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Statistical arbitrage in the US treasury futures market PDF

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Statistical arbitrage in the U.S. treasury futures market Wale Dare September 2017 Discussion Paper no. 2017-16 School of Economics and Political Science, University of St.Gallen Department of Economics Editor: Martina Flockerzi University of St.Gallen School of Economics and Political Science Department of Economics Müller-Friedberg-Strasse 6/8 CH-9000 St. Gallen Phone +41 71 224 23 25 Email [email protected] Publisher: School of Economics and Political Science Department of Economics University of St.Gallen Müller-Friedberg-Strasse 6/8 CH-9000 St. Gallen Phone +41 71 224 23 25 Electronic Publication: http://www.seps.unisg.ch Statistical arbitrage in the U.S. treasury futures market Wale Dare Author’s address: Wale Dare Faculty of Mathematics and Statistics Bodanstrasse 6 CH-9000 St. Gallen Phone +41 71 224 2413 Email [email protected] Abstract We argue empirically that the U.S. treasury futures market is informational inefficient. We show that an intraday strategy based on the assumption of cointegrated treasury futures prices earns statistically significant excess return over the equally weighted portfolio of treasury futures. We also provide empirical backing for the claim that the same strategy, financed by taking a short position in the 2-Year treasury futures contract, gives rise to a statistical arbitrage. Keywords Market efficiency, U.S. treasury futures, statistical arbitrage, joint-hypothesis. JEL Classification C12, G13, G14. 1 Introduction Is the U.S. treasury bond futures market informational efficient? Weak-form informational efficiency requires all strategies that rely solely on historical price data to be dominated by the passive strategy of holding single traded assets or a weighted portfolio of traded assets. The notion of dominance as it relates to asset pricing was introduced by Merton (1973) to study option pricing formulas that are consistent with rational investor behavior. More recently, Jarrow & Larsson (2012) obtained a characterization of informa- tional efficiency in terms of the no dominance condition (ND) and the No Free Lunch with Vanishing Risk condition (NFLVR) of Delbaen & Schacher- mayer (1994). Accordingly, market inefficiency can be asserted as soon as either the ND or NFLVR fails. This result simplifies considerably the task of verifying market efficiency; it belies the long held belief that in order to test for violations of market efficiency, one must first specify a model of equilibrium prices such as the CAPM and then test for efficiency in relation to the estimated equilibrium model. Unfortunately, this two step procedure runs quickly into difficulties, since it may not be possible to tell apart errors due to model misspecification and those that are solely due to market inefficiency. This is the well-known joint-hypothesis problem discussed in (Fama, 1969). Moreover, the No Dominance condition itself could be dispensed with as soonasachangeofnumeraireisperformed. IndeedletB := (Ω,F,(F ) ,P) t t≥0 denotea probability basis, and letS denoteann-dimensionalsemimartingale whose components Si,0 ≤ i < n, represent the price of n distinct assets, expressed in units of the zeroth asset. For the sake of convenience, also assumethatattimezero, eachassetispricedatone, i.e. Si = 1for0 ≤ i < n. 0 Now, let γ denote a positive number between zero and one, i.e. 0 < γ < 1, and define Zγ,i := (S,Sγ,i)(Sγ,i)−1, where Sγ,i = γ +(1−γ)Si. According to Dare (2017, Proposition 2.1), the efficiencyof(S,B)isequivalenttotheexistenceofalocalmartingalemeasure for the markets (Zγ,i,B), for 0 ≤ i < n and 0 < γ < 1. In fact, a stronger statement can be made achieve a slightly provided prices are expressed in units of a portfolio constructed on the basis of a strictly positive weight vector α = (α ,··· ,α ), i.e. α > 0 for 0 ≤ i < n. 0 n−1 i Indeed if Zα := (S,Sα)(Sα)−1, 3 then according to Dare (2017, Corollary 2.2), the market (S,B) is efficient if and only if (Zα,B) admits a local martingale measure. The choice of a market portfolio is irrelevant so long as it assigns positive weight to each traded asset. WewillargueforaviolationofmarketefficiencyusingDare(2017, Propo- sition 2.1) with Si representing the price of the 2-Year U.S. Treasury futures contract. Fortunately, since the NFLVR condition is specified in terms of the physical measure, the joint-hypothesis issue may be avoided by evaluating trading rule for violations of NFLVR. Using this testing approach, we make and emprically support the claim that between April 1, 2010 and Decem- ber 31, 2015, the equally weighted buy-and-hold strategy was out-performed by a simple cointegration-based trading rule. Moreover, the hypothesis of the existence of a statistical arbitrage, in the sense of (Hogan et al., 2004), achieves a p-value less than 2%. The trading rule we examine takes as starting point the hypothesis that treasury bond futures are cointegrated and then attempts to profit from deviationsfromthecointegratingrelationships. Thecointegrationhypothesis assumes, among other things, that even though prices of individual contracts may be non-stationary, there exists at least one linear combination of these contracts that results in a stationary price process. That is to say, it is possible to put together a portfolio of long and short positions in individual contracts such that the resulting market value of the portfolio is stationary. The hypothesis of cointegrated bond prices has been examined by Bradley & Lumpkin (1992), Zhang (1993), and many others. In these studies, the data employedwassampledatlowfrequency, dailyormonthly, andthehypothesis ofintegratedbondpricescouldnotberejected. Wecarryoutsimilaranalysis and find empirical support for cointegration using data sampled intra day at one-minute intervals. We obtain theoretical motivation for the cointegration-based trading rule by embedding our analysis within the literature devoted to the study of the term structure of bonds using factor models. Starting with Litterman & Scheinkman (1991) and later Bouchaud et al. (1999) and many others, it has been noted that between 96% and 98% of overall variance of the entire family of treasury securities may be explained by the variance of just three factors, the so-called level, slope, and curvature factors. The factors are so named because of how they affect the shape of the yield curve. A shock emanating from the first factor has nearly the same impact on contracts of all maturities; the resulting effect is a vertical shift, upward or downward, of the entire yield curve. The second factor affects bonds of different maturities in such a manner as to change the steepness or slope of the curve; it does so by affecting securities at one end of the maturity spectrum more or less 4 than those at the other end. Finally, the third factor has the effect of making the yield curve curvier; it does so by having more or less pronounced effects on medium term bonds than on bonds situated either ends of the maturity spectrum. We argue that a strategy based on a cointegration hypothesis is natural within the context of a term structure driven by common stochastic trends or factors. Infact, theoppositeisalsotrue, thatis, acommonfactorstructureis a natural consequence of cointegrated yields. This line of argument provides support based on economic theory for our strategy and helps explain its performance. Our results suggests that the futures market may be inefficient. Market inefficiency is clearly not a desired outcome. It implies the existence of a free lunch. Put another way, our results points to possible misallocation of resources. The rest of the paper proceeds as follows: in section 2, we provide a description of the data used. Futures price data usually does not come in continuous form for extended periods of time, so we had to make certain choices about how available historical price data is transformed into a state suitable for our analysis. These choices can be implemented in real-time and are, therefore, to be considered as part of the trading rule. In section 3, we provide theoretical foundation for our trading rule. This foundation allows us to reach beyond our data and assert that the profitability of the trading rule is very likely not confined to the period for which we have data. Section 4 is devoted to the implementation details of the trading rule. Section 5 summarizes our empirical results, and section 6 concludes. 2 Data 2.1 Treasury futures CBOT Treasury futures are standardized foreward contracts for selling and buying US government debt obligations for future delivery or settlement. Theywereintroducedinthenineteen-seventiesattheChicagoBoardofTrade (CBOT), now part of the Chincago Merchantile Exchange (CME), for hedg- ing short-term risks on U.S. treasury yields. They come in four tenors or maturities: 2, 5, 10, and 30 years. In reality, each contract type is written on a basket of U.S. treasury notes and bonds with a range of maturities and coupon rates. For instance, the 30-Year Treasury Bond Futures contract is written on a basket of bonds with maturities ranging from 15 to 25 years. It is, therefore, worth keeping in mind that a study of the dynamics of the yield curve using futures data reflects influences from a range of maturities. 5 Every contract listed above except the 2-Year T-Note Futures contract, which has a face value of $200,000, has a face value of $100,000. That is each contract affords the buyer the right to buy an underlying treasury note or bond with a face value of $100,000 or $200,000 in the case of the 2-Year contract. In practice, the price of these contracts are quoted as percentages of their par value. The minimum tick size of the 2-Year T-Note Futures is 1/128%, that of the 5-Year T-Note Futures is 1/128%, that of the 10-Year T-NoteFuturesis1/64%, andthatofthe 30-YearT-BondFuturescontractis 1/32%. In Dollar terms, this comes to $15.625, $7.8125, $15.625, and $31.25, respectively, per tick movement. 1 These tick sizes are orders of maginitude larger than those typically encounted in the equity markets. Even though most futures contracts are settled in cash at the expiration of the contract, for a small percentage of open interests, delivery of the underlying bond actually takes place. Given that the futures contract is written on a basket of notes and bonds, the actual bond or note delivered is at the discretion of the seller of the contract. In practice, the seller merely selects the cheapest bond in the basket to deliever. For our purposes, we shall focus on only the above listed tenors, but it is worth keeping in mind that there is is also a 30-Year Ultra contract that is also traded at the CME. For our analysis, we use quote data, prices and sizes, from April 1, 2010 through December 31, 2015. Even though we have at our disposal data rich enough to allow resolution down to the nearest millisecond, we opted, arbi- trarily, to aggregate the data into one-minute time bars. The representative quotedpriceandsizeforeachtimebaristhelastrecordedquotefallingwithin that interval. Our use of quotes , bids and offers, instead of transaction data allows the computation of a proxy for the unobserved true price, by means of the mid-quote, at a higher frequency than transaction prices might have allowed. Using quotes, we are also able to reflect directly a major portion of the execution costs associated with any transaction, i.e. the bid-ask spread. Trading in these markets primarily takes place electronically via CME ClearPort Clearing virtually around the clock between the hours of 18:00 and 17:00 (Chicago Time), Sunday through Friday. But, the markets are at their most active during the daytime trading hours of 7:20 and 14:00 (Chicago Time), Monday through Friday. This also the opening hours of the open outcry trading pits. For our analysis, We use exclusively data from the daytime trading hours. This ensures that the strategy is able to benefit from the best liquidity these markets can offer, while mitigating the effects of slippage (orders not getting filled at the stated price) and costs associated 1We refer the reader to more detailed information about the features of each contract to Labuszewski et al. (2014). 6 with breaking through the Level 1 bid and ask sizes. 2.2 Continuous prices Unlike stocks and long bonds, futures contracts tend to be short-lived, with price histories extending over a few weeks or months. This stems from the traditionaluseoffuturescontractsasshort-termhedginginstrumentsagainst price/interest rate fluctuations. Treasury futures contracts, in particular, have a quarterly expiration cycle in March, June, September, and December. At any given point in time, several contracts written on the same underlying bond, differentiated only by their expiration dates, may trade side by side. Usually, the next contract due to expire, the so-called front-month con- tract, offers the most liquidity. As the front-month approaches expiration, liquidity is gradually transferred to the next contract in line to expire, the deferred month contract. At anyrate, a given contractis only actively traded for a few months or weeks before it expires. Hence, holding a long-term po- sition in a futures contract actually entails actively trading in and out of the front month contract as it nears its expiration date. The implementation of this process is known as rolling the front month forward. For the purpose of evaluating a trading strategy over a historical period of more than a few months, the roll can be retroactively implemented to generate a continuous price data. The usual way to go about the roll is to trade out of the front-month a given number of days before it expires. In the extreme case, the roll takes place on the expiration date of the front month contract. The downside of this type of approach is that the roll may take place at a date when liquidity in the deferred month is not yet plentiful. The result is that a backtest may not necessarily capture the increased trading cost associated with the lower liquidity level. Our preferred approach for implementing the roll is to start trading out of the front month contract at any point during its expiration month as soon as the open interest in the deferred month contract exceeds the open interest in the front month contract. The data used in our backtest is spliced together thisway; theprocedureisimplementableinreal-timeandmustbeconsidered part of the trading strategy discussed in this paper. Now, while retroactive contract rolling may solve the problem of creat- ing an unbroken long-term price history, it creates another: splicing prices together as described above would invariably introduce artificial price jumps into historical prices. To see this, consider a futures contract with price F written on a bond with price B. Using an arbitrage argument and ignor- ing accrued interest, the price of a futures contract at any time t may be 7 expressed as: F = B e(r−c)d, (2.1) t t where c is the continuously compounded rate of discounted coupon payments on the underlying bond, d is the number of time units before the futures contract expires, and r is the repo rate. Now, assumingtherolltakesplaceintheexpirationmonth, dforthefront month is less than 30 days, whereas for the deferred month contract, d is at least 90 days. This results in a price differential between the two contracts, which shows up in the price data as a jump. In reality, and assuming a self-financing strategy, the price differential would necessitate a change in the number of contracts held, so that overall, the return on the portfolio is unaffected by the roll. Hence, in order to avoid fictitious gains and losses, the price series must be adjusted to remove the roll-induced price jumps. The most often used methods in practice applies an adjustment to prices either prior or subsequent to the roll date. When the adjustment is applied to prices recorded after the contract is rolled forward, the price history is said to be adjusted forward; if on the other hand, the adjustment is applied to prices recorded prior to the roll date then the prices are said to be adjusted backward. The actual price adjustment, in the case of a backward adjust- ment, is most commonly carried out in one of two ways: in the first instance, the roll-induced price gap (price after roll minus price right after roll) is sub- tracted from all prices recorded prior to the roll date; in the second instance, all prices preceding the roll date are multiplied by a factor representing the relative price level before and after the roll. The second approach is remi- niscent of how stock prices are adjusted after a stock split. We will refer to the first approach as the backward difference adjustment method and to the second as the backward ratio adjustment method. Forward ratio adjustment and forward difference adjustment are implemented similarly with the ad- justments applied to prices recorded after the roll date. In our analysis, we will only consider backward adjusted prices, as they appear to be the more intuitive approach. Both types of backward price adjustment methods are widely-used in practice, but the ratio adjustment method has the advantage of guaranteeing that prices, however early in the price series, always remain positive. In the- ory, the difference adjustment approach may generate negative prices given enough roll-induced price gaps. We mention these adjustment procedures because they tend to affect the performance of most strategies, including the one we study in this paper. The price adjustment procedures cannot be con- sidered as part of a real-time trading strategy, so we report results using both the backward ratio adjustment and the backward difference adjustment. 8

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Keywords. Market efficiency, U.S. treasury futures, statistical arbitrage, joint-hypothesis. pricing formulas that are consistent with rational investor behavior. More glimpse of what the original factors might look like. Using the
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