also satisfy his needs by purchasing 1 million marks
spot (or six months forward) and buying an option to sell 1 million marks
spot which expires in six months. (If he discovers he will need the marks
during the first six months, he will let his option expire, otherwise it will
be executed.) Of course, the transactions cost may be greater for this
operation (particularly, if the importer faces a ‘borrowing constraint’), but
it will always be true that an option on a futures contract is equivalent to
an option for a spot transaction and a spot or forward transaction. In addi-
tion, the importer could also purchase an option to buy 1 million marks which
expires in six months. In short, given the existence of a complete set of
spot and futures markets, and spot options, prices of options to purchase
futures contracts will be determined by arbitrage given spot and forward
exchange rates, spot option prices, and interest rate differentials.

The existing options on gold futures and Treasury bond futures have
expiration dates which are exactly one month prior to the delivery date on the
futures contract (thus, only the delivery date is quoted in the market). The
set of options prices generated by the one month ahead expiration clause is
equally suited to the task of giving the importer the flexibility that would
be allowed by having a set of options which expire on the delivery date (that
is, spot options). In other words the arbitrage condition only requires that
one vector of the matrix of expiration dates and delivery dates for call and
put options on a commodity is necessary to determine the prices of all other
options with alternate expiration and delivery dates, if complete markets

exist for underlying futures (and spot) contracts. Increasing the variability


-5-

then an arbitrager could write calls and buy puts, while simultaneously
buying the foreign currency spot (or forward). If the end-of-period spot rate
exceeds the forward rate, the purchaser of the call will exercise the option,
and the arbitrager will sell his foreign currency. If not, the call will
expire and the arbitrager will exercise his put. In either case, the arbi-
trager earns the difference between the call option and put option prices and
has no foreign currency position when the options expire. When the strike
price is greater than the forward rate, the call option is priced less than
the put option (and conversely when the strike price is less than the forward
rate).

An option to purchase (sell) foreign currency spot may equivalently
be thought of as a forward purchase (sale) with an option to back out of the
agreement at any time up to the delivery date (that is, the expiration date of
the option). The importer in effect is buying an insurance provision to back
out of his forward contract at any date up to the expiration date of the
option. However, this insurance protection might be more than ample (and thus
More expensive) than what the importer actually needs. Suppose that the
importer is uncertain whether he will need 1 million marks one year hence, but
will receive information within the next six months that will enable him to
determine with certainty his needs at the end of the year. In this case, the
importer only needs an insurance provision to void a 1 year forward purchase
of 1 million marks which expires after six months rather than 1 year. An
option to purchase a futures contract (with delivery one year after the
contract date) which expires in six months will satisfy this need. It is
intuitively clear (and required by a simple 'dominance' theorem) that the
price of an option which expires in six months to purchase currency at the end

of a year will cost less than the option which expires at the end of a year.


-8-

less the cost of the option as compared to the cost of complete hedging
using the forward market. Compared to a completely unhedged position, the
importer loses the cost of the option. If the future spot rate exceeds the
forward rate, the option is exercised. In this case the importer loses the
cost of the option relative to his position had he entered a forward
contract, but is better off than a completely unhedged position if the actual
additional cost of foreign exchange exceeds the option price.

If the importer purchased a call option with a strike price, say, 2
cents/mark higher than the forward rate, the option price would be much
lower, but his loss would be higher in the case of an adverse rate movement.
This would be analogous to an insurance policy with a lower premium but a
higher deductible.

b) A speculator expects highly variable exchange rates, but has no view
of the direction of movement.

A strategy for this speculator would be to purchase both call and
put options for a particular strike price (and same expiration date). The
speculator will have a gain on one option and will let the other option
expire. If there is a big movement in price, the gain on one option will
exceed the cost of the options.

This strategy could be modified by having different strike prices
for the put and call options, if the speculator expects a greater likelihood
of an exchange rate movement in a certain direction.

c) A speculator expects little movement in exchange rates.

This speculator could write a put and a call at the same strike
price. He expects that the price he receives for selling the two options
will exceed the loss which will occur when one or the other option is

exercised.


International Finance Niscussion Papers
Number 220
April 1983

SPECULATION AND HEDGING USING OPTIONS ON FUTURES CONTRACTS

by

Laurence R. Jacobson

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished
material) should be cleared with the author or authors.


-ll-

Two types of options have been adopted in markets. American
options allow the option to be exercised at any time between the contract
date and the expiration date. European options can be exercised only on the
expiration date. Clearly, an American option can be priced no lower than an
otherwise identical European option. However, American call options will
generally not be exercised before the expiration date if a secondary market
exists. This occurs because the option price in the secondary market must be
at least as great as the intrinsic value of the option if immediately
exercised. Holders of American options will generally close out open
positions by selling options rather than by exercising them. Certain
problems (such as uncertain dividend payments in the case of equity options)
might cause early exercise of American options.

American put options, however, have a positive probability of pre-
mature exercise even if technical difficulties concerning dividends, etc. are
eliminated. This assymmetry is due to the fact that asset prices are bounded
below. Suppose a holder of a gold futures put option at $400/oz. observes
that the futures price falls to $1/oz. The option could immediately be
exercised for a profit of $399; if not exercised, the price of the option
could never exceed $400. If the (risk free) interest that could be earned on
$399 prior to the expiration date exceeds the expected additional gain from
holding the option (most likely), the option would be prematurely exercised.

Call options prices must be non-increasing functions of the
exercise price, while put options must be non-decreasing functions of the
exercise price. Both puts and calls on American options must be priced as
non-decreasing functions of the expiration date. As noted in Section 2, if

futures prices exist, then the call price will be greater/equal/less than the


-10-

The institution could write call options, using its currency
holdings to satisfy any margin requirements. The institution will gain if
the currency does not appreciate enough to offset the premium received for
writing its options. Losses are unlimited until the position is closed out.
Of course, as with stock options, a common public misperception is that the
institution cannot lose if it is holding the securities to back the call
options it is writing. In fact, it may lose an unlimited amount on the

option, although this will be offset by the gain on its security holdings.

4.) Some notes on option pricing.

A number of useful boundary conditions on option prices have been
derived through dominance arguments. A dominant asset or portfolio is one in
which the return on the asset is greater or equal to the return on another
asset under all possible states of the world, and strictly greater for at
least one state of the world. Such an asset cannot exist, since the relative
price of that asset must be bid up until dominance disappears. Arbitrage must
eliminate dominance. Interest rate parity and the correspondence between spot
options prices and futures options prices discussed in Section 2 are dominance
conditions. While dominance arguments place useful limitations on put and
call prices, the level of options prices cannot be determined without a model
of option price determination. In this section I will summarize some key
dominance arguments described in detail by Merton, and will briefly describe
the general equilibrium option pricing model developed by Black and Scholes
(1973) and some extensions and applications made by other authors. A useful

review of the literature through 1975 is found in Smith (1976).


-7-

of expiration dates relative to delivery dates will not add additional
information to the market (although, might reduce transactions costs for some
individuals). For a commodity which is costly to deliver (like gold), or when
trading volume in near term futures contracts is high, it may be institu-
tionally preferrable to have options on futures contracts (which expire at
some set time prior to delivery) rather than options for spot transactions.
The low volume of trading thus far for foreign currency options on the
Philadelphia Exchange (which require spot delivery) may be indicative of a

less than optimal institutional framework.

3.) Strategies for speculation using options

A wide range of speculation strategies exist for using options,
sometimes in conjunction with futures contracts. Several possibilities are
outlined in this section. Note that some popular strategies involve
partially hedging speculative positions, and hence could be considered some-
thing between pure hedging and pure speculation.

a) An importer with a view on exchange rates desires a partial hedge.

Suppose an importer needs to acquire marks at some point in the
future but expects the future spot rate to be less than or equal to the
forward rate. Since he expects to be able to purchase the currency at a
cheaper rate in the future, he would rather not acquire a forward contract.
However, the importer may desire some protection against an adverse exchange
rate movement (i.e., a mark appreciation). The importer may buy a call option
with the strike price equal to the forward rate to limit his losses in case of
a mark appreciation. If marks in fact are cheaper in the future as the
importer expected, the option will not be exercised. The importer gains the

difference between the actual future spot rate and the initial forward rate


-3-

the opportunity cost of (1) must equal the cost of (2), so that interest

rate differentials determine the forward discount or premium:
(1) (14rg)/(1tr¢) °S =F

where S = spot rate ($/mark), F = forward rate, and tq = interest rate on
dollar denominated assets, and rp = interest rate on mark assets. If

Tg > Yer the forward rate must have a discount (a higher dollar price of
marks) equivalent to the interest rate differential.

Consider now an importer who is unsure whether or not he will need
1 million marks in one year. Suppose he will either require 1 million marks
or no marks at all. Even with complete spot and futures markets (no discrete
time problems, etc.) he will not be able to perfectly hedge his exchange rate
risk. Under these conditions, the optimal hedge will be a function of the
mean and variance of his probabilistic exchange needs and should be between
zero and a million marks, but his actual need one year hence will result in
an under-hedged or over-hedged position. Unlike the previous case, the
importer has uncertainty about the magnitude of his foreign exchange needs as
well as about the movement in exchange rates.

On the other hand, if an options market exists for spot foreign
currency transactions, the importer could buy a call (an option to purchase)
for 1 million marks which expires one year hence, and can let the option
expire unused if the million marks are not required. At the cost of the
option, the importer is now able to perfectly hedge his exchange rate risk.
In fact, even if the importer does not require the million marks, the value of
the option may well exceed zero at the termination date and the importer could
sell or exercise the option and offset the initial option cost. The option

price thus represents the maximum cost of hedging. Of course, an equivalent


-13-

option pricing problem is a function of five variables: the asset price,
the variance rate on the asset price, the strike price, the expiration date
of the option, and the risk-free interest rate (which is assumed constant).
In addition to yielding a solution which can be explicitly solved given the
values of the five variables, several qualitative results are derived which
conform to the dominance arguments already described. The price of calls and
puts are also shown to be increasing functions of the variance of the asset
price. The call price rises with an increase in the risk free interest
rate, while put prices fall, since the present value of the exercise price
falls. Note that options pricing models usually are concerned with European
options (those which can be exercised only on the expiration date) or
American call options. As a result of the possibility of premature exercise
discussed earlier, no closed form solution exists for the valuation of
American put options. However Brennan and Schwartz (1977) derive an
algorithm for solving the put pricing problem under certain circumstances.

The Black-Scholes option pricing model assumes a continuous
stochasic process for describing price movements. A solution using a jump
stochastic process is derived in Cox and Ross (1977). Cox, Ross, and
Rubinstein (1979) simplify the mathematical derivations by using a discrete
binomial formula, which contains the Black-Scholes and jump process formulas
as special limiting cases. The qualitative characteristics of options prices
in relation to changes in the asset price, expiration date and other
variables are not affected by the type of stochastic process assumed,
although numerical results and forecasts may differ to some extent.

A number of empirical studies of options prices have been under-

taken, using data for equity options which have been in existence for ten


BIBLIOGRAPHY

Beckers, Stan. Standard Deviations Implied in Option Prices as Predictors of
Future Stock Price Variability. Journal of Banking and Finance. 1981.

Black, F. and M. Scholes. The Pricing of Options and Corporate Liabilities.
Journal of Political Economy. June 1973.

Brennan, Michael and Schwartz, Eduardo. The Valuation of American Put
Options. Journal of Finance. May 1977.

Brenner, Menachem and Galai, Dan. A Note on Measuring the Risk of Common
Stocks Implied by Options Prices. Working Paper #278. NYU Graduate School
of Business Administration. 1982.

Chicago Board of Trade. Strategies for Buying and Writing Options on
Treasury Bond Futures. 1982.

COMEX. Options on Comex Gold Futures. 1982.

Cox, John and Ross, Stephen. A Survey of Some New Results in Financial
Option Pricing Theory. Journal of Finance. May 1976.

Cox, John and Ross, Stephen. The Valuation of Options for Alternative
Stochastic Processes. Journal of Financial Economics. May 1977.

Cox, John, Ross, Stephen, and Rubinstein, Mark. Option Pricing: A
Simplified Approach. Journal of Financial Economics. 1979.

Emanuel, David and MacBeth, James. Further Results on the Constant
Elasticity of Variance Call Option Pricing Model. Journal of Financial and
Quantitative Analysis. November 1982.

Feiger, George and Jacquillat, Bertrand. Currency Option Bonds, Puts, and
Calls on Spot Exchange and the Hedging of Contingent Foreign Earnings.
Journal of Finance. December 1979.

Garman, M. and Kohlhagen, Stephen. Foreign Currency Option Values. School
of Business Administration, Berkeley. Working Paper. December 1982.

Giddy, Ian. Foreign Exchange Options. Working Paper Series #CSFM-47.
Columbia Business School. December 1982.

Goodman, Laurie. Interest Rate Options and Financial Institutions. Draft
paper. Federal Reserve Bank of New York. 1982.

Latane, Henry and Rendleman, Richard. Standard Deviations of Stock Price
Rates Implied in Options Prices. Journal of Finance. May 1976.


-2-

The third section gives a brief tutorial of different speculative
opportunities provided by options, and describes how options can be used to
partially hedge speculative positions. Since these strategies are described
in detail elsewhere*, this section is intended mainly to serve as a tutorial
for those unfamiliar with the market.

The final section of the paper is a description of some of the
theory of options pricing which has been developed in the literature, and how
the new options on financial instruments might provide useful information for
economic analysis. Since the variance on the price of the underlying asset
is a key variable affecting the price of an option, options may provide useful
information as to the market's expected variance of exchange rates, interest

rates, etc.

2.) Hedging with options on spot and forward transactions

Consider a U.S. importer who requires 1 million marks one year from
now. Hedging of the exchange rate risk can take place by either of two
methods if spot and futures markets exist: 1) The importer could purchase
marks spot and invest the proceeds in mark denominated assets for one year, or
2) The importer could buy 1 million marks in a futures contract for delivery
in one year. Of course, incomplete markets or institutional constraints
(fixed dates for delivery, minimal size of transactions, etc.) will in general
limit the ability of an importer to hedge exchange rate risk, and it would not
generally be optimal to hedge all risk even if possible (see McKinnon, 1979).

Moreover, covered interest arbitrage (under certain assumptions) assures that

—
* See the reports prepared by the exchanges, listed in the bibliography.


Speculation and Hedging Using Options on Futures Contracts
by Laurence R. Jacobson*

1.) Background

Last year, a number of options contracts were initiated in several
exchanges, and several others have been started in 1983. Trading began in
the fourth quarter of 1982 for options to purchase Treasury bond futures,
options for gold futures, and options for spot purchases of five major
currencies. Options on stock index futures were initiated early this year.
The rapid expansion of options contracts is probably related to the increased
volatility (variance) of interest rates, exchange rates, stock prices, etc.,
as well as to the enlarged opportunity set for hedging and speculation. Some
of the new option contracts may become moribund due to thinness of the
market, although any prognosis at this time may be premature. As will be
explained further below, the existing options on futures (gold, Treasury
bonds) are only a subset of possible options on futures contracts, since the
expiration date of available options always bear a fixed relation-
ship to the delivery date of the futures contract. However, all other
possible futures options prices would be determined by arbitrage possibilities,

The next section of this paper will explain the economic purpose of
options on futures contracts in expanding the set of hedging opportunities
available to entrepreneurs facing exchange rate risk (or interest rate risk,
etc.). The relationship beweeen spot transactions, futures contracts,
options on spot transactions, and options on future contracts will be
described; in the case of currency markets, this yields a generalized
interest rate parity condition.

*This paper represents the views of the author and should not be inter-

preted as reflecting the views of the Board of Governors of the Federal
Reserve System or other members of its staff.


-12-

Put price whenever the exercise price is less/equal/greater than the futures
price. The table shows these relationships for actual trading in gold
futures. This restriction also implies that the Price of both call and put
options less their intrinsic value will reach a maximum at the current
futures price, as shown in the lower panel of the table.

Note, that arbitrage would also suggest that for any particular
strike price, puts and calls should be priced identically once the intrinsic
value is subtracted. If not, an arbitrager could write the higher priced
call and fully cover himself by purchasing the lower priced call, earning the
difference in the premiums. However, since the price paid for the option
includes the intrinsic value, the interest cost of purchasing the higher
priced option exceeds the interest received from selling the lower priced
option. Thus, the price of an in-the-money call (a call with intrinsic value)
less the intrinsic value should be lower than the put at the same exercise
price, and conversely for in-the-money puts. This is generally true in the
case shown in the table, although there are some slight discrepancies.

As can be observed in the table, the option price (both put and
call) less the intrinsic value approaches zero as the strike price approaches
zero or infinity. This results from the fact that such options will always
or never be exercised. For example, I will gladly sell you an option to give
you gold futures (exercise price of zero) at any margin above the futures
price, since I could cover myself immediately and earn the margin. However,
you obviously would be unwilling to pay a penny more than the futures price
for this gift.

Black and Scholes use a general equilibrium asset model to derive

options prices expressed as differential equations. The solution to the


-14-

years. (See Beckers (1981), Brenner and Galai (1982), Latane and Rendleman
(1976), and Schmalensee and Trippi (1978).) The Black-Scholes model, or
variants thereof, can be used to derive the implied variance of the asset
price given option prices, other known variables, and some assumption about
the appropriate 'risk free' interest rate. The variance implied in options
price can be empirically tested as a predictor of future volatility and
compared with historical variance as a predictor. In general, actual option
prices are reasonably predicted using the general equilibrium pricing model.
Implied variances are most accurate for at-the-money options; for deep in-
the-money or out-of-the-money options, the option price is minimally affected
by the variance so that a small price differential implies a large difference
in the implied variance.

Foreign currency options have recently been examined by Giddy
(1982) and Garman and Kohlhagen (1982). Garman and Kohlhagen show formally
that the currency spot call option price increases with the domestic interest
rate and rises with the forward exchange rate. Although the option price
also varies positively with the spot rate and negatively with the foreign
interest rate, these are not independent conditions because of interest rate
parity. Hedging using interest rate options (Treasury bonds, bills) has been
discussed by Goodman (1982).

In summary, option pricing models may be useful empirically to
derive estimates of the market's expected variance of interest rates, gold
prices, exchange rates, etc., or to examine how quickly market participants
react to actual changes in volatility of asset prices. In addition to the
possible use of options prices in empirical work, direct observation of
changes in option prices may provide additional insight into ongoing trends

in financial markets.