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15

IR futures are traded at the Chicago Mercantile Exchange (CME),14 the EU-REX (formerly the DTB, MATIF, etc.), the London International Financial Futures Exchange (LIFFE), the Singapore International Monetary Exchange (SIMEX), and other exchanges. In the first three quarters of 1997, the CME Eurodollar time deposit had a mean daily volume of 461,098 contracts,15 with each contract corresponding to a notional 1 million dollar 3-month deposit. IR futures are exchange-traded contracts and this entails several differences with respect to over-the-counter (OTC) instruments, as already explained in Section 2.1.2. IR futures are linked to a specific exchange, except when a fungibility agreement is in effect.16 Trading is typically limited to opening hours. Most IR futures exchanges have replaced or are replacing floor trading (open outcry) with electronic trading. In studies of long data samples, we may be forced to use a first half of data originating from floor trading and a second half from electronic trading.

IR futures are traditionally known under the name Eurofutures; the contracts were called "Eurolira," "Euroyen," and similar after the underlying currency. In statistical studies based on historical samples, these names may still be used, but nowadays, they lead to confusion with the new currency named the Euro.

Information on IR futures is particularly valuable for financial institutions and above all for banks. A quick analysis of a typical balance sheet would often reveal a higher exposure to IR risk than, for example, to foreign exchange risk. IR futures can also be used as hedging instruments. From a practical point of view, there is widespread need for a better understanding of the empirical behavior of IR futures on an intraday basis; nevertheless, in the literature there is little material on intraday IR futures markets. The studies by Ballocchi et al. (1999a,b), and Ballocchi et al. (2001) offer deep insights. The impact of scheduled news releases has been investigated by Ederington and Lee (1993, 1995).

IR futures refer to an underlying deposit (usually a 3-month deposit). An IR future price / (bid, ask, or transaction price) is quoted as a number slightly below 100 according to the following formulas

where r is the annualized forward interest rate17 with a forward period of usually 3 months. There are four main settlement months in a year (March, June, September, and December), known as quarterly expiries. Serial expiry contracts (i.e., contracts expiring in months that do not correspond to the quarterly sequence) have been introduced more recently and typically have lower liquidity.

Unlike the spot IRs, the futures prices are not affected by the individual credit ratings of clients, because the collateral account required by the exchange already

14 They are traded at the International Monetary Market (IMM) Division of the CME.

15 All expiries combined, as reported in the January 1, 1998, issue of Futures magazine.

16 One example of a fungibility agreement is the mutual offset system between CME and SIMEX, through which contracts opened in one exchange can bc liquidated on the other one.

17 For instance, a value r - 3% implies a futures price of 97.00.

(2.3)



covers the credit risk. The existence of the collateral account can usually be ignored in studies of IR futures data.

Unlike other futures markets contracts, IR futures contracts are settled in cash. The notional deliverable asset is a (3-month) deposit starting at expiry, but the exchange or the party with the short position does not deliver such a contract at expiry. Instead, a cash payment corresponding to the value of the notional deposit is made. This value is determined by the short-term offered rate (e.g., EURIBOR or LIBOR) at expiry.

As for futures in general, single contracts have a nonstationary time series of limited lifetime (e.g., Fung and Leung, 1993). A typical nonstationary effect of IR future contracts is the systematic decrease of mean volatility when moving closer to the expiry (which is fixed in calendar time). In order to study long time series, we have to connect the data from several contracts. Rollover schemes as suggested in Section 2.1.2 are not the most suitable method in the case of IR futures, because several contracts with different expiries trade at the same time with comparable liquidity, unlike what happens in bond futures and other futures markets, where basically only one contract (or at most two) are actively traded at any given time.

In the case of IR futures, the problem is inherently multivariate such that the interplay of several contracts with different expiries but simultaneous high open interest levels cannot be neglected. The method followed here is to infer implied interest rates from the prices of futures as discussed in the next section.

2.4.2 Implied Forward Interest Rates and Yield Curves

Implied interest rates have some advantages over IR futures prices. They can be studied in long time series. The behavior of their returns is closer to stationary, although more subtle effects such as local volatility peaks before expiry dates may still remain in time series of implied rates. Implied interest rates can be studied in two different forms:

Forward interest rates: The interest rate for a period of usually 3 months, with a starting point always shifted to the future by a fixed time interval;

Yield curve: Spot interest rates for periods starting now. The yield curve is the full term structure of interest rates of different maturities.

Both forms need to be computed as discussed next. Futures prices alone are not sufficient to construct implied spot rates (points on the yield curve), because the IR futures market does not convey any information about the applicable spot rate for the period from the current time to the next futures expiry. The necessity to use data from other instruments (spot IRs) can be avoided by studying forward IRs, with a minimum starting point of 3 months in the future.

There are many methods to construct forward interest rates or full yield curves from IR futures. Instead of presenting the wide field of methods in the literature, we explain the method that was actually used to obtain the results presented in this book, the polynomial method. To use the information from IR futures to construct a yield curve of forward (or spot) rates, a timing problem needs to be dealt with.



Futures are denned in terms of the contract expiry date (a fixed quarterly date, every 3 months for the major contracts) and the maturity period of the underlying reference rate (a fixed time interval, usually 3 months), whereas the implied forward rates are defined in terms of fixed time intervals, not in terms of fixed calendar dates. Those time intervals (forward periods) can be written as [/exp, exp + ATm]. They start at time rexp = t + (t is the current time) where is called the time-to-start for the implied forward rate and ATm is the maturity period of the notional deposit.

The polynomial method presented here is based on the interpolation of rates between points on the expiry time axis. Polynomials of degree 2 are used for the interpolation. The choice of polynomial rather than linear interpolation is motivated by the seasonal behavior of returns. Forward IR series generated by linear interpolation exhibit some 3-month seasonalities, which are weak but distinctly stronger than those of forward IR series generated by polynomial interpolation. The seasonality of linearly interpolated data turns out to be an artifact due to insufficient modeling within the 3-month interpolation intervals.

A continuously compounded forward interest rate q is assumed and modeled as a function of the time T where T is the size of the time interval from the time when the quote was issued to the time point of interest. The annualized implied forward rate r, whose relation to futures prices / is given by Equation 2.3, can be expressed by

Tend

1 year f fTe"d

cxp / q(T)dT

1} 100% (2.4)

where the forward period is from Tstan to Tmd- For a futures contract, Tslm = 7eXp is the expiry time and rend = 7mat is the maturity time, which terminates at the maturity period, Tmat = Texp + ATm, with the forward period ATm (often 3 months). The inverse formula is

= 1 (1+

Tmat

which gives the mean value q of q(T) within the forward period. The function q(T) consists of piecewise, continuously connected, quadratic polynomials

6{T) = at2+bt + c, withr = 2 T ~ T ~ T (2.6)

Tmat - Texp

for Texp <T< rmat (i.e., - 1 < t < 1). The polynomial coefficients should obey the requirements of Equation 2.4 and all quoted forward rates are reproduced by integration of Equation 2.6. The other requirement is the continuity of q(T) at the

18 We do not use the term "maturity" as a synonym of "expiry," but we reserve it to denote the duration of deposits, including the underlying deposits of futures contracts. The maturity period of a futures contract thus starts at expiry.



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