Statistical Arbitrage with Genetic Programming

36 Pages·2008·2.86 MB·English

Centre for WP020-08
Computational
Finance and
Economic
Agents
Working
Paper
Series
Philip Saks
Dietmar Maringer
Statistical Arbitrage with
Genetic Programming
May 2008
www.essex.ac.uk/ccfea
Statistical Arbitrage with Genetic Programming
PhilipSaks∗ DietmarMaringer
May5,2008
Abstract
This paper employs genetic programming to discover statistical arbi-
trage strategieson thebanking sector inthe EuroStoxx universe. Binary
decisionrulesareevolvedusingtwodifferentrepresentations. Thefirstis
theclassicalsingletreeapproach,whereonedecisiontreeforbuyandsell
ordersisdeveloped. Thesecondversionusesadual tree structure where
two decision trees are generated and the evaluation is contingent on the
currentmarketposition. Hence,buyandsellrulesareco-evolvedforlong
andshortpositions. Appliedtoempiricalhigh-frequencypricesforstocks
ofbankingcompanies,bothmethodsarecapableofdiscoveringsignificant
statisticalarbitragestrategies,eveninthepresenceofrealisticmarketim-
pact. Thisimpliestheexistenceofmarketinefficiencieswithinthechosen
universe.However,theperformanceofthesuccessfulstrategiesdeteriorate
overtimeandtheinefficiencieshavedisappearedinthesecondhalfofthe
out-of-sampleperiod.
Astransactioncostsareincreasedthereisaclearasymmetricresponse
between single and dual trees. Naturally, increased costs have a negative
impactonperformance,butthedualtreesaremuchmorerobustandcan
adapttothechangedenvironment,whereasthesingletreescannot.
1 Introduction
During thelastdecades, theEfficient MarketHypothesis(EMH) hasbeen putto
trial,especiallywiththeemergenceofbehavioralfinanceandagent-basedcom-
putationaleconomics. Theriseoftheabovementionedfieldsprovidesatheoret-
ical justification for attacking the EMH, which in turn stimulates the empirical
forecastingliterature.
A basic premise for efficiency is the existence of homo economicus, that the
marketsconsistsofhomogeneousrationalagents,drivenbyutilitymaximization.
However,cognitivepsychologyhasrevealedthatpeoplearefarfromrational,in-
steadtheyrelyonheuristicsindecisionmakingtosimplifyagivenproblem[17].
∗both:CentreforComputationalFinanceandEconomicAgents,UniversityofEssex,Wivenhoe
Park,ColchesterCO43SQ,UK. . Theauthorsaregratefulfor
{psaks,dmaring}[at℄essex.a
.uk
valuablecommentsandsuggestionsbyconferenceandseminarparticipantsinCambridge,Naples
andEssex.FinancialsupportfromtheEUCommissionthroughMRTN-CT-2006-034270COMISEF
isgratefullyacknowledged.
1
1 Introduction 2
This is both useful and necessary in everyday life, but in certain situations it
can lead to biases such as overconfidence, base-rate neglect, sample-sizeneglect,
gamblersfallacy, conservatismandaversiontoambiguity[3]. Whatismoreim-
portantisthatthesebiasesmanifestthemselvesonanaggregatelevelinthemar-
kets as momentum and mean-reversion effects [14, 7]. Accepting the existence
ofheterogenousagentshavepronounced effectsonatheoretical level. Insucha
scenario the market clearing price cannot be determined formally, since agents
needtoformexpectationsaboutotheragents’expectations. Thisleadstoan“in-
finite regressin subjectivity” where agents cannot form expectations by deduc-
tivemeans, regardlessoftheir reasoningpowers. Thus perfect rationalityisnot
well-defined. Instead, investors are forced to hypothesize expectational models
where the only means of verification is to observe the models’ performance in
practice[2]. Insuchaworlditisindeedsensibletodevelopexpectational mod-
elsbeyondtraditionalequilibrium analysis. Inthispaper,suchmodelsarebuilt
usinggeneticprogramming (GP).
The majority of existing applications of GP in financial forecasting have fo-
cused on foreign exchange. Here, the general consensus is that GP can dis-
coverprofitabletradingrulesathighfrequenciesinpresenceoftransactioncosts
[16,9,6]. Forthestockmarketresultsaremixed. Thebuy-and-holdstrategyon
daily S&P500 data is not outperformed [1], while it is on a monthly frequency
[5]. Besides changing the frequency, a reduced grammar is considered. More-
over,buyandsellrulesareco-evolvedseparately.
Inthispaper,weconsidergeneticprogrammingforstatisticalarbitrage. Ar-
bitrageinthetraditional senseisconcerned withidentifying situations wherea
self-funding is generated that will provide only non-negative cash flows at any
pointintime. Obviously,suchportfoliosarepossibleonlyinout-of-equilibrium
situations. Statistical arbitrage is a wider concept where, again, self-funding
portfolios are sought where one can expect non-negative pay outs at any point
in time. Here one accepts negative pay-outs with a small probability as long as
the expected positive payouts are high enough and the probability of losses is
small enough; ideally this shortfall probability converges to zero. In practice,
such a situation can occur when price processes are closely linked. In the clas-
sical story of Royal Dutch and Shell [3], the pair of stocks are cointegrated due
totheirfundamentallinkviatheirmergerin1907. Inmostcases,however,such
linksarenotasobvious,butthatdoesnoteliminatethepossibilitythatsuchrela-
tionshipsmightexistandcanbedetectedbystatisticalanalysis. Inthefollowing
stockswithinthesameindustrysectorareconsidered,sinceitcanbearguedthat
these stocks are exposed to many of the same risk factors and should therefore
havesimilarbehavior.
As mentioned previously, an arbitrage portfolio is constructed by using the
proceedings from short selling some stocks to initiate long positions in other
stocks. Moreformally,thecumulative discountedvalue(v )ofastatisticalarbi-
t
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
2 ClusteringofFinancialData 3
tragestrategyhastosatisfythefollowingconditions[13],
v =0 (1)
0
lim E(v )>0 (2)
t
t→∞
lim prob(v <0)=0 (3)
t
t→∞
Var(v )
lim t =0 if prob(v <0)>0 ∀t <∞ (4)
t
t→∞ t
Thismeansthatithastohavezeroinitialcostandbeself-financing(1);apositive
discountedvalue(2);andaprobabilityoflossconverging tozero(3). Condition
(4) states that a statistical arbitrageproduces riskless incremental profits in the
limit.
Bytakingrelativevaluebetsbetweenhighlycorrelatedstocksfromthesame
industry, much of the market uncertainty is hedged away. Hence, profits made
from this strategy are virtually uncorrelated with the market index. Further-
more,bymodelingtherelationships betweenstocks,theattentionisfocusedon
adirection wheremorestablepatterns shouldexistrather thanmaking specific
predictions about future developments. This statement defies the EMH in its
weakestform,thatnotradingsystembasedonhistoricalpriceandvolumeinfor-
mationshouldgenerateexcessreturns[10].
The rest of the paper is organized as follows. Section 2 analyzes the Euro
Stoxx universe and provides evidence of significant clustering between sectors.
Section 3 introduces the data, model and framework. Sections 4 and 5 present
resultsundertheassumptionsoffrictionlesstradingandrealisticmarketimpact,
respectively. InSection 6thetransaction costisgraduallyincreasedandtheef-
fects on the single and dual trees are investigated. Finally, Section 7 concludes
andgivespointerstopossiblefutureresearch.
2 ClusteringofFinancialData
It is frequently argued that stocks within the same industry sector are exposed
tomanyof thesameriskfactors andshouldtherefore havesimilar behavior. In
order toclarify this, weinvestigate the majority of stocks in theEuro Stoxx600
index. The data is gathered from Bloomberg and includes information such as
company name, ticker symbol, industry sector andindustry group. Inaddition
hereto, we obtain the adjusted closing prices in the period from 21-Jan-2002 to
26-Jun-2007. Since the index composition is changing over time, we only con-
sider stocks where data exists for the last two years for both price and volume
series. Takingthisintoaccount,theuniversecomprisesofatotalof477stocks.
Thenotionthatstockshavesimilarbehaviorneedstobespecifiedinorderto
conduct a proper analysis. Anobvious measurefor price data isthecorrelation
ofreturns,whereahighercorrelationimpliesstrongersimilarity. Figure1shows
the maximum spanning tree for the undirected graphdefined by the upper tri-
angleofthereturn’scorrelationmatrix,adjustedforcountryeffects-inpractice
weestimatetheminimumspanningtree,wherethecorrelationsaretransformed,
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
2 ClusteringofFinancialData 4
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RSL LN
Basic Materials Energy
Communications Financial
Consumer, Cyclical Industrial
Consumer, Non-cyclical Technology
Diversified Utilities
Figure1: Maximum spanning tree of Euro Stoxx 600 correlation matrix (graph
createdwithPajek)
ρi,j →|ρi,j−1|. An undirected graphis a collection of vertices and edges, and
a spanning tree is a subgraph in which all the vertices are connected. In our
case,eachvertexrepresentsastock, andwehaveafullyconnected graphwhere
theedgescompriseofthetransformedcorrelations. Thismeasurecanbeviewed
as thecost of linking twostocks, and theminimum spanning tree issimply the
spanningtreewhichconnectsallthenodes,suchthatthecombinedcostismini-
mal.Theminimumspanningtreeisunique,giveneachedgehasadistinctweight
andcanbeestimatedefficientlyusingthegreedyKruskal’salgorithm.
The color of the vertices represents the industry sector and it appears that
there is a pronounced clustering, especially for the basic material, communica-
tion and financial sector. It is worth noting that retail giants such as Unilever
(UNA NA), Nestle (NESN VX) and Cadbury Schweppes (CBRY LN) are tightly
linked. Another interesting feature isthat three major basic material stocks are
connected: BHPBilliton(BLTLN),RioTinto(RIOLN)andAntofagasta(ANTO
LN). This is not surprising since these stocks in addition to being in the same
industrysector,alsosharethesameindustrygroup,mining.
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
3 Framework 5
The maximum spanning tree merely provides a graphical representation of
the relationships within the index, but is not in itself a statistical test for the
hypothesis that stocks within a sector tend to be clustered together. Clustering
analysis is nothing novel within statistics, and various methods such as the k-
meansalgorithmcanefficiently tacklethisproblem. Thus, ifthestatisticalclus-
teringisindependentofthefundamentalclustering,dictatedbytheindustrysec-
tors,thenonecanreject thehypothesis. LetS andF betwostochasticvariables
whichdescribethestatisticalandfundamentalclusters, respectively. Thestatis-
ticalclustersareobtainedbyusingk-meansonthecorrelationmatrix,whilethe
fundamentalclusterarebuiltaccordingtotheindustriestheassetsbelongto. The
maximumvaluestheycanattainisdenotedbytheintegersk andk . IfN isthe
s f
number of stocksintheuniverse, thehypothesisof independence canbetested
bycomputing
N
Vj,m= I{si=j}·I{fi=m} ∀ j =1,2,...,ks m=1,2,...,kf (5)
i=1
X
viaaχ2 statisticforcontingency tables. Settingk =k =10,thisteststatisticis
s f
χ2 =1314.1 while the critical value is χ2 (81)=103.0; hence, thus strongly re-
0.05
jectingthenullhypothesisthatstatisticalandfundamentalclustersareindepen-
dent. In other words, significant clustering within the sectors can be expected,
anditisreasonabletopre-selectassetsfromonefundamentalcluster(here: be-
longingtothesameindustry)fortheactualstatisticalarbitrageapplication.
3 Framework
Asmentionedpreviously,theobjectiveistodevelopatradingstrategyforstatis-
tical arbitragebased on price and volume information, and in the following we
elaborateondata,preprocessingandmodelconstruction.
3.1 Data
The data comprises of Volume-Weighted Average Prices (VWAP) and volume,
sampledatanhourlyfrequencyforthebanksintheEuroStoxx600index. Itcov-
ersthetimeperiodfrom01-Apr-2003to29-Jun-2007,correspondingtoatotalof
8648 observations. Again, we only consider stocks for which we have enough
data, which limits the portfolio to 30 assets. The components and summary
statistics are documented in Table 7 and the VWAP prices are depicted in Fig-
ure2.
Whenanalyzing highfrequency data, itisimportant totakeintradayeffects
intoaccount. Figure3showstheaverageintradayvolumeforABNAMROinthe
periodfrom04-Jul-2006to29-Jun-2007. Clearly,thevolumeishigherafteropen
andbeforeclosethanduringthemiddleoftheday. Inthecontextoftradingrule
induction it is important to remove this bias, which is basically a proxy for the
timeofday,andprohibitsreasonableconditioningonintradayvolume.
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
3 Framework 6
Banking Stocks
103
102
101
2003 2004 2004 2005 2005 2006 2006 2007
Year
Figure2: HourlyVWAPpricesforbankingstockswithintheEuroStoxxindex
Volume Profile, ABN AMRO
x 106
5
4
3
2
1
0
16
14 07
12 07
06
10
06
Time 8 06 Year
Figure3: ExpectedintradayvolumeforABNAMROintheperiodfrom17-May-
2007to29-Jun-2007
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
3 Framework 7
3.2 Preprocessing
Thereturnseriesfor eachstockisstandardized withrespect toitsvolatility, es-
timated using simple exponential smoothing. Likewise, a volume indicator is
constructedthatremovestheintradaybias,andmeasurestheextenttowhichthe
level isloweror higher thanexpected. Specifically, wetakethelogarithmof the
ratiobetweentherealizedandexpected volume, wherethisratio hasbeenhard
limitedintherangebetween0.2and5.
Since the main focus is on cross-sectional relationships between stocks,
ratherthantheirdirection,wesubtractthecross-sectionalaveragefromthenor-
malizedreturnsandvolumeseriesforeachstock.Basedontheseseries,wecalcu-
latethemovingaveragesoverthelast8,40and80periods;atanhourlyfrequency
this corresponds to one day, a week and two weeks, respectively. All indicators
are transformed to quantiles, using a Gaussian distribution function estimated
ona3-monthrollingwindowbasis.
3.3 Model
There are two approaches for modeling trading rules; either as decision trees
wheremarketpositionsoractionsarerepresentedintheterminalnodes[18],or
asasinglerulewheretheconditioningisexogenoustotheprogram[6].
Weconsider thelatterapproachinthecontextofabinarydecisionproblem,
which corresponds to long and short positions. As mentioned previously, the
focus of this paper is on arbitrage portfolios, where the purchase of stocks is
financedbyshort sellingothers. Naturally,aprecondition forthisisthatnotall
the forecasts across the 30 stocks are the same. For example if the trading rule
takes a bullish view across the board, then short-selling opportunities have not
been identified and proper arbitrage portfolios cannot be constructed. In this
casenostocksareheld.However,whenforecastsfacilitateportfolioconstruction,
thisisdoneonavolatilityadjusted basis. Letoi ∈{−1,1} denote theforecast on
t
stock i at time t, such that -1 and 1 corresponds to a bearish and bullish view,
respectively. Thentheholdingisgivenby
1
hi =oi· σit (6)
t t n 1 ·I
j=1σtj {otj=oit}
P
whereσ isthevolatility, n isthenumber of stocks intheuniverse, and I
t {oj=oi}
t t
is an indicator variable that ensures that forecasts are normalized correctly i.e.
itdiscriminatesbetweenlongandshortpositions. Byconstruction thelongand
shortpositionssumto-1and1,respectively.
n
hjI = 1
t {hj>0}
j=1 t
X
n
hjI = −1 (7)
t {hj<0}
j=1 t
X
Bydownweightingmorevolatilestockstheportfoliobecomesmorestable.
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
3 Framework 8
Function Arguments ReturnType
, ( , )
< > qrtn q
onst bool
, ( , )
< > qvol q
onst bool
( , , )
BTWN qrtn q
onst q
onst bool
( , , )
BTWN qvol q
onst q
onst bool
, , ( , )
AND OR XOR bool bool bool
( )
NOT bool bool
( , , )
ITE bool bool bool bool
Table1: Statisticalarbitragegrammar,where checksifthefirstargumentis
BTWN
betweenthesecondandthird. representstheif-then-elsestatement
ITE
We employ two different methods for solving the binary decision problem.
Thefirstusesastandardsingletreestructure,whilethesecondconsiders adual
treestructureinconjunctionwithcooperativeco-evolution[4].Inbothmethods,
the trees return boolean values. For a more rigorous analysis of the dual tree
approach,seeSaksandMaringer[20]
Forthedual treestructure, programevaluation iscontingent onthecurrent
marketpositionforthatparticularstock,i.e.,thefirsttreedictatesthelongentry,
whilethesecondentersashortposition. Inotherwords,whichofthesetwotrees
isevaluated,dependsonthepreviousposition;ifstocki attimet wasinashort
position (oi <0), then tree k =1 is evaluated and dictates if a long position
t−1
should be initiated. Alternatively, tree k =2 is evaluated to decide whether to
enter ashort position. Let bk,i ∈{0,1} bethetruth valuefor tree k onstock i at
t
time t, whether or not to switch positions, then b1,i =1 (b2,i =1) indicates to
t t
enteralong(short)position, whileb1,i =0(b2,i =0)leavesthecurrent position
t t
unchanged. Thenthenewforecastisgivenas
2·b1,i−1 ifoi <0
oi = t t−1 (8)
t (−2·b2,i+1 otherwise
t
Alltreesareconstructed fromthesamegrammar,whichinadditiontotype
constraints also introduces semantic restrictions. This improves the search ef-
ficiency significantly, since computational resources are not wasted on non-
sensical solutions [6]. We consider a fairly restricted grammar, which is docu-
mented in Table 1. It consists of numeric comparators, boolean operators and
if-then-elsestatements( ).Furthermore,aspecialfunction hasbeenin-
ITE BTWN
troduced,thattakesthreeargumentsandevaluatesifthefirstisbetweenthesec-
ondandthird. Theterminalscompriseofthesixindicatorswherethereisadis-
tinction betweenreturn( )andvolume ( )information, andnumerical
qrtn qvol
real-valued constants ( ) ranging from 0 to 1. The parsimonious gram-
q
onst
mar reduces therisk ofoverfitting, andenhances interpretabilityof theevolved
solutions.
The choice of a suitableobjective function is essential in evolutionary com-
putation. Previous studies suggest that a risk-adjusted measure improves out-
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.
4 FrictionlessTrading 9
of-sample performance when compared to an absolute return measure [6]. In
this context, the ratio between the average profits and their volatility would be
anobviouscandidate. However,underthismeasurestrategiesmightevolvethat
do extremely well only on a subset of the in-sample data and mediocre on the
remainder; from a practical point of view, this can lead to additional vulnera-
bility to market timing as the overall success might depend more on the entry
and exit points than on the overall time. Instead, the t-statistic of the linear fit
between cumulated profits and time is employed, since it maximizes the slope
whileminimizing thedeviation from theideal straightlineperformance graph.
Thismeasurefavorsasteadyincreaseinwealth–Figure13intheAppendixillus-
tratesthisconcept.
3.4 ParameterSettings
In the following computational experiments a population of 250 individuals is
initializedusingtherampedhalf-and-half method. Itevolvesforamaximumof
51generations, but is stopped after 15generations if no newelitist (best-so-far)
individual has been found. A normal tournament selection is used with a size
of 5, and the crossover and mutation probabilities are 0.9 and 0.1, respectively.
Moreover, the probability of selecting a function node during reproduction is
0.9, and the programs are constrained to a maximum complexity of 50 nodes.
Again, thisconstraint isimposed tominimize theriskofoverfitting, but alsoto
facilitate interpretability. If themodels losetractability, it defies the purpose of
geneticprogrammingasaknowledgediscoverytool.
Thedataissplitintoatrainingandtestset.Theformercontains6000samples
andcovers theperiodfrom01-Apr-2003to10-Mar-2006,andthelatterhas2647
samplesintheperiodfrom13-Mar-2006to29-Jun-2007.
4 FrictionlessTrading
4.1 Performance
As will bediscussed in more detail inSection 5, placing a buy or sell order will
haveanimpact onthemarketprice, inparticular onahighfrequency level ina
market withcontinuous auctions. This impact will besmaller for anarbitrarily
small order size in relation to the market volume. For the sake of simplicity,
this section assumes the absence of market impact, i.e., that all placed orders
areexecutedontherealizedvolumeweightedaverageprice(VWAP).Tradingon
the VWAP differs from a traditional market order, where a trade is executed at
the current observed price. The VWAP is a backward looking measure, and it
is therefore not possible to trade on the observed VWAP at time t. Instead the
executionoccursgraduallybetweent andt+1,resultingintheVWAPatt+1.In
summary, a trading decision is formed based on theVWAP at time t, the entry
priceisobservedattimet+1andtheoneperiodprofitisevaluatedatt+2.
PSaks,DMaringer,Statisti
alArbitragewithGeneti
Programming,CCFEAWP020-08,University
ofEssex2008.