Mean-Reverting Portfolio Design with Budget Constraint

0.47 MB·

by  Ziping Zhao

Submittedpaper 1
Mean-Reverting Portfolio Design with Budget
Constraint
Ziping Zhao, Student Member, IEEE, and Daniel P. Palomar, Fellow, IEEE
Abstract—This paper considers the mean-reverting portfolio series which are named to be cointegrated. Later, the cointe-
7 design problem arising from statistical arbitrage in the finan- gratedvectorautoregressivemodel[13]–[17] was proposedto
1 cial markets. We first propose a general problem formulation
describe the cointegration relations. Empirical and technical
0 aimed at finding a portfolio of underlying component assets
analyses [18]–[21] show that cointegration can be used to get
2 by optimizing a mean-reversion criterion characterizing the
mean-reversion strength, taking into consideration the variance statisticalarbitrageopportunitiesandsuchrelationsreallyexist
n
of the portfolio and an investment budget constraint. Then in financial markets. Taking the prices of common stocks for
a
J several specific problems are considered based on the general example, it is generally known that a stock price is observed
formulation, and efficient algorithms are proposed. Numerical
8 and modeled as a nonstationary random walk process that
resultsonbothsyntheticandmarketdatashowthatourproposed
1 can be hard to predict efficiently. However, companies in
mean-reverting portfoliodesign methodscan generate consistent
profits and outperform the traditional design methods and the the same financial sector or industry usually share similar
]
M benchmark methods in the literature. fundamentalcharacteristics, then their stock prices may move
in company with each other under the same trend, based
Index Terms—Portfolio optimization, mean-reversion, cointe-
P
gration, pairs trading, statistical arbitrage, algorithmic trading, on which cointegration relations can be established. Two
.
n quantitative trading. examples are the stock prices of the two American famous
i consumerstaple companiesCoca-ColaandPepsiCo andthose
f
q- I. INTRODUCTION of the two energy companies Ensco and Noble Corporation.
Some examples for other financial assets, to name a few, are
[ PAIRS trading [2]–[6] is a well-known trading strategy
thefuturecontractpricesofE-miniS&P500andE-miniDow,
that was pioneered by scientists Gerry Bamberger and
1
the ETF prices of SPDR S&P 500 and SPDR DJIA, the US
v DavidShaw,andthequantitativetradinggroupled byNunzio
dollar foreign exchange rates for different countries, and the
6 TartagliaatMorganStanleyinthemid1980s.Asindicatedby
1 thename,itisaninvestmentstrategythatfocusesonapairof swap rates for US interest rates of different maturities.
0 Mean-reversion is a classic indicator of predictability in
assets at the same time. Investors or arbitrageurs embracing
5 financial markets and used to obtain arbitrage opportunities.
thisstrategydonotneedtoforecasttheabsolutepriceofevery
0
Assets in a cointegration relation can be used to form a
. single asset in one pair, which by natureis hardto assess, but
1 portfolio or basket and traded based upon their stationary
only the relative price of this pair. As a contrarianinvestment
0 mean-reversionproperty.We call such a designedportfolioor
7 strategy, in order to arbitrage from the market, investors need
basket of underlying assets a mean-revertingportfolio (MRP)
1 to buy the under-priced asset and short-sell the over-priced
or sometimes a long-short portfolio which is also called a
: one. Then profits are locked in after trading positions are
v “spread”. An asset whose price shows naturally stationarity
i unwound when the relative mispricing corrects itself in the
X is a spread as well. The profits of statistical arbitrage come
future.
directly from trading on the mean-reversion of the spread
r More generally, pairs trading with only two trading assets
a aroundthelong-runequilibrium.MRPsinpracticeareusually
falls into the umbrella of statistical arbitrage [7], [8], where
constructed using heuristic or statistical methods. Traditional
the underlying trading basket in general consists of three or
statisticalcointegrationestimationmethodsareEngle-Granger
more assets. Since profits from such arbitrage strategies do
ordinary least squares (OLS) method [12] and Johansen
not depend on the movements and conditions of the general
model-based method [14]. In practice, inherent correlations
financial markets, statistical arbitrage is referred to as a kind
mayexistamongdifferentMRPs.However,whenhavingmul-
of market neutral strategies [9], [10]. Nowadays, statistical
tiple MRPs, they are commonly traded separately with their
arbitrage is widely used by institutionalinvestors, hedge fund
possible connections neglected. So a natural and interesting
companies, and many individual investors in the financial
question is whether we can design an optimized MRP based
markets.
on the underlying spreads that could outperform every single
In [11], [12], the authors first came up with the concept
one. In this paper, this issue is clearly addressed.
of cointegration to describe the linear stationary and hence
Designing one MRP by choosing proportions of various
mean-reverting relationship of underlying nonstationary time
assets in generalis a portfoliooptimizationor asset allocation
The authors are with the Department of Electronic and Computer Engi- problem [22]. Portfolio optimization today is considered to
neering, The Hong Kong University of Science and Technology (HKUST), be an important part in portfolio management as well as in
ClearWaterBay,Kowloon,HongKong(e-mail:[email protected];
algorithmic trading. The seminal paper [23] by Markowitz in
[email protected]).
Partoftheresultsinthispaperwerepreliminary presentedat[1]. 1952 laid on the foundations of what is now popularly re-
Submittedpaper 2
ferred to as mean-varianceportfolio optimization and modern solving methods for GEVP and GTRS. The MM framework
portfolio theory. Given a collection of financial assets, the and MM-based solving algorithms are elaborated in Section
traditional mean-variance portfolio design problem is aimed V. The performanceof the proposedalgorithmsare evaluated
at finding a tradeoff between the expected return and the risk numericallyinSectionVIand,finally,theconcludingremarks
measuredbythevariance.Differentfromtherequirementsfor are drawn in Section VII.
mean-variance portfolio design, in order to design a mean- Notation:Boldfaceuppercaselettersdenotematrices,bold-
reverting portfolio, there are two main factors to consider: face lower case letters denote column vectors, and italics
i) the designed MRP should exhibit a strong mean-reversion denotescalars.Thenotation1andIdenoteanall-onecolumn
indicating that it should have frequent mean-crossing points vectorandan identitymatrixwith propersize, respectively.R
and hence bring in trading opportunities, and ii) the designed denotesthe real field with R+ denotingpositive real numbers
MRP should exhibit sufficient but controlled variance so and RN denoting the N-dimensional real vector space. N
that each trade can provide enough profit while controlling denotesthe naturalfield. Z denotesthe integercircle with Z+
the probability that the believed mean-reversion equilibrium denoting positive integer numbers. SK denotes the K K-
breaks down could be reduced. dimensional symmetric matrices. The superscripts ()T×and
In [24], the author first proposed to design an MRP by op- ()−1 denotethematrixtransposeandinverseoperator,·respec-
·
timizingacriterioncharacterizingthemean-reversionstrength tively.Duetothecommutationoftheinverseandthetranspose
which is a model-free method. Later, authors in [25] realized for nonsingular matrices, the superscript ()−T denotes the
·
that solving the problem in [24] could result in a portfolio matrix inverse and transpose operator. x denotes the (ith,
i,j
with very low variance, then the variance control was taken jth) element of matrix X and x denotes the ith element of
i
into consideration and also new criteria to characterize the vector x. Tr() denotes the trace of a matrix. vec() denotes
· ·
mean-reversion property were proposed for the MRP design the vectorization of a matrix, i.e., vec(X) is a column vector
problem. In [24], [25], semidefinite programming (SDP) re- consisting of all the columns of X stacked. denotes the
⊗
laxation methods were used to solve the nonconvex problem Kronecker product of two matrices.
formulations; however, these methods are very computation-
allycostlyingeneral.Besidesthat,thedesignmethodsin[24],
II. MEAN-REVERTING PORTFOLIO (MRP)
[25]wereallcarriedoutbyimposinganℓ -normconstrainton
2
the portfolio weights. This constraint brings mathematically Forafinancialasset,e.g.,acommonstock,afuturecontract,
convenience to the optimization problem, but its practical an ETF, or a portfolio of them, its price at time index or
significance in financial applications is dubious since the ℓ2- holding period t Z+ is denoted by pt R+, and the cor-
norm is not meaningful in a financial context. In this paper, respondinglogari∈thmic price or log-price∈y R is computed
t
∈
we proposeto use investmentbudgetconstraintsin the design as y =log(p ), where log() is the natural logarithm.
t t
·
problems. If we consider a collection of M assets in a bas-
The contributions of this paper can be summarized as ket, their log-prices can be accordingly denoted by y =
t
follows. First, a general problem formulation for MRP de- [y ,y ,...,y ]T RM. Based on this basket, an MRP
1,t 2,t M,t
sign problem is proposed based on which several specific is accordingly defined∈by the portfolio weight or hedge ratio
problem formulations are elaborated by considering different w = [w ,w ,...,w ]T RM and its (log-price)
s s,1 s,2 s,M
mean-reversion criteria. Second, Two classes of commonly spreads isdefinedass =wTy∈= M w y .Vector
t t s t m=1 s,m m,t
used investment budget constraints on portfolio weights are w indicates the market value proportion invested on the
s
considered, namely, dollar neutral constraint and net budget underlyingasset1.Form=1,2,...,MP,w >0,w <0,
s,m s,m
constraint. Third, efficient algorithms are proposed for the and w =0 mean a long position (i.e., the asset is bought),
s,m
proposed problem formulations, it is shown that some prob- a short position (i.e., the asset is short-sold, or, more plainly,
lems after reformulations can be tackled readily by solving borrowed and sold), and no position, respectively.
the well-known generalized eigenvalue problem (GEVP) and In Figure 1, the spread of a designed MRP together with
the generalized trust region subproblem (GTRS). The other the log-prices of the two underlying assets is given. It is
problems can be easily solved based on the majorization- worth notingthat an MRP can be interpretedas a synthesized
minimization (MM) framework by solving a sequence of stationary asset. The spread accordingly means its log-price
GEVPs and GTRSs, which are named iteratively reweighted which could be easier to predict and to make profits from in
generalized eigenvalue problem (IRGEVP) and iteratively comparisonwiththeunderlyingcomponentassetsinthisMRP.
reweighted generalized trust region subproblem (IRGTRS),
Suppose there exist N MRPs with their spreads denoted
respectively. An extension for IRGEVP with closed-form so- by s = [s ,s ,...,s ]T RN. Different spreads
t 1,t 2,t N,t
lutionineveryiterationnamedEIRGEVP(extendedIRGEVP) ∈
may possess different mean-reversion and variance properties
is also proposed.
in nature. Our objective is to design an MRP to combine
The remaining sections of this paper are organized as
such spreads into an improved overall spread with better
follows. In Section II, we introduce the design of mean-
properties. In particular, we denote the portfolio by w =
reverting portfolios. In Section III, we give out some mean- [w ,w ,...,w ]T RN, where w denotes the market value
1 2 N
reversioncriteriaforanMRP,andageneralformulationforthe ∈
MRPdesignproblemisproposedtogetherwithtwocommonly
1Ifthespreadisdesignedbasedonassetpricept insteadofthelog-price,
used investment budget constraint. Section IV develops the ws indicates theassetamountproportionmeasuredinshares.
Submittedpaper 3
σ2
ces 46 prez = σz2ˆ, (4)
g-pri 2 y1 z
Lo 0 y2 where σ2 = E z2 and σ2 = E zˆ2 . If we define
-2 z t zˆ t−1
σ2 =E ǫ2 , then from (4), we can have σ2 =σ2+σ2 in the
ǫ t (cid:2) (cid:3) (cid:2) z (cid:3) zˆ ǫ
denominator.Whenpre issmall,thevarianceofǫ dominates
(cid:2) (cid:3) z t
1 that of zˆ , and z behaves like a white noise; when pre is
t−1 t z
d y1-0.8y2 large,the varianceofzˆ dominatesthatofǫ , andz can be
ea 0 t−1 t t
Spr well predicted by zˆt−1. The predictability statistics is usually
-1 used to measure how close a random process is to a white
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 noise.
Time index Under this criterion, in order to design a spread z as close
t
aspossibletoawhitenoiseprocess,weneedtominimizepre .
Fig. 1. An illustrative example of log-prices of two assets and a designed z
For a spread z = wTs , we assume the spread s follows a
spread. t t t
centered vector autoregressive model of order 1 (VAR(1)) as
follows:
on the underlyingspread. The resulting overallspread is then
s =As +e , (5)
given by t t−1 t
N where A is the autoregressive coefficient and et denotes a
z =wTs = w s . (1) white noise independent from s . We can get A from the
t t n n,t t−1
nX=1 autocorrelationmatricesasA=M1M−01. Multiplying(5) by
w and further defining zˆ = wTAs and ǫ = wTe ,
III. MRPDESIGN PROBLEM FORMULATIONS t−1 t−1 t t
we can get σ2 = wTM w, and σ2 = wTTw, where T =
z 0 zˆ
Traditional portfolio design problems are based on the No- AM AT = M M−1MT. High order models VAR(p), with
0 1 0 1
bel prize-winningMarkowitz portfolio theory [23], [26]–[28].
p>1, can be trivially reformulated into VAR(1) with proper
Theyaimatfindingadesiredtrade-offbetweenreturnandrisk,
reparametrization [33]. Then the estimator of predictability
the latter being measuredtraditionallyby the varianceor, in a
statistics for z is computed as
t
moresophisticatedway,byvalue-at-riskandconditionalvalue-
at-risk. The recently proposed risk-parity portfolios [29]–[31] wTTw
pre (w)= . (6)
can also be categorized into this design problem. z wTM w
0
For the mean-reverting portfolio, we can formulate the
design problem by optimizing some mean-reversion criterion 2) Portmanteau Statistics por (p,w): The portmanteau
z
quantifyingthemean-reversionstrengthofthespreadz ,while statistics of order p [34] for a centered univariate stationary
t
controlling its variance and imposing an investment budget process zt is defined as
constraint.
p
por (p)=T ρ2, (7)
z i
A. Mean-Reversion Criteria
i=1
X
Inthissection,weintroduceseveralmean-reversioncriteria
where ρ is the ith order autocorrelation (autocorrelation for
that can characterize the mean-reversion strength of the de- lagi)ofiz definedasρ = E[ztzt+i].Theportmanteaustatistics
signed spread z . We start by defining the ith order (lag-i) t i E[z2]
t is used to test whether a randotm process is close to a white
autocovariance matrix for a stochastic process s as
t noise. From the above definition, we have por (p) 0 and
z ≥
theminimumofpor (p)is attainedbyawhite noiseprocess,
M = Cov(s ,s ) z
i t t+i i.e., the portmanteau statistics for a white noise process is 0
(2)
= E (st E[st])(st+i E[st+i])T , for any p.
− −
where i N. Spechifically, when i = 0, M standis for the Under this criterion, in order to get a spread zt close to
0
∈ a white noise process, we need to minimize por (p) for a
(positive definite) covariance matrix of y . z
t prespecified order p. For an MRP z = wTs , the ith order
Since for any random process s , we can get its centered t t
t
autocorrelation is given by
counterpart ˜s as ˜s = s E[s ], in the following, we will
t t t t
−
use s to denote its centered form ˜s .
1) tPredictability Statistics pre (wt ): Consider a centered ρ = E[ztzt+i] = wTE stsTt+i w = wTMiw. (8)
z i E[z2] wTE s sT w wTM w
univariatestationaryautoregressiveprocesswrittenasfollows: t (cid:2) t t (cid:3) 0
z =zˆ +ǫ , (3) Then we can get the expression(cid:2)for p(cid:3)or (p,w) as
t t−1 t z
where zˆt−1 is the prediction of zt based on the information p wTM w 2
up to time t 1, and ǫ denotes a white noise independent por (p,w)=T i . (9)
from zˆt−1. Th−e predictabtility statistics [32] is defined as z Xi=1(cid:18)wTM0w(cid:19)
Submittedpaper 4
3) Crossing Statistics cro (w) and Penalized Crossing investment budget constraint by , the general MRP design
z
W
Statisticspcro (p,w): Crossingstatistics(zero-crossingrate) problem can be formulated as follows:
z
of a centered univariate stationary process z is defined as
t
T mininize ξwTHw+ζ wTM w 2+η p wTM w 2
1 w 1 i=2 i
cro = E 1 , (10)
z T 1 " {ztzt−1≤0}# subjectto wTM0w =ν(cid:0) (cid:1) P (cid:0) (cid:1)
− Xt=2 w ,
where 1 (z ) is the indicator function defined as 1 (z ) = ∈W (15)
E t E t
1, if zt ∈E , and the event here is E = z z 0 . where the objective function is denoted by fz(w) in the
t t−1 following.Theproblemin(15)isanonconvexproblemdueto
(0, if zt / E { ≤ }
∈ the nonconvexity of the objective function and the constraint
Crossing statistics is used to test the probability that a sta-
set.
tionary process crosses its mean per unit of time and it is
easyto noticethatcro [0,1].Accordingto [35],[36], fora
z
∈
centeredstationaryGaussianprocessz ,wehavethefollowing C. Investment Budget Constraint
t
W
relationship:
In portfolio optimization, constraints are usually imposed
to represent the specific investment guidelines. In this paper,
1
croz = arccos(ρ1). (11) we use to denote it and we focus on two types of budget
π W
constraints:dollarneutralconstraintandnetbudgetconstraint.
Remark 1. As a special case, if z is a centered stationary
t Dollar neutral constraint, denoted by , means the net
0
AR(1), W
investmentornetportfoliopositioniszero;inotherwords,all
z =φz +ǫ , (12)
t t−1 t the long positions are financed by the short positions, com-
monlytermedself-financing.2Itisrepresentedmathematically
where φ <1 and ǫ is a Gaussian white noise, then φ=ρ
t 1
and acc|or|dinglythe crossing statistics is cro = 1 arccos(φ). by
z π = 1Tw =0 . (16)
Usingthiscriterion,inordertogetaspreadz havingmany W0
t
zero-crossings, instead of directly maximizing cro , we can Net budget constraint, d(cid:8)enoted by(cid:9) , means the net
z 1
W
minimizeρ .Foraspreadz =wTs ,wecantrytominimize investmentornetportfoliopositionisnonzeroandequaltothe
1 t t
thefirstorderautocorrelationofz givenin(8).In[25],besides current budget which is normalized to one.3 It is represented
t
minimizing the first order autocorrelation, it is also proposed mathematically by
to ensure the absolute autocorrelations of high orders ρ s
| i| = 1Tw =1 . (17)
(i=2,...,p) are small at the same time which can result in W1
good performance. In this paper, we also adopt this criterion It is worth noting that, fo(cid:8)r two trad(cid:9)ing spreads defined by
and call it penalized crossing statistics of order p defined by wTy and wTy , they are naturally the same, because in
t t
−
statistical arbitrage the actual investmentnot only dependson
wTM w p wTM w 2 w,whichdefinesaspread,butalsoonwhetheralongorshort
pcro (p,w)= 1 +η i , (13) position is taken on this spread in the trading.
z wTM w wTM w
0 i=2(cid:18) 0 (cid:19)
X
where η is a positive prespecified penalization factor. IV. PROBLEM SOLVING ALGORITHMS VIA GEVPAND
GTRSALGORITHMS
B. General MRP Design Problem Formulation In this section, solving methods for the MRP design prob-
lem formulationsusingpre (w) and cro (w) (i.e., (15) with
TheMRP designproblemisformulatedastheoptimization z z
ζ =η =0) are introduced.
of a mean-reversion criterion denoted in general as F (w),
z
whichcanbetakentobeanyofthecriteriamentionedbefore.
This unified criterion can be written into a compact form as A. GEVP - Solving Algorithm for MRP Design Using
pre (w) and cro (w) with w
Fz(w)=ξwwTTMH0ww +ζ wwTTMM10ww 2+η pi=2 wwTTMM0iww 2, Fzor notationalzsimplicity, w∈e Wde0note the matrices T in
(cid:16) (cid:17) P (cid:16) (1(cid:17)4) prez(w) and M1 in croz(w) by matrix H in general and
which particularizes to i) pre (w), when ξ =1, H=T, and recast the problem as follows:
z
ζ = η = 0; ii) por (p,w), when ξ = 0, and ζ = η = 1; iii)
z
cro (w), when ξ = 1, H = M , and ζ = η = 0; and iv) minimize wTHw
z 1 w
pcroz(p,w), when ξ = 1, H = M1, ζ = 0, and η > 0. The subjectto wTM0w=ν (18)
matrices Mis in (14) are assumed symmetric without loss of 1Tw =0,
generality since they can be symmetrized.
The variance of the spread should also be controlled to 2Dollar neutral constraint generally cannot be satisfied by the traditional
a certain level which can be represented as Var wTs = designmethods,like methodsin[12]and[14],andthemethodsin[25].
t 3The net portfolio position can be positive or negative under net budget
wTM w = ν. Due to this variance constraint, the denom-
0 (cid:2) (cid:3) constraint. Since the problem formulation in (15) is invariant to the sign of
inators of Fz(w) can be removed. Denoting the portfolio w,onlythecasethatbudgetisnormalized topositive 1isconsidered.
Submittedpaper 5
whereν isapositiveconstant.Theaboveproblemisequivalent withusingtailoredalgorithms.Here,wewillapplythesteepest
tothefollowingnonconvexquadraticallyconstrainedquadratic descent algorithm [41] to solve it. The procedure to solve
programming (QCQP) [37] formulation: problem (18) is summarized in Algorithm 1.
minimize wTHw Algorithm 1 GEVP - Algorithm for MRP design problems
w
subjectto wTM0w=ν (19) using prez(w) and croz(w) with w∈W0.
wTEw=0, Require: N, N , and ν >0.
0
where E = 11T. By using the matrix lifting technique, i.e., 1: Set k =0, and choose x(k) ∈ x:xTN0x=ν ;
definingW=wwT, the aboveproblemcanbe solvedby the 2: repeat (cid:8) (cid:9)
following convex SDP relaxation problem: 3: Compute R x(k) =x(k)TNx(k)/x(k)TN0x(k);
4: Compute d((cid:0)k) =N(cid:1) x(k)−R x(k) N0x(k);
minimize Tr(HW) 5: x = x(k) + τd(k) with τ chosen to minimize
W R x(k)+τd(k) ; (cid:0) (cid:1)
subjectto Tr(M W)=ν
0 (20)
Tr(EW)=0 6: x(k(cid:0)+1) =√νx/(cid:1) xTN0x;
W 0. 7: k =k+1; p
(cid:23) 8: until convergence
Thefollowingtheoremgivesausefulrelationshipbetweenthe
number of variables and the number of equality constraints.
Theorem 2 ( [38, Theorem 3.2]). Given a separable SDP as
B. GTRS-SolvingAlgorithmforMRPDesignUsingpre (w)
follows: z
and cro (w) with w
z 1
mX1in,.i.m.,XizLe Ll=1Tr(AlXl) Asbefore,forgener∈aliWty,wedenotematricesTinprez(w)
subjectto PLl=1Tr(BmlXl)=bm, m=1,...,M aasndM1 incroz(w)asH.Thentheproblemscanberewritten
X 0, l=1,...,L.
l
P (cid:23) (21)
minimize wTHw
SupposethattheseparableSDParestrictlyfeasible.Then,the w
problem has always an optimal solution (X⋆1,...,X⋆L) such subjectto wTM0w=ν (23)
that 1Tw =1,
where ν is a positive constant. As before, rewriting the
L
[rank(X⋆)]2 M. constraint 1Tw = 1 as wTEw = 1 (since the problem is
l ≤ invariant with respect to a sign change in w) and using the
l=1
X
matrixlifting technique,the problemin (23) can be solvedby
Observe that if there is only one variable X, that is to say,
the following convex SDP problem:
L=1,wecangetrank(X⋆) √M.Further,ifthenumberof
≤
constraintsM 3,arank-1solutioncanalwaysbeattainable.
≤ minimize Tr(HW)
W
Lemma 3. The nonconvex problem in (18) or (19) has no subjectto Tr(M W)=ν
0 (24)
duality gap. Tr(EW)=1
W 0.
Proof: This lemma directly follows from Theorem 2 and
(cid:23)
the equivalence of problems (18) and (19). Like before, the nonconvex problem in (23) has no duality
In other words, by solving the convex SDP in (20), there gap. Besides the above SDP method, here we introduce
alwaysexistsarank-1solutionforWwhichisthesolutionfor an efficient solving approach by reformulating (23) into a
the originalproblem (18), however,in practice, to find such a generalizedtrustregionsubproblem(GTRS)[42].Considering
solution,rankreductionmethods[39]shouldbeappliedwhich w = w +Fx where w is any vector satisfying 1Tw =1
0 0 0
could be computationally expensive. and F is the kernel of 1T satisfying 1TF = 0 and a semi-
As an alternative to the SDP procedure mentioned above, unitarymatrixsatisfyingFTF=I.LetusdefineN=FTHF,
we find the problem in (18) can be efficiently solved as a p=FTHw ,b=wTHw ,N =FTM Fwhichispositive
0 0 0 0 0
generalized eigenvalue problem (GEVP) [40] by reformula- definite, p = FTM w , and b = wTM w , then the
0 0 0 0 0 0 0
tion. Considering w = Fx, where F is the kernel that spans problem in (23) is equivalent to the following problem:
the null space of 1T, i.e., 1TF = 0, and also required to be
semi-unitary, i.e., FTF = I, we can define N = FTHF and minimize xTNx+2pTx+b
x (25)
N0 = FTM0F which is positive definite, then the problem subjectto xTN0x+2pT0x+b0 =ν,
(18) is equivalent to the following problem:
which is a nonconvex QCQP and QCQPs of this type are
minimize xTNx speciallynamedGTRSs.Suchproblemsareusuallynonconvex
x (22)
subjectto xTN x=ν, but possess necessary and sufficient optimality conditions
0
based on which efficient solving methods can be derived. We
which is still a nonconvex QCQP. However, this problem
first introduce the following useful theorem.
becomes the classical GEVP problem and can be easily dealt
Submittedpaper 6
Theorem 4 ( [42, Theorem 3.2]). Consider the following Algorithm 2 GTRS - Algorithm for MRP design problems
QCQP: using prez(w) and croz(w) with w∈W1.
Require: N, N , p, p , b , λ (N,N ), and ν >0.
minimize q(x),xTAx+2aTx+a 0 0 0 min 0
x (26) 1: Set k =0, and choose ξ(k) ( (N,N0), );
subjectto c(x),xTBx+2bTx+b=0. 2: repeat ∈ − ∞
3: Compute φ ξ(k) according to (30);
Assume that the constraint set c(x) is nonempty and that
2c(x) = 2B = 0. A vector x⋆ is a global minimizer of 4: Update ξ(k+(cid:0)1) ac(cid:1)cording to the value of φ ξ(k) by a
∇ 6 line search algorithm;
the problem (26) together with a multiplier ξ⋆ if and only if (cid:0) (cid:1)
5: k =k+1;
the following conditions are satisfied:
6: until convergence
q(x⋆)+ξ⋆ c(x⋆)=0 7: Compute x according to (28).
∇ ∇
c(x⋆)=0

 2q(x⋆)+ξ⋆ 2c(x⋆) 0,
∇ ∇ (cid:23) V. PROBLEM SOLVING ALGORITHMS VIA
and the interval set defined by MAJORIZATION-MINIMIZATION METHOD
In this section, we first discuss the majorization-
= ξ A+ξB 0
I { | ≻ } minimization or minorization-maximization (MM) method
is not empty. briefly, and then solving algorithms for the MRP design
problem formulations using por (p,w) (i.e., (15) with ξ =0
According to Theorem 4, the optimality conditions for the z
and ζ = η = 1) and pcro (p,w) (i.e., (15) with ξ = 1,
primal and dual variables (x⋆,ξ⋆) of problem (25) are given z
H = M , ζ = 0 and η > 0) are derived based on the MM
as follows: 1
frameworkandtheGEVPandGTRSalgorithmsmentionedin
the previous section.
(N+ξ⋆N )x⋆+p+ξ⋆p =0
0 0
x⋆TN x⋆+2pTx⋆+b ν =0 (27)
 0 0 0− A. The MM Method
N+ξ⋆N0 0.
(cid:23) The MM method [43]–[45] refers to the majorization-
We assumeN+ξN0 04, then we can see that the optimal minimization or minorization-maximization which is a gen-
≻
solution is given by eralizationof the well-knownexpectation-maximization(EM)
algorithm.TheideabehindMMisthatinsteadofdealingwith
x(ξ)= (N+ξN )−1(p+ξp ), (28)
− 0 0 the original optimization problem which could be difficult to
and ξ is the unique solution of the following equation with tackle directly, it solves a series of simple surrogate subprob-
definition on the interval : lems.
I Suppose the optimization problem is
φ(ξ)=0, ξ , (29)
∈I minimize f(x)
x (32)
where the function φ(ξ) is defined by subjectto x ,
∈X
φ(ξ)=x(ξ)T N0x(ξ)+2pT0x(ξ)+b0−ν, (30) where the constraint set X ⊆ RN. In general, there is no
assumption aboutthe convexityand differentiabilityon f(x).
and the interval consists of all ξ for which N+ξN 0,
I 0 ≻ The MM method aims to solve this problem by optimizing a
which implies that
sequence of surrogate functions that majorize the objective
=( λ (N,N ), ), (31) function f(x) over the set . More specifically, starting
I − min 0 ∞ from an initial feasible point Xx(0), the algorithm produces a
where λmin(N,N0) is the minimum generalized eigenvalue sequence x(k) according to the following update rule:
of matrix pair (N,N ).
0
(cid:8) x(cid:9)(k+1) argminu x,x(k) , (33)
Theorem5([42,Theorem5.2]). Assume isnotempty,then ∈ x∈X
I
the function φ(ξ) is strictly decreasing on unless x(ξ) is (cid:16) (cid:17)
I where x(k) is the point generated by the update rule at the
constant on .
I kth iteration and the surrogate function u x,x(k) is the
In practice, the case x(ξ) is constant on cannot hap- corresponding majorizing function of f(x) at point x(k). A
I (cid:0) (cid:1)
pen. So from Theorem (5), we know when φ(ξ) is strictly surrogate function is called a majorizing function of f(x) at
decreasing on , then a simple line search algorithm like point x(k) if it satisfies the following properties:
I
bisection algorithm can be used to find the optimal ξ over .
I u x,x(k) f(x), x ,
The algorithm for problem (23) is summarized in Algorithm ≥ ∀ ∈X (34)
u x(k),x(k) =f x(k) .
2. (cid:0) (cid:1)
Thatis to say,(cid:0)the surrog(cid:1)atefun(cid:0)ction(cid:1)u x,x(k) shouldbe an
4The limiting case N+ξN0 being singular (i.e., ξ =−λmin(N,N0)) upperboundoftheoriginalfunctionf(x)over andcoincide
can be treated separately. The assumption here is reasonable since the case (cid:0) X(cid:1)
whenξ=−λmin(N,N0)isveryraretooccurtheoreticallyandpractically. withf(x)atpointx(k).Althoughthedefinitionofu x,x(k)
(cid:0) (cid:1)
Submittedpaper 7
gives us a great deal of flexibility for choosing it, in practice, Specifically, we can have the expressions for portmanteau
the surrogatefunctionu x,x(k) must be properlychosen so statistics por (p,w) (i.e., ζ = 1 and η = 1) and penalized
z
as to make the iterative update in (33) easy to compute while crossing statistics pcro (p,w) (i.e., ζ = 0 and η > 0) as
(cid:0) (cid:1) z
maintaining a fast convergenceover the iterations. follows:
The MM method iteratively runs until some convergence M¯ = p vec M¯ vec M¯ T
criterionismet.UnderthisMMmethod,theobjectivefunction porz i=1 i i
= (L L)−1 p vec(M )
value is decreased monotonically in every iteration, i.e.,  P⊗ (cid:0) i=(cid:1)1 (cid:0) (cid:1)i ·
f x(k+1) ≤u x(k+1),x(k) ≤u x(k),x(k) =f x((k3)5).  M¯pcroz = vηec(Mpi=2i)vTe(cPLM⊗¯iL)v−eTc M¯i T (40)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) = η(L L)−1 p vec(M )
The first inequality and the third equality follow from the P⊗ (cid:0) i=(cid:1)2 (cid:0) (cid:1)i ·
firersspteacntidvesleycoannddpthroepseerctoiensdoifnethqeuamliatyjofroizlilnowgsfufnrocmtion(3i3n).(34) Now,theobjectivevfuencc(tMioni)iTn((L3P8⊗)iLsa)−qTua.draticfunctionof
W¯ ,however,thisproblemisstillhardtosolveduetotherank-
B. IRGEVP and IRGTRS - Solving Algorithms for MRP De- 1 constraint W¯ =w¯w¯T. We then consider the application of
sign Using por (p,w) and pcro (p,w) the MM trick on this problem (38) based on the following
z z
simple result.
Werewritetheproblemsusingpor (p,w)andpcro (p,w)
z z
in the general formulation as follows: Lemma 6 ( [41, Lemma 1]). Let A SK and B SK such
that B A. Then for any point x∈ RK, the∈quadratic
minwimize ξwTM1w+ζ wTM1w 2 function(cid:23)xTAx is majorized by xTB0x∈+2xT0 (A−B)x+
+η pi=2 wTM(cid:0) iw 2 (cid:1) (36) xT0 (B−A)x0 at x0.
subjectto wTM0w =ν AccordingtoLemma6,givenW¯ (k) atthekthiteration,we
P (cid:0) (cid:1)
w , knowthesecondpartintheobjectivefunctionofproblem(38)
∈W
is majorized by the following majorizing function at W¯ (k):
where the specific portfolio weight constraints are implicitly
replaced by .
W
To solve the problems in (36) via majorization- u W¯ ,W¯ (k)
1
minimization, the key step is to find a majorizing function = ψ M¯ vec W¯ T vec W¯
of the objective function such that the majorized subproblem +2(cid:0)vec W¯ (k)(cid:1)T M¯ ψ M¯ I vec W¯ (41)
is easy to solve. Observe that the objective function is (cid:0) (cid:1) (cid:0) (cid:1) −(cid:0) (cid:1)
quartic in w. The following mathematical manipulations are +vec W(cid:0)¯ (k) T(cid:1) ψ(cid:0) M¯ I(cid:0)−M¯(cid:1) (cid:1)vec W(cid:0)¯ (k(cid:1)) ,
necessary. We first compute the Cholesky decomposition of whereψ M¯ (cid:0)isasca(cid:1)lar(cid:0)nu(cid:0)mbe(cid:1)rdepend(cid:1)ingo(cid:0)nM¯ a(cid:1)ndsatisfy-
M0 which is M0 = LLT, where L is a lower triangular ingψ M¯ I M¯. Sincethefirsttermvec W¯ T vec W¯ =
with positive diagonal elements. Let us define w¯ = LTw, (cid:0) (cid:1)(cid:23)
M¯i = L−1MiL−T, and W¯ = w¯w¯T. The portfolio weight w¯Tw¯(cid:0) 2 =(cid:1) ν2 and the last term only depe(cid:0)nds(cid:1)on W¯ ((cid:0)k), t(cid:1)hey
set is mapped to ¯ under the linear transformation L. are just two constants.
ThenWproblem (36) canWbe written as (cid:0) On t(cid:1)he choice of ψ M¯ , according to Lemma 6, it is
obvious to see that ψ M¯ can be easily chosen to be
λ M¯ = M¯ . In(cid:0)the(cid:1)implementation of the algorithm,
minimize ξTr M¯ 1W¯ +ζ Tr M¯ 1W¯ 2 almthaoxugh M¯ on2ly ne(cid:0)eds(cid:1)to be computed once for the
w¯,W¯ (cid:0) (cid:1) (cid:13)2 (cid:13)
subjectto W+¯η=(cid:0) wpi¯=w1¯TT(cid:1)r M¯(cid:0)iW¯ (cid:0) 2 (cid:1)(cid:1) (37) wInhvoileewalogf(cid:13)(cid:13)ortihthi(cid:13)(cid:13)(cid:13)sm, ,w(cid:13)iet iinstrsotdilulcneotthceofmolpluowtaitniognalellmymeaastyotoobtgaeitn.
w¯TwP¯ =ν(cid:0) (cid:0) (cid:1)(cid:1) more possibilities for ψ M¯ which could be relatively easy
w¯ ¯. to compute. (cid:0) (cid:1)
∈W
Lemma 7 ( [40]). For any matrix B RP×Q, the following
Since Tr M¯iW¯ = vec M¯i T vec W¯ (recall the Mis are inequalities about B hold: ∈
assumed symmetric), problem (36) can be reformulated as k k2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
B
minimize ξvec M¯ T vec W¯ +vec W¯ T M¯vec W¯ k k2
wshuebw¯rje,eW¯cintttohe oWww¯¯¯bTj∈e=w¯c(cid:0)Wt¯iw=v¯,ew¯1νf(cid:1)Tunction(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (38(cid:1)) ≤k√√√BPQPkBQkkFBBk=Bkk∞1qkB=m=Pa√x√Pi=Q=P1m√PmaPaQjx=xQj=i1=m1|1b,.,ia..j..x,.|Q,2iP=P1P,.Pi.Qj=.,=P11|mb|biajij|x|j=1,...,Q|bij|
k k∞k k1
M¯ =ζvec(cid:0)M¯1(cid:1)vec(cid:0)M¯ 1(cid:1)T +ηPpi=2vec(cid:0)M¯ i(cid:1)vec(cid:0)M¯(i3(cid:1)9T). =pr(cid:16)maxi=1,...,P PQj=1|bij|(cid:17)(cid:16)maxj=1,...,QPPi=1|bij|(cid:17).
Submittedpaper 8
According to the above relations, ψ M¯ can be chosen directly which could be difficult, we just need to iteratively
to be any number is larger than M¯ but much easier to solve a sequence of GEVPs or GTRSs. We call these MM-
2(cid:0) (cid:1)
compute. based algorithmsiterativelyreweightedGEVP (IRGEVP) and
(cid:13) (cid:13)
After ignoringthe constantsin (4(cid:13)1),(cid:13)the majorizedproblem iterativelyreweightedGTRS(IRGTRS)respectivelywhichare
of problem (38) is given by summarized in Algorithm 3.
Algorithm 3 IRGEVP and IRGTRS - Algorithms for MRP
minimize ξvec M¯ T vec W¯
1 design problems using por (p,w) and pcro (p,w).
w¯,W¯ z z
+2ve(cid:0)c W¯(cid:1)(k) T(cid:0)M¯ (cid:1) ψ M¯ I vec W¯ Require: p, Mi with i=1,...,p, and ν >0.
subjectto W¯ =w¯w¯T − 1: Set k =0, and choose initial value w(k) ;
w¯Tw¯ =(cid:0) ν (cid:1) (cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1) 2: Compute M¯ according to (39) and ψ M¯∈;W
w¯ ¯, 3: repeat
(cid:0) (cid:1)
∈W (42) 4: Compute H(k) according to (46);
which can be further written as 5: Update w(k+1) by solving the GEVP in (22) or the
GTRS in (25);
minimize ξTr M¯ W¯ +2ζTr M¯ W¯ (k) Tr M¯ W¯ 6: k =k+1;
w¯,W¯ 1 1 1 7: until convergence
+2η(cid:0) p T(cid:1)r M¯ W¯(cid:0)(k) Tr M¯(cid:1)W¯(cid:0) (cid:1)
i=2 i i
2ψ M¯ Tr W¯ (k)W¯
subjectto −W¯ =Pw¯w¯T (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) C. EIRGEVP - An Extended Algorithm for IRGEVP
w¯Tw¯ =ν
w¯ ¯. In the MM-based algorithmsmentioned above, it would be
∈W (43) much desirable if we could get a closed-formsolution for the
By changing W¯ back to w¯, problem (43) becomes subproblemsineveryiteration.Infact,forIRGEVPs,applying
the MM trick once again, a closed-form solution is attainable
minimize w¯TH¯(k)w¯
w¯ ateveryiteration.Toillustratethis,werewritethesubproblem
subjectto w¯Tw¯ =ν (44) (44) of IRGEVP again as follows:
w¯ ¯,
∈W minimize wTH(k)w
where in the objective function, H¯(k) is defined in w
subjectto wTM w=ν (48)
this way H¯(k) = ξM¯ + 2ζ w¯(k)TM¯ w¯(k) M¯ + 0
1 1 1 1Tw =0.
2η p w¯(k)TM¯ w¯(k) M¯ 2ψ M¯ w¯(k)w¯(k)T. Finally,
we cain=2undo the chiange of viar−iable w(cid:0)¯ =LTw, obta(cid:1)ining Considering the trick used to eliminate the linear constraint
P (cid:0) (cid:1) (cid:0) (cid:1) to get problem (22), we can get the following equivalent
minimize wTH(k)w formulation:
w
subjectto wTM w=ν (45)
0 minimize xTN(k)x
w , x (49)
∈W subjectto xTN x=ν,
0
where in the objective function
where F and N are defined as before; N(k) = FTH(k)F.
H(k) = ξM +2ζ w(k)TM w(k) M 0
1 1 1 Considering the Cholesky decomposition N = RRT with
+2η p w(k)TM w(k) M (46) 0
i=2(cid:0) i (cid:1) i R to be a lower triangular with positive diagonal elements,
2ψ M¯ M w(k)w(k)TM .
− P (cid:0) 0 (cid:1)0 we can have the variable transformation x¯ =RTx. Then the
More specifically, for po(cid:0)rtm(cid:1)anteau statistics por (p,w) (i.e., problem (48) becomes
z
ξ = 0, ζ = 1 and η = 1) and penalized crossing statistics
minimize x¯TN¯(k)x¯
pcroz(p,w) (i.e., ξ = 1, ζ = 0 and η > 0), we have the x¯ (50)
following expressions: subjectto x¯Tx¯ =ν,
H(k) = 2 p w(k)TM w(k) M where N¯(k) =R−1N(k)R−T.
porz i=1 i i
2ψ M¯ M w(k)w(k)TM , ApplyingLemma6again,theobjectivefunctionofproblem
 −P (cid:0) 0 (cid:1) 0 (50)ismajorizedbythefollowingmajorizingfunctionatx¯(k):
 H(pkc)roz = M1+(cid:0) 2η(cid:1) pi=2 w(k)TMiw(k) Mi
2ψ M¯ M w(k)w(k)TM . u x¯,x¯(k)
− P 0 (cid:0) 0 (cid:1) 2
Finally, in the majori(cid:0)zati(cid:1)on problems (44) and (45),(4th7e) = ψ+2(cid:0)N¯(Nk¯)(kx)¯(cid:1)Tx¯ψ N¯(k) I x¯(k) T x¯ (51)
objective functions become quadratic in the variable rather +x¯(cid:0)(k)T (cid:1)ψ−N¯(k) I N¯(k) x¯(k),
than quartic in the variable as in the original problem (36). (cid:2)(cid:0) (cid:0) −(cid:1) (cid:1) (cid:3)
Depending on the specific form of , problem (45) is either where ψ N¯(k) can be(cid:2)cho(cid:0)sen u(cid:1)sing the re(cid:3)sults fromLemma
W
the GEVP or GTRS problems discussed in the previous 7. The first and last parts are just two constants. Note that
(cid:0) (cid:1)
sections. So, in order to handle the original problem (36) although in the derivation we have applied the MM scheme
Submittedpaper 9
twice, it can be viewed as a direct majorization for the MRP design methods in Sections IV and V using both
objective of the original problem at w(k). The following synthetic data and real market data are shown accordingly.
lemma summarizes the overall majorizing function.
Lemma 8. For problem (36) with w , the majorization
0 A. Mean-Reversion Trading Design
∈W
in (41) together with (51) can be shown to be a majorization
fortheobjectivefunctionoftheoriginalproblematw(k) over In this paper, we use a simple trading strategy where
the constraint set by the following function: the trading signals, i.e., to buy, to sell, or simply to hold,
are designed based on simple event triggers. Mean-reversion
trading is carried out on the designed spread z which is
u w,w(k) t
2
tested to be unit-root stationary. A trading position (either a
= 2 H(k) ψ R−1FTH(k)FR−T M w(k) T w
(cid:0) − (cid:1) 0 long position denoted by 1 or a short position denoted by
+2ψ R−1FTH(k)FR−T ν w(k)TH(k)w(k).
(cid:2)(cid:0) (cid:0) − (cid:1) (cid:1) (cid:3) 1) denotes a state for investment and it is opened when
(52) −
where the la(cid:0)st two terms are cons(cid:1)tants. the spread zt is away from its long-run equilibrium µz by
a predefined trading threshold ∆ and closed (denoted by 0)
Proof: See Appendix A. when z crosses its equilibrium µ . (A common variation is
t z
Then, the majorized problem of (50) becomes to close the position after the spread crossed the equilibrium
by more than another threshold ∆′.) The time period from
minimize e(k)Tx¯
position opening to position closing is defined as a trading
x¯ (53)
subjectto x¯Tx¯ =ν, period.
In order to get a standard trading rule, we introduce a
wheree(k) =2 N¯(k) ψ N¯(k) I x¯(k) forthe majorization
− standardization technique by defining z score which is a
in (51). By Cauchy-Schwartz inequality, we have eTx¯ −
(cid:0) (cid:0) (cid:1) (cid:1) ≥ normalized spread as follows:
e x¯ = ν e , and the equality holds only when
−k k2k k2 − k k2
x¯ and e are aligned in the opposite direction. Considering
z µ
the constraint, we can get the optimal solution of (53) as z˜ = t− z, (54)
t
x¯ = √ν e . We call this algorithm extended IRGEVP σz
(EIRG−EVP)kewkh2ich is summarized in Algorithm 4. whereµ andσ arethemeanandthestandarddeviationofthe
z z
spread z and computed over an in-sample look-back period
t
Algorithm4EIRGEVP-AnextendedalgorithmforIRGEVP. in practice. For z˜, it follows that E[z˜]=0 and Std[z˜]=1.
t t t
Require: p, Mi with i=1,...,p, and ν >0. Then, we can define ∆ = d σz, for some value of d (e.g.,
1: Set k =0, and choose initial value w(k) ; d=1). ×
2: Compute M¯ and ψ M¯ ; ∈W In a trading stage, based on the trading position and ob-
3: repeat served (normalized) spread value at holding period t, we can
4: Compute N¯(k) an(cid:0)d ψ(cid:1) N¯(k) ; get the trading actions at the next consecutive holding period
5: Update w(k+1) with a closed-form solution; t+1. The mean-reversion trading strategy is summarized in
(cid:0) (cid:1)
6: k =k+1; Table I anda simple tradingexamplebased onthis strategyis
7: until convergence illustrated in Figure 2.
VI. NUMERICAL EXPERIMENTS σ
z
A statistical arbitrage strategy involves several steps of
which the MRP design is a central one. Here, we divide the 0.5σz
˜zt
cwohnoslteruscttriaotneg,yMiRntPodfoesuirgnse,quuneint-triaolotstteepsst,, nanamdemlye:anas-rseevtserpsoioonl Spread-0.5σ0z
trading. In the first step, we select a collection of possibly
-σz
cointegrated asset candidates to construct an asset pool, on
which we will not elaborate in this paper. In the second step, -1.5σz
based on the candidate assets from the asset pool, MRPs are
1
designed using either traditional design methods like Engle-
ns
GprroapnogseerdOmLeSthmodesthoindt[h1i2s]paanpderJ.oIhnantsheentmhierdthostdep[,13u]niotr-rothoet g positio 0
n
test procedures like Augmented Dickey-Fuller test [46] and di
a
Phillips-Perron test [47] are applied to test the stationarity or Tr
mean-reversion property of the designed MRPs. In the fourth -11 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
step, MRPs passing theunit-roottests will be tradedbasedon Trading time
a designed mean-reversion trading strategy.
Fig.2. Asimpleexampleformean-reversiontradingstrategydesign(trading
In this section, we first illustrate a mean-reversion trading threshold∆=σz).
strategy and based on that the performance of our proposed
Submittedpaper 10
TABLEI
TRADINGPOSITIONS,NORMALIZEDSPREAD,ANDTRADINGACTIONSOFAMEAN-REVERSIONTRADINGSTRATEGY
TradingPositionatt NormalizedSpreadz˜t Action(s)TakenwithinHoldingPeriodt+1 TradingPositionatt+1
+d≤z˜t Closethelongpos.&Openashortpos. -1
1 0≤z˜t<+d Closethelongpos. 0
z˜t<0 Noaction 1
+d≤z˜t Openashortpos. -1
0 −d<z˜t<+d Noaction 0
z˜t≤−d Openalongpos. 1
0<z˜t Noaction -1
-1 −d<z˜t≤0 Closetheshortpos. 0
z˜t≤−d Closetheshortpos.&Openalongpos. 1
B. Performance Metrics the portfolio P&L calculation within the trading periods, we
have the following lemma.
AfteranMRPisconstructed,we needtodefinetherelation
between the designed MRP with the underlying financial Lemma9(P&LCalculationforMean-ReversionTrading).
assets. Recall that the spread for the designed MRP is z = Within one trading period, if the price change of every asset
t
wTs , where s = [s ,s ,...,s ]T with s = wT y in an MRP is small enough, then the P&L in (55) can be
t t 1,t 2,t N,t n,t sn t
for n = 1,2,...,N. By defining s = WTy with W = approximatelycalculatedbythechangeofthelog-pricespread
t s t s
[ws1,ws2,...,wsN], we get the spread zt = wpTyt, where zt. Specifically,
wp = Wsw denotes the portfolio weight directly defined on 1) for a long position opened on the MRP, P&Lt(τ)
≈
the underlying assets. z z ; and
t t−τ
−
Based on the mean-reversion trading strategy introduced 2) for a short position opened on the MRP, P&L (τ)
t
≈
before and the MRP defined by w here, we employ the z z .
p t−τ t
−
following performance metrics in the numerical experiments.
Proof: See Appendix B.
1) Portfolio Return Measures: In the following, we first
This lemma reveals the philosophy of the MRP design and
give the return definition for one single asset, and after that,
also the mean-reversion trading by showing the connection
severaldifferentreturnmeasuresforanMRParetalkedabout.
between the log-price spread value and the portfolio return.
For onesingle asset, the returnorcumulativereturnat time
Since there is no trading conduct between two trading
t for τ holding periods is defined as r (τ)= pt−pt−τ, where
t pt−τ periods, the P&L measures (both the multi-period P&L and
τ in the parentheses denotes the period length and is usually
single-period P&L) are simply defined to be 0.
omittedwhenthelengthisone.Here,thereturnr (τ)asarate
t b) CumulativeP&L: Inorderto measurethecumulative
ofreturnisusedtomeasuretheaggregateamountofprofitsor
returnperformanceforanMRP,wedefinethecumulativeP&L
losses (in percentage) of an investment strategy on one asset
in one trading from time t to t as
over a time period τ. 1 2
a) Profit and Loss (P&L): The profit and loss (P&L) Cum. P&L(t ,t )= t2 P&L . (57)
1 2 t=t1 t
measures the amount of profits or losses (in units of dollars)
c) Return on Investment (ROPI): Since different MRPs
of an investment on the portfolio for some holding periods.
mayhavedifferentleveragepropertiesduetow ,weintroduce
Within one trading period, if a long position is opened on p
another portfolio return measure (rate of return) called return
anMRPattimet andclosedattimet ,thenthemulti-period
o c on investment (ROI).
P&L of this MRP at time t (t t t ) accumulated from
to is computed as P&Lt(τ) =o ≤wpT≤rt(τc) = wpTrt(t−to), is Wdeifithniendotnoebteratdhiengsipnegrlieo-dp,etrhioedRPO&ILatattimtime et (ttono≤rmta≤liztecd)
where τ = t t denotes the length of the holding periods,
and rt(τ) =−[r1o,t(τ),r2,t(τ),...,rM,t(τ)]T is the return bgyrotshseingvroessstminevnetsetmxpeonstudreeptlooytehdewmhaicrkheitsiknwclupdki1ng(thtahteislotnhge
vector. More generally, the cumulative P&L of this MRP at
position investmentand the short position investment)written
time t for τ (0 τ t t ) holding periods is defined as
≤ ≤ − o as
P&L (τ)=wTr (t t ) wTr (t τ t ), (55)
t p t − o − p t−τ − − o ROI =P&L / w . (58)
t t k pk1
where we define r (0) = 0. Then we have the single-period
t
Like the P&L measures, between two trading periods, ROI
P&L (e.g., daily P&L, monthly P&L) denoted by P&L at t
t
is defined to be 0.
time t (i.e., τ =1) is computed as
2) Sharpe Ratio (SR): The Sharpe ratio (SR) [48] is a
P&L =wTr (t t ) wTr (t 1 t ). (56) measure for calculating risk-adjusted return. It describes how
t p t − o − p t−1 − − o much excess return one can receive for the extra volatility
Likewise, within one trading period, if a short position is (square root of variance).
opened on this MRP, then multi-period P&L is P&L (τ) = Here,theSharperatioofROI(or,equivalently,Sharperatio
t
wTr (t τ t ) wTr (t t ) and the single-period of P&L) for a trading stage from time t to t is defined as
p t−τ − − o − p t − o 1 2
P&L is P&L =wTr (t 1 t ) wTr (t t ). About follows:
t p t−1 − − o − p t − o