Statistical arbitrage in the US treasury futures market

38 Pages·2017·1.05 MB·English

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 [phone]
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 [phone]
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 [phone]
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