It is generally considered that capitalization weighted indices give the most accurat indication of the collective movement in corporate asset or liability prices. However two alternative methods of weighting the constituents of equity indices are fowl^,‘equally weighted‘ and ‘price weighted‘. In the case of equally weighted indices, a’ equal amount of money is assumed to be invested in each security in the index Changes in the index thus represent changes in the value of the portfolio. Pri weighted indices reflect the average price of the securities in the index and Chang: in the index represent the average price change of the securities in the index. The Dow—Jones and the Major Market Index of the American Stock Exchange (MMI) a. both of the price weighted form. The FT 30 is a geometric average equally weight index. Only the Major Market Index has a futures contract based upon it.

Equally weighted and price weighted indices give greater relative weight to small company constituents than do capitalization weighted indices. However, in t Dow—Jones, the FT30 and the MMI this influence is negligible because only a f• stocks — all relatively large — are included.

Typically the indices upon which futures and options are based are market capitalization weighted arithmetic means of prices. The two notable exceptions are t Major Market Index, mentioned above, and the Value Line Composite Index in USA. This latter index is calculated as an unweighted geometric mean of constitute share prices. This has a major drawback in terms of derivative valuation, in that it not possible to derive an arbitrage strategy that replicates the returns to this type I index. We will therefore limit our discussion to derivatives on indices that are calculated using market capitalization weighted arithmetic means.

Price indices and total rate of return

It should be noted that the published indices are price indices only. The moveme in the index only reflect movement in the share prices; they take no account of dividends received by shareholders and therefore do not calculate the total rate return.

The exclusion of dividend payments results in the return as measured by movements in the index understating the actual returns to holding a portfolio t replicates that index. To see how this occurs, imagine that company C paid a dividend of 5. On the ex-dividen date the share price will fall by 5 (assuming no tax effects) to 95. The index will to 99.16. Yet the investor gets the 5 in dividend and the overall value of the portfolio has not changed. Thus, when comparing the returns to an index with the returns a portfolio, the influence of dividends upon the index must be taken into account.

Unlike futures contracts which require physical delivery of the underlying financial instrument, equity index futures require cash settlement of the difference between the current and future levels of the index. In practice, because the futures contract is marked to market daily, there is one final marking to market based upon the level of the index when the future expires.

Thus a futures contract based upon an equity index may be defined as: an agreement to make or receive a cash payment based upon the difference between the current and future values of a specified index.

The process of cash settlement avoids the difficulties of delivering the many individual securities that constitute the particular index. It also avoids the risk of the securities being cornered or squeezed. However, it also requires that the index calculation is transparent, reliably accurate and free from manipulation. Moreover, it does introduce difficulties in hedging futures positions with the underlying equities, and thus difficulties in the arbitrage process that is the theoretical basis of valuing such futures.

In order to avoid manipulation of the index for the purposes of calculating the futures expiry price, the procedure adopted by LIFFE is typical. The settlement price, the Exchange Delivery Settlement Price (EDSP), is calculated as the arithmetic mean, after the exclusion of the highest and the lowest levels, of the index levels between 10.10 and 10.30 a.m. on the expiry day. The index itself is recalculated by the Stock Exchange each minute, giving timely adjustments to reflect the movements of the stock market.

Conceptually, the various equity index futures contracts traded around the world are all very similar; therefore, we shall here only describe in detail the LIFFE futures contract on the FTSE 100 index.

The futures contract represents a future claim on a notional index fund with a market value of £25 per index point. Thus with the future standing at, say, 2500 the notional index fund is valued at 2500 x £25 = £62 500. The market quotations have a minimum price movement or tick size of 0.5 of an index point. This tick size represents £12.50 in terms of the monetary value of the FTSE 100 index.

The futures trade on a March, June, September and December cycle, with the expiry dates being the last business day of each delivery month; although as from the June 1992 delivery month, the delivery day will be the third Friday in the delivery month. The contracts are cash settled, with settlement being the first business day after expiry.

1. Options on the spot index itself such as the contract traded in the London International Financial Futures and Options Exchange — like the futures, these options are settled in cash, rather than by delivery of the underlying securities. Sometimes both European and American options are traded on the same index.
2. Options on the index futures are American options that call for delivery of an equity index futures contract at expiry.
3. Over-the-counter equity index warrants, issued by a number of securities houses, are effectively longer-term options on the spot index, with a different credit risk than the market traded options. Similar valuation considerations apply to these instruments as to other equity index options. However, the problems of forecasting volatility become greater as maturity increases.

Options on spot equity indices

Options on the spot value of equity indices are available on a number of the major indices. In some instances, both European and American options are traded simultaneously. Typical contracts are the American and the European options traded on the FTSE 100 spot index and listed on the London Traded Options Market. Both contracts represent an option on a notional index fund equal to £10 multiplied by the level of the index. The American option contract has an expiry cycle of four nearby consecutive months, plus two additional months; whereas the European option has, a March, June, September and December cycle, plus the two nearest months. These contracts are cash settled.

The Black—Scholes model adjusted for dividends can be used for valuing European options on indices that pay discrete dividends; the binomial model can be used; for valuing all American options; or the Barone-Adesi and Whaley model can be used for American options where the index is characterized as paying continuous dividends. In all these cases, the index level has to be adjusted for the amount and type of the dividend flow. Moreover, the use of these models does assume that the diffusion process of the index is identical to that of the individual securities.

Strictly speaking, if the assumption of a log-normal distribution of asset price relatives holds for individual equities, it will not hold for a weighted average of those equities — i.e. the index. However, the log-normal assumption is considered to be reasonably accurate, and the models discussed make that assumption.

The assumptions about flow of dividends, discussed above in the context of futures, also have important implications for the valuation of both European and American options. Brenner, Courtadon and Subrahmanyam found that for a broad-based index, such as the New York Stock Exchange Composite Index, the assumption of continuous dividends was not unreasonable but for the narrower MMI such an assumption could not be supported. They reported considerable mispricing of options on the MMI using a model that assumes a continuous dividend flow. Thus the correct assumption of the dividend flow is highly important in index option valuation, as indeed it is for index futures valuation.

Option valuation theory using the risk-neutral assumptions does assume that a riskless hedge can costlessly be created by combining the underlying asset with fairly priced options. One problem with index options is that the underlying asset itself is not traded, only the individual constituents. Thus it is more difficult (costly) to establish the risk-free portfolio. When there is a liquid futures market in the same underlying index, market participants may prefer to hedge the options with transactions in the futures market rather than in the cash market. Yet we know from our earlier discussions of equity index futures pricing that arbitrage in the futures market is also difficult and value basis can be considerable. Thus an option on the spot index may be priced not off the cash index, but off the index future, which itself may be mispriced. Moreover, the volatility of that future will be influenced by the volatility of the index, the volatility of short-term interest rates and of dividends (i.e. net cost of carry) and the volatility of the value basis. Thus it may be different to the volatility of the index itself. These factors may account for some of the mispricing that has been observed in empirical studies of equity index options

As with index futures, the procedure of cash settlement makes arbitrage less than riskless. Although the writer of the option may try to hedge the written options in the spot equity market, if exercised, settlement is in cash. Thus the option writer suffers the execution risk of adjusting the equity position. This problem is greatest with American options because of the multiple potential for early exercise and the fact that exercise notices are issued after the spot equity market has closed and is not communicated until the next business day.

Although it is to be expected that (because an index is a diversified portfolio of individual assets) the volatility of the index will be lower than that of the individual equity constituents, the volatility of the index will not be constant if the volatility of the constituents is not constant.

In addition to the direct or indirect quotation, forward exchange rates can be expressed in one of three ways. First, the forward rate can be quoted as an outright rate — i.e. the actual forward rate of exchange.

Secondly, it can be quoted as forward exchange margins or points (also called swap rates). These latter are either discounts or premiums depending on the interest differentials between the home and foreign currency. If the foreign currency interest rate is higher than the home currency interest rate, the foreign currency will be at a forward discount to its spot rate. If, on the other hand, the foreign interest rate is below the home currency interest rate, the foreign currency will be at a forward premium to its spot value. The magnitude of the discount or premium is dependent upon the size of the differential in home and foreign interest rates and the time to maturity of the forward contract.

With indirect quotes, discounts are added and premiums deducted from the spot rate to calculate the outright forward rate. In the case of direct quotes, the discounts are deducted and the premiums added to the spot rate to arrive at the outright forward rate.

Thirdly, the swap rates can be quoted in percentage terms. This method is sometimes employed to facilitate comparison with the interest rate differential between two currencies. Examples of how the outright forward rate and the forward margins are determined will now be given.

The determination of forward exchange rates

We have shown in the previous section that forward exchange rates differ from spot rates by the amount of the forward margin. This section explains why those differences occur and thereby explains the determination of forward exchange rates and the pricing of forward exchange contracts.

Currency can be considered as a carryable asset that pays a continuous and known distribution. This is because it can be purchased in the spot market and invested in the foreign risk-free asset. Consequently, the hedging transaction for the sale of forward currency will be to borrow domestic currency, buy the foreign currency a the current spot rate and place the funds in the foreign risk-free asset. Although in reality they will usually be placed in a eurocurrency deposit, earning the foreign rate of interest, the principle is clear.

This strategy is analogous to the cash and carry arbitrage found in the futures markets. Remembering that the net cost of carry consists of storage costs, transportation costs and financing costs, then the storage costs are the foreign currency interest earnings, the financing costs are the domestic interest costs and the other costs are insignificant in respect to currencies. Consequently, the net cost of carry for currency forwards, and for futures, is determined by the interest rate differential. Thus the forward exchange rate will be equal to the spot rate plus the net cost of carry, which is equal to the difference between the domestic and foreign interest rates.

The transactions in the FX market emanate from international trade, international investment, the hedging of exchange risks, the establishment of speculative positions or arbitraging between mispriced sections of what is a vast market.

The various participants can therefore be classified as hedgers, speculators or arbitragers. The hedgers are trying to lay off risk and therefore to provide just those risks that the speculators wish to take on. The arbitragers ensure that mispricing is quickly eliminated, thereby ensuring that the price at which the risk is transferred is a fair one.

A foreign exchange transaction can be classified as a spot, a forward or a swap transaction. Spot transactions are those that require delivery of the currency within two working days of the transaction date. Forward transactions require delivery at some previously agreed point in time, more than two working days hence, at a rate of exchange agreed when the transaction is initiated. Swap transactions are the simultaneous combination of a spot transaction and a forward transaction in the reverse direction for the same amount.

The quotation of exchange rates

There are four types of exchange rate that are relevant to this: the spot rate, the forward rate, the cross rate and the swap rate. Spot rates relate to spot transactions; forward rates relate to forward transactions; cross rates are exchange rates between two currencies determined by each currency’s relationship with a third currency — e.g. if one French franc is worth $0.1838 and one pound Sterling is worth $1.7900, the £/Ffr rate is 9.7300 ($1.7900/Ffr 0.1838); and swap rates, also known as forward margins or points, are the amounts by which the spot rate is adjusted in order to derive the forward rate. This is illustrated later; it is the market practice to quote exchange rates to four decimal places.

It is also market practice to quote spot and forward rates for most currencies bilaterally against the US dollar, and to derive the exchange rate between two non- US currencies as a cross rate. Exceptions to this are the DM/yen and the £/DM exchange rates.

Methods of quoting exchange rates

There are two methods of quoting exchange rates: the direct method quotes the foreign exchange as so much domestic currency per unit of foreign currency. This is the method of quotation used in Asia, continental Europe and North America in the interbank market. For example, the Deutschmark may be quoted in New York as 0.61-0.63 dollars per DM. Thus traders would buy one DM at 0.61 dollars and sell one DM at 0.63 dollars. They therefore make their profit by buying low and selling high.

The indirect method gives the quotation in terms of an amount of foreign currency per unit of home currency. This is the method of quotation used in London. For example, the US dollar may be quoted as, say, 1.9988-2.0000 per pound Sterling. Traders would bid for dollars at 2.0000 per pound and would sell dollars at 1.9988 to the pound. Thus they make a profit by buying high and selling low. The bid—offer spread represents the gross profit margin for the market maker, and a transactions cost to other market participants.

When not available separately, direct quotations can easily be derived from indirect quotes. For example, if the forward DM is quoted indirectly as 2.98 DM per pound, the direct quote is 1/2.98 = 0.3355 or £0.3555 per DM.

Although both direct and indirect quotations are used in the forward market according to the practice of the particular market, futures generally and options always use direct quotations for the exchange rate data input.

The reader will note that if Sterling rises above $1.8900, the written call position will make a loss. This is off-set by the rising value in dollar terms of the underlying Sterling investment. Conversely, if Sterling falls below $1.8850, the puts make a profit which off-sets the currency losses on the investment in the Sterling Treasury bills.

This strategy, although a zero cost one, does have limited benefits and should not be considered as a free lunch. The investor is able to benefit from a small rise in Sterling, up to 1.8900, but at the cost of incomplete protection from a fall. This is because the put strike price of $1.8500 is below the current spot price of $1.8750.

It is possible to develop cylinder options with greater potential for gains, but a brief consideration of the principles involved in the above example will indicate that this potential can only be attained at the cost of greater premium or less downside protection.

As traded currency options use the US dollar as the base currency, a Sterling-based investor wishing to hedge, say US Treasury bills would buy a call on Sterling (a put on the dollar) and sell a put on Sterling (a call on the dollar) in order to create a similar hedge against a fall in the dollar. Thus, if the dollar fell against Sterling, the investor would exercise the call using the investment proceeds to acquire Sterling at the exercise price which would be below the then spot price. If, on the other hand, the dollar rose against Sterling (i.e. Sterling fell), the investment manager will have the put on Sterling exercised against him, but the loss would be off-set by the higher Sterling proceeds of the dollar Treasury bills.

Forward participation agreements

A strategy that has been developed in the OTC market is the forward participation agreement. This gives the buyer of the agreement the benefits of an outright forward contract — i.e. a guaranteed floor level and no up-front fee — while avoiding the main drawback of the outright forward, which is the symmetrical pay-off.

We have already seen that the outright forward exchange agreement is costless to establish because the buyer pays for the protection against any unfavourable movement in exchange rates by foregoing any potential favourable movement. The forward participation agreement enables the buyer to participate in the favourable movement albeit at a reduced rate, while ensuring a guaranteed floor rate, although this will be below the current forward rate.

An hypothetical example of a forward participation purchase agreement to purchase dollars for Sterling will illustrate this. Assume that the Sterling/dollar spot rate is 1.7500 and the three-month forward rate is 1.7000, and the buyer agrees the floor exchange rate of, say, 1.6500. This is the rate that will apply at expiry. In effect, the investor has purchased an out-of-the-money put with an exercise price at this level.

As the investor has chosen the floor rate, the investment bank specifies the participation rate. Assume that the chosen participation rate is 75%. The investor will participate in 75% of any favourable movement in the exchange rate.

If Sterling falls below 1.6500, he will benefit from the guaranteed floor rate of 1.6500. If, on the other hand, Sterling rises to 1.8000, he will benefit from 75% of any movement in the rate above 1.6500 — i.e. 75% of 15 cents or 11.25 for an exchange rate of 1.7625.

The analytical problem in valuing these contracts is to determine the participation rate that compensates for the non-payment of an up-front premium and the asymmetry of the pay-off. Remembering that this strategy is a zero cost strategy, and the investor has bought a put option, then he must have to sell an option in order to earn the off-setting revenue. He effectively sells a call on a proportion of the rise in the exchange rate above the exercise price of the purchased put.

The early exercise potential of American options on futures is different to that of options on the spot. Futures prices do not exhibit the discrete jumps that accompany spot market assets when the underlying spot asset makes discrete distributions. However, as the carry basis of the future converges to zero at delivery, the futures price converges to the spot price in an orderly manner. Thus, if carry basis is positive (i.e. F, > P,), the futures price will decline relative to the spot rate in a predictable manner. In effect, because of the daily marking to market, the future would be behaving like an asset that made a continuous distribution. Merton (1973) has shown that it may be optimal to exercise, at any time, an option on an asset that pays a continuous dividend. Thus when the carry basis is positive, it may be optimal to exercise American options on those futures at any time.

It will be recalled that carry basis is positive when the underlying asset pays no dividend or interest, or when the distribution that is paid is less than the borrowing and storage costs incurred in acquiring the underlying asset. Thus, in the case of currency futures, carry basis is positive if the domestic interest rate is above the foreign rate. Accordingly, in such circumstances the currency future can be treated as an asset that pays a continuous distribution, and it may be optimal to exercise American call options on such futures early. However, if the foreign interest rate is greater than the domestic rate, the futures price would be at a discount to the spot rate and the future would behave like an asset that continuously receives an injection of value. Accordingly, it would not be optimal to exercise call options on such futures prematurely, although it may be optimal to so exercise put options.

Thus, if r > r_f, it may be optimal to exercise call options on currency futures prematurely, but not puts, and the early exercise premium on calls should be valuable. If, on the other hand, r < r_f, it would not be optimal to exercise calls prematurely and therefore they can be valued as European options. However, puts may be optimally exercised prematurely and should exhibit an early exercise premium.

The relationship between American options on the future and American options on the spot currency also depends upon the relationship between the domestic and foreign interest rates. If domestic interest rates are above foreign rates, the forward currency will be at a premium and the futures price will be above the spot rate (remember direct quotes). American calls on the futures will be worth more than the otherwise identical American calls on the spot currency. On the other hand, American puts on the future will be worth less than American puts on the spot currency.

If the foreign interest rates were above the domestic rates, the forward currency would be at a discount to the spot and the futures price will be below the spot price. In such circumstances, American calls on the future must be worth less than otherwise identical American calls on the spot currency. By the same reasoning, American puts on the future must be worth more than puts on the spot currency.

Applications of currency options to currency risk management

Currency options, like their counterparts on other underlying assets, can be used for a variety of purposes and a great variety of strategies can be created. The basic strategies of buying or writing of calls and puts, the vertical, diagonal and calendar spreads, as well as the volatility spreads, are all applicable to the currency option market.

The relative cost of options

It is often stated that currency options are expensive instruments for use in risk management. This view is usually expressed relative to forward rates or futures prices, and is directed at the premium that is paid for options compared with the lack of premium for forwards or futures.

The view as to whether a currency option is expensive or not depends upon one’s view of expected volatility compared with that implied in the option’s price. It is not correct to consider an option to be expensive for risk management just because a premium is payable.

To understand this, we have only to compare the returns distribution that results from hedging a currency exposure with forwards, with that from using options.

Note that the exposure hedged with forwards has a much higher probability of achieving the expected return compared with the unhedged exposure. However, although there is no downside with the forward hedged position, there is no upside either. The risk of any downside has been removed without any payment for this service simply because the potential of any upside has been passed over to the counterparty in compensation.

If we look at the position of the exposure insured with an option, there is a very high probability of achieving the expected return and some probability of achieving an even higher return, but virtually no probability of suffering a lower return.

It is this asymmetry in the pay-offs from an option that results in the premium being payable. That premium clearly is a form of insurance premium.

The need to pay a premium for the asymmetrical pay-off does suggest that currency options, like other options, should be purchased in those situations where asymmetry is valuable. Examples would be where insurance against losses is preferable to hedging against all rate changes.

Purchasing options as a form of insurance

If a US investment manager has strong expectations of a rise in the value of Sterling but wishes to insure against being totally wrong, slightly out-of-the money puts will provide the required insurance.

Take, for example, a situation where the current US dollar spot rate is 1.8500 and a 1.825 put with 103 days to maturity costs 3.15 cents per pound Sterling. The put protects against any fall below 1.8250, a fall of 2.5/185.0 = 1.35%. Such a fall would result in a total loss, including option premium, of 3.15 + 2.5 = 5.65/185.0 = 3.05% over 103 days.

The break-even rate for this strategy is 1.9065; if the currency were to rise above that figure, the option would expire worthless and the currency sold in the market for a profit. Clearly strong expectations of a substantial rise in the currency will be required before the case for spending the option premium can be justified.

Hedging the currency risk in a tender application or a fixed price application that may be oversubscribed

When an investor tenders for a foreign currency investment, there is a risk that he or she may not be successful in the tender and also that if successful there is the risk that the foreign currency appreciates vis-à-vis the home currency in the period between tendering and when payment is due.

There are two distinct time periods in the tendering process: first, the period between delivering the tender and the announcement of the tender outcome; and secondly, the period between the announcement and the payment by the successful tenderers. During the first period the tenderer does not know whether there will be a need for foreign currency. During the second period, if there is a need to pay foreign currency, there is the risk that the currency will appreciate.

If the tenderer were to hedge the potential currency risk with forwards or futures, there is exposure to the risk of not being successful in the tender. If in the meantime the currency has depreciated, there will be a loss if the hedging transaction is not ultimately required.

An alternative hedging strategy is to buy call options that mature at the time payment is due. This will lock in the exchange rate at which the payment will be made if indeed one has to be made.

On the date that the results of the tender are announced, the tenderer has a choice. If the tender has been unsuccessful, the options can be sold in the market in order to recoup some of the cost of the hedge. If the currency has appreciated sufficiently, a profit can be realized. If the tender was successful, the investor can choose between maintaining the current option hedge or selling the option and covering in the forward market. This decision will be influenced by the amount of time value remaining in the option premium. If it is substantial, it may be better to sell the options and enter into a forward hedge.

Average rate currency options are based upon the average exchange rate of the underlying currency as distinct from the exchange rate on a single date — the expiry date.

The advantage of an average rate option is that the volatility of a moving average of a variable is less than the volatility of individual observations of that same variable. With daily observations, and with the volatility levels seen in the currency markets, the volatility of the moving average is in the order of 60% of that of the raw observations. Consequently, the price of an average rate option with a given exercise price will be less than an otherwise identical standard European currency option.

Average rate options are available from the OTC market, are European in form and cash settled. The majority have fixed exercise prices and the asset price (currency rate) is averaged. However, some options have the exercise price set as the average of the exchange rate over time, and the exchange rate at expiry is not averaged. These options are known as floating strike options. Average rate options where the exercise price is fixed are the most common. The option contract must specify the source and the frequency with which the exchange rate will be observed during the life of the option. The market practice is to calculate the arithmetic mean of the exchange rate; however, early average rate option models assumed a geometric mean of rates.

In the currency options markets, these options are developing a following because they genuinely provide an economic service. They provide insurance against losses, on a periodic stream of cash flows, caused by the exchange rate falling below (for puts) or rising above (for calls) an average rate for the whole period.

For example, a company may have a regular stream of currency receipts or payments, and in the budgeting process assumes an average rate of exchange over the accounting period. Some cash flows will be exchanged above the budgeted rate, while others will be exchanged below it.

In order to insure against the exchange rate falling below this average level, there is a choice between buying options with specific expiry dates that match the periodic cash flows, and exercise prices set to match the average. Alternatively, a single put option can be purchased, with an amount covering the total cash flows, an expiry equal to the date of the final cash flow and an exercise price equal to the average of the periodic exchange rates — an average rate option.

As the pricing of the average rate option will be based upon the volatility of the moving average of the exchange rates, it will therefore be cheaper than a standard currency option. Yet over the entire period of the strategy, it will provide the same level of insurance as a more expensive standard option with a similar exercise price.

Clearly, then, the first step in calculating the fair value is to calculate the 135-day forward rate on a three-month deposit. We will assume that the life of that deposit is actually 90 days, the current 135-day LIBOR is 10% p.a., the 225-day LIBOR is 10.25% p.a. and rates are quoted on an actual over 360-day basis.

The market practice for calculating the 90-day forward rate (referred to in futures market parlance as the 90-day forward-forward) is to use simple interest rates 10.24.

With the forward rate calculated as 10.24% and the market convention of quoting the futures price as 100 minus the forward rate, the theoretical futures price is 100 - 10.24 = 89.76.

In this example, the forward rate is actually below the 225-day rate. This result seems, at first sight, to be counter-intuitive. The forward rate should be such that a sum of money invested for the first time period at the current rate for that period, and the principal and interest reinvested for the second time period at the forward rate, will produce a total return equal to that earned by investing for the whole term at the longer-term rate of interest. If the 135-day rate is below the 225-day rate, it may be thought that the funds must be invested for the 90-day period beginning on day 136, at a rate above the 225-day rate in order for the combination of 135-day and 90-day investments to produce the same return as the 225-day investment. T answer is, not necessarily. The reason is that when the 135-day deposit is ro over, with interest, that interest is compounded. The interest rate on the 225- instrument has no compounding, but the interest on the comparable combination the 135-day and 90-day instruments is compounded after 135 days.

When the yield curve is positively sloped, the longer the time period before th notional roll over (135 days in this example), the greater will be the interest element that is being multiplied and thus the less likely it is that the forward rate will be above the longer-term deposit rate. When the yield curve is negatively sloped, the longer the period before the notional roll over, then the lower will become the forward rate.

Given that the forward deposit, and therefore the forward interest rate on which the future is based, does not yet exist, what is the arbitrage process that keeps th traded futures price at its fair price?

The answer is that forward rates can be locked i by the creation of forward deposits or loans.

By borrowing for 135 days and depositing for 225 days, the investor has created neutral position for the first 135 days (i.e. deposit and loans off-set each other) and net deposit position for the final 90 days. Thus a forward deposit has been create and the net return on that deposit will be the forward rate. Arbitragers will trad between these forward deposits and the futures contract, thereby keeping the future trading at close to its fair price. As there are bid—offer spreads in the money market and the futures markets, there will be an arbitrage-free channel similar to that found in other futures contracts.

It has given the 90-day forward rate 135 days hence as 10.24%. If the future were implying a yield of, say, 10.5%, arbitragers would set up a forward deposit by depositing for 225 days and borrowing for 135 days, thus being a net depositor for 90 days in 135 days‘ time. The arbitrager would then short the future to lock in a yield differential of 26 basis points per annum. If the future was quoted at, say, 10.0%, arbitragers would buy the cheap future, and set up a forward loan by borrowing for 225 days and redepositing the funds for 135 days. The result would be a net 90-day loan in 135 days‘ time hedged by a long futures position.

As forward interest rates reflect the markets’ consensus expectations of what interest rates will be in the future, money market interest rate futures contracts also reflect those same market expectations.

Futures prices and forward rate information

It was stated above that forward rates could not be directly observed in the marketplace. This is true; nevertheless, there is a good proxy for short-term forward rates in the form of the yield implicit in prices of money market interest rate futures. Here it shows a strip of three-month short Sterling futures with deliveries covering the period from late June 1991 to December 1992, a period of 18 months. The forward rates that are derived from these price quotations reflect the market’s current (June 1991) expectations of what the three-month bank deposit rates will be at each respective future point in time.

These rates will not be expected future spot rates, although such rates can be derived in a similar manner from Treasury bill futures contracts in the countries where these contracts exist.

Theoretically, the equality between forward rates and the rates implicit in futures prices depends upon the interest rate being constant. In such circumstances, there will be no margin payments, thus no margin funding cost or interest income, to complicate the comparison between the forward and the future. However, when interest rates are stochastic the futures and forward prices will differ; nevertheless, empirical studies find the rate implicit in short-term interest rate futures and forward rates to be virtually identical. Thus, the rates implicit in money market futures contracts may be taken as good proxies for forward rates.

Conversely, if the implied forward rate were below expectations, investors would borrow for the longer term, raising that rate, and deposit for the shorter term, lowering that rate, until the implied forward rate matched expectations.

From this analysis it is clear that forward rates, in an informationally efficient, competitive market free of restriction on the inflow and outflow of funds, will reflect the market’s consensus expectations of future interest rates. This does not imply that forward rates are good forecasters of future spot rates. In the intervening period new information will affect the market and the rates will change.

Futures on short-term interest rates

Futures on short-term interest rates are based upon the rates applicable to bank deposits, Treasury bills, certificates of deposit or the differential between eurocurrency interest rates. The CD futures have declined in popularity in recent years and, at the time of writing, the CD future traded on the IMM of the Chicago Mercantile Exchange is moribund.

It should be noted that the price quotations of the short-term interest-rate futures are not in terms of interest rates. They are actually similar to those traditionally seen in the secondary market for debt, and the movements in these quotations are similar. When interest rates rise, the futures price falls, similar to the price response of a Treasury bill, a Certificate of Deposit or indeed a bond. This price reaction is created by quoting the futures price as an index of 100 minus the market yield.

Futures contracts on interbank interest rates

A number of futures exchanges trade futures on short-term bank deposit