OPTIMAL PORTFOLIO DESIGN FOR STATISTICAL ARBITRAGE IN FINANCE

Ziping Zhao∗, Rui Zhou∗, Zhongju Wang† and Daniel P. Palomar∗

∗Department of Electronic and Computer Engineering,The Hong Kong University of Science and Technology (HKUST), Hong Kong

†Hong Kong Applied Science and Technology Research Institute (ASTRI), Hong Kong

ABSTRACT

In this paper, the optimal mean-reverting portfolio (MRP) de-sign problem is considered, which plays an important role forthe statistical arbitrage (a.k.a. pairs trading) strategy in finan-cial markets. The target of the optimal MRP design is to con-struct a portfolio from the underlying assets that can exhibita satisfactory mean reversion property and a desirable vari-ance property. A general problem formulation is proposed byconsidering these two targets and an investment leverage con-straint. To solve this problem, a successive convex approx-imation method is used. The performance of the proposedmodel and algorithms are verified by numerical simulations.

Index Terms— Portfolio optimization, mean reversion,quantitative trading, nonconvex problem, convex approxima-tion.

1. INTRODUCTIONStatistical arbitrage [1] is a general quantitative investmentand trading strategy widely used by many parties in the fi-nancial markets, e.g., institutional investors, hedge funds, andindividual investors [2]. Since it can hedge the overall marketrisk, it is also referred to as a market neutral strategy [3]. Instatistical arbitrage, the underlying trading basket can consistof many financial assets of different kinds such as equities,options, bonds, futures, commodities, etc. In order to arbi-trage from the market, investors should buy the under-pricedassets and short-sell the over-priced ones and profits will bemade after the trading positions are unwound when the “mis-pricing” corrects itself. The statistical arbitrage can be tracedback to the famous pairs trading [4] strategy, a.k.a. spreadtrading, where only two assets are considered.

In statistical arbitrage, the trading basket is used to form a“spread” characterizing the “mis-pricing” of the assets whichis stationary, hence mean-reverting. To make arbitrage, trad-ing is carried out on the mean reversion (MR) property of thespread, i.e., to buy it when it is below some statistical equi-librium and sell it when it is above the statistical equilibrium.There are many ways to design a spread, like the distancemethod [5], factor analysis [6], and the cointegration method

This work was supported by the Hong Kong RGC 16208917 researchgrant.

[7]. In this paper, we focus on the cointegration method wherethe spread is discovered by time series analysis like the ordi-nary least squares method in [8] and the model-based methodsin [9, 10]. In practice, an asset that naturally shows stationar-ity is also a spread [11].

The spreads from the statistical estimation methods es-sentially form a “cointegration subspace”. In terms of invest-ment, a natural question is whether we can design an opti-mized portfolio from this subspace. Such a portfolio is namedmean-reverting portfolio (MRP). To design an MRP, there aretwo objectives to consider: firstly the MRP should exhibit astrong MR so that it has frequent mean-crossing points andhence brings in more trading opportunities; and secondly thedesigned MRP should exhibit sufficient variance so that eachtrade can provide enough profit. These two targets naturallyresult in a multi-objective optimization problem, i.e., to find adesirable trade-off between MR and variance.

In [12], the author first proposed to design an MRP byoptimizing an MR criterion. Later, authors in [13, 14] foundthat the method in [12] can result in an MRP with very lowvariance, then the variance control was taken into consider-ation. But all these works were carried out by using an `2-norm constraint on the portfolio weights which do not carrya physical meaning in finance. To explicitly represent thebudget allocation for different assets, the investment budgetconstraints were considered in [15, 16]. However, in somecases the methods in [15, 16] can lead to very large leverage(i.e., the dollar values employed) which makes it unaccept-able to use for real investment. Besides that, when the vari-ance is changed, although the investment leverage can changeaccordingly, the MR property of the portfolio is insensitivewhich makes it really hard to find a desirable trade-off be-tween the MR and the variance properties in practice.

In this paper, a new optimal MRP design method is pro-posed that takes two design objectives and an explicit leverageconstraint into consideration. The objective in this methodcan suffice to find a desirable trade-off between the MR andthe variance for an MRP. Different MR criteria are consideredand the portfolio constraint takes two cases into consideration.The design problem finally becomes a nonconvex constrainedproblem. A general algorithm based on the successive con-vex approximation method (SCA) is proposed. An efficient

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acceleration scheme is further discussed. Numerical simula-tions are carried out to address the efficiency of the proposedproblem model and the solving algorithms.

2. MEAN-REVERTING PORTFOLIO (MRP) DESIGNFor a financial asset, e.g., a stock, its price at time t is de-noted by pt, and its log-price is given by yt = log (pt), wherelog (·) is the natural logarithm. For M assets with log-pricesyt = [y1,t, y2,t, . . . , yM,t]

T , one (log-price) spread can be de-signed by the weights β = [β1, β2, . . . , βM ]

T (say, from thecointegration model) and given by st = βTyt. Suppose thereexists a cointegration subspace with N (N < M ) cointegra-tion relations, i.e., B = [β1;β2; . . . ,βN ], then we can have

st = BTyt, (1)

where st denote N spreads. Specifically, if the log-prices arestationary in nature, we get st = yt with B = I (N = M).

The objective of mean-reverting portfolio (MRP) designis to construct a portfolio of the underlying spreads to attaindesirable trading properties. An MRP is defined by its portfo-lio weights w = [w1, w2, . . . , wN ]

T , with its resulting spreadgiven by zt = wT st =

∑Nn=1 wnsn,t. Due to (1), we can get

zt = wTp yt =

∑Mm=1 wp,mym,t, (2)

where wp = Bw are the MRP weights indicating the marketvalue on different assets. For m = 1, 2, . . . ,M , wp,m > 0,wp,m < 0, and wp,m = 0 mean a long position (i.e., it isbought), a short position (i.e., it is short-sold or, more plainly,borrowed and sold), and no position on the asset, respectively.

Considering the two design objectives, i.e., MR and vari-ance, we formulate the optimal MRP design problem as

minimizew

F (w) , U (w) + µV (w)

subject to W = {w | ‖Bw‖1 ≤ B} .(3)

The MR criterion term U (w) is jointly represented asU (w)

= ξ wTHwwTM0w

+ ζ(

wTM1wwTM0w

)2+ η

∑pi=2

(wTMiwwTM0w

)2,

which particularizes to the predictability statistics pre (w)with ξ = 1, H = MT

1 M−10 M1, and ζ = η = 0; the portman-

teau statistics por (p,w) with ξ = 0, and ζ = η = 1;the crossing statistics pre (w) with ξ = 1, H = M1,and ζ = η = 0; and the penalized crossing statisticspcro (p,w) with ξ = 1, H = M1, ζ = 0, and η > 0,where Mi = Cov (st, st+i) for i = 1, . . . , p [13, 16]. Thevariance term V (w) is represented by

V (w) = 1wTM0w

.

And µ ≥ 0 defines the trade-off between the MR and vari-ance. Specially, when µ = 0, the designed MRP has thebest MR property; and likewise when µ → ∞, the prob-lem leads to the MRP with best variance. In the constraintsetW , B means the total leverage deployed on all the assetsin an investment. The problem in (3) is a nonconvex con-strained problem with a nonconvex smooth objective and aconvex nonsmooth constraint.

3. PROBLEM SOLVING VIA THE SCA METHOD

3.1. The Successive Convex Approximation Method

The successive convex approximation (SCA) method [17] is ageneral optimization method especially for nonconvex prob-lems. In this paper, a variant of SCA in [18] is used, whichsolves the original problem by solving a sequence of stronglyconvex problems and can also preserve feasibility of the iter-ates. Specifically, a problem is given as follows:

minimizex

f (x) subject to x ∈ X , (4)

where X ⊆ RN and no assumption is on the convexity andsmoothness of f (x) and X . Instead of tackling (4) directly,starting from an initial point x(0), the SCA method solvesa series of subproblems with surrogate functions f

(x;x(k)

)approximating the original objective f (x) and a sequence{x(k)

}is generated by the following rules:{

x(k+1) = arg minx∈X

f(x;x(k)

)x(k+1) = x(k) + γ(k)

(x(k+1) − x(k)

).

(5)

The first step is to generate the descent direction (i.e.,x(k+1) − x(k)) by solving a best-response problem, andthe second step is the variable update with step-size γ(k). Forf(x;x(k)

), the following conditions are needed:

A1) given x(k), f(x;x(k)

)is c-strongly convex on X for

some c > 0, i.e.,∇2xf(x;x(k)

)� cI;

A2)∇xf(x(k);x(k)

)= ∇xf

(x(k)

)for all x(k) ∈ X

A3)∇xf (x;x) is continuous for all x ∈ X .It is easy to see that the key point in SCA is to find a good

approximation f(x;x(k)

)and to choose a proper step-size

γ(k) for a fast convergence.

3.2. Optimal MRP Design Based on The SCA Method

Applying the SCA method to solve problem (3), we can firsthave the convex approximation function F (w;w(k)) given by

F(w;w(k)

)= U

(w;w(k)

)+ µV

(w;w(k)

)+ τ‖w −w(k)‖22,

(6)

with the parameter τ ≥ 0 on the proximal term is addedfor convergence reason. To get the convex approximationU(w;w(k)

), the second and the third nonconvex terms in

U (w) are convexified by linearizing each term inside thesquares (·)2. This approximation technique ensures the samegradient for U

(w;w(k)

)with U (w) and naturally keep the

convex structure U (w). Then U(w;w(k)

)is given by

U(w;w(k)

)= wTA

(k)U w + b

(k)TU w, (7)

where A(k)U , 4ζ(d

(k)0,1d

(k)T0,1 + d

(k)1,0d

(k)T1,0 − d

(k)0,1d

(k)T1,0 −

d(k)1,0d

(k)T0,1 )+4η

∑pi=2(d

(k)0,i d

(k)T0,i +d

(k)i,0 d

(k)Ti,0 −d

(k)0,i d

(k)Ti,0 −

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d(k)i,0 d

(k)T0,i ), and b

(k)U , 2ξ(d

(k)0,h − d

(k)h,0) + 2ζr

(k)1 (d

(k)0,1 −

d(k)1,0)+2η

∑pi=2 r

(k)i (d

(k)0,i−d

(k)i,0 ), with r(k)h = (w(k)THw(k))

/(w(k)TM0w(k)), r(k)i = (w(k)TMiw

(k))/(w(k)TM0w(k)),

d(k)0,h = Hw(k)/(w(k)TM0w

(k)), d(k)h,0 = r

(k)h M0w

(k),

d(k)0,i = Miw

(k) /(w(k)TM0w(k)), and d

(k)i,0 = r

(k)i M0w

(k),with i = 1, . . . , p. Likewise, the V

(w;w(k)

)is the convex

approximation for the variance term V (w) which is given by

V(w;w(k)

)= b

(k)TV w, (8)

where b(k)V , −2

(w(k)TM0w

(k))−2

M0w(k).

Then by combining U(w;w(k)

)with V

(w;w(k)

)and

dropping some constants, the F(w;w(k)

)in (6) becomes

F(w;w(k)

)= wTA(k)w + b(k)Tw, (9)

where A(k) , A(k)U +τI, and b(k) , b

(k)U +µb

(k)V −2τw(k).

And the subproblem to solve becomes

minimizew

wTA(k)w + b(k)Tw

subject to ‖Bw‖1 ≤ B,(10)

which is a convex problem. Summarizing, in order to solvethe original problem (3), we just need to iteratively solve a se-quence of convex problems (10). The SCA-based algorithmis named SCA-MRP and given in Algorithm 1. The conver-gence of Algorithm 1 can be guaranteed if the step-size γ(k)

is chosen as a (suitably small) constant or alternatively chosenaccording to the following diminishing step-size rule:

Given γ(0) ∈ (0, 1] ,Let γ(k+1) = γ(k)

(1− θγ(k)

)for k = 0, 1, 2, . . . ,

where θ ∈ (0, 1) is a given constant.

Algorithm 1 SCA-MRP Algorithm for Optimal MRP DesignRequire: H, Mi (i = 0, . . . , p), µ, B, B and τ

1: Set k = 0, γ(0) and w(0).2: repeat3: Compute A(k) and b(k).4: w(k+1) = arg min

w∈WwTA(k)w + b(k)Tw

5: w(k+1) = w(k) + γ(k)(w(k+1) −w(k))6: k ← k + 17: until convergence

The inner convex problem (see Step 5 in Algorithm 1)has no closed-form solution, but we can resort to the off-the-shelf solvers like MOSEK [19] or the popular scripting lan-guage CVX [20]. However, as an alternative to the general-purpose methods, we can also develop problem-specific algo-rithms to solve this problem efficiently.

3.3. Solving The Inner Subproblem in SCA Using ADMM

The alternating direction method of multipliers (ADMM) iswidely used to solve convex problems by breaking them intosmaller parts, each of which is then easier to handle [21]. Tosolve problem (10) by ADMM, we first rewrite problem (10)(for notational simplicity, superscripts (k) are omitted) by in-troducing an auxiliary variable z = Bw as follows:

minimizew,z

wTAw + bTw

subject to ‖z‖1 ≤ B, Bw − z = 0.(11)

Then the augmented Lagrangian is

Lρ (w, z,u (y))

= wTAw + bTw + IC(z) + ρ2 ‖Pw − z + u‖22 ,

where IC (z) =

{0, z ∈ C+∞, otherwise with C ,

{z∣∣ ‖z‖1 ≤ B}

is the indicator function and the penalty parameter ρ > 0serves as the dual update step-size with the scaled dual vari-able u = 1

ρy. Then, the ADMM updates are given in threevariable blocks, i.e., (w, z,u (y)), by

w(k+1) = arg minw

{wTAw + bTw

+ρ2

∥∥Bw − z(k) + u(k)∥∥22

}z(k+1) = arg min

z

{IC(z) + ρ

2

∥∥z−Bw(k+1) − u(k)∥∥22

}u(k+1) = u(k) + Bw(k+1) − z(k+1).

Specifically, for variable w, it is to solve a convex quadraticprogramming with a closed-form solution as follows:

w(k+1) = −(2A + ρBTB

)−1 (b + ρBT

(u(k) − z(k)

)).

By defining h(k) = Bw(k+1) + u(k), the variable z update isequivalent to solve

z(k+1) = arg minz∈C

∥∥z− h(k)∥∥22

= ΠC(h(k)

), (12)

which is the classical projection onto the `1-ball problem [22,23] with ΠC (·) denoting the projection operator. This prob-lem has a closed-form solution given in the following.

if ||h||1 ≤ B thenz = h, return z

elsea = sign(h) and b = abs(h)Sort b in order: b(1) ≥ b(2) ≥ · · · ≥ b(N)

ρ = arg max1≤j≤N

{b(j) − 1

j

(∑ji=1 b(i) −B

)> 0

}θ = 1

ρ

(∑ρi=1 b(i) −B

)zj = aj max{bj − θ, 0}, 1 ≤ j ≤ N , return z

end ifIn the solution for z-update, sgn (·) is the “sign function”;

abs (·) is the absolute value function; and b(j) (1 ≤ j ≤ N)denotes the j-th largest element in b. Then, the overallADMM-based algorithm can be summarized in Algorithm 2.

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Algorithm 2 An ADMM-Based Algorithm for Problem (10)Require: A, b, B, B and ρ

1: Set w(0), z(0), u(0) and k = 0.2: repeat3: w(k+1) = −(2A+ρBTB)−1(b+ρBT (u(k)−z(k)))4: h(k) = Bw(k+1) + u(k)

5: z(k+1) = ΠC(h(k))

6: u(k+1) = u(k) + Bw(k+1) − z(k+1)

7: k ← k + 18: until convergence

3.4. Acceleration Scheme for The SCA-MRP Algorithm

Besides the diminishing step-size rules for step-size γ(k), it ispossible to get a better convergence speed for Algorithm 1 byusing some acceleration method. One Armijo-like backtrack-ing line search rule for the step-size [24] is given as follows:

Given α, β ∈ (0, 1) , l = 0While ∆F

(w(k)

)> −αβl‖∆w(k)‖22

l = l + 1Let γ(k) = βl for k = 0, 1, 2, . . . ,

where ∆F(w(k)

)= F

(w(k) + βl∆w(k)

)− F

(w(k)

)with

∆w(k) = w(k+1) −w(k).4. NUMERICAL SIMULATIONS

In this section, we first test the proposed problem formulationand algorithms using synthetic data generated from a time se-ries model. We show the convergence property over iterationsof the objective function value in problem (3) by using theproposed SCA-MRP algorithm with and without accelerationin comparison to some benchmark algorithms.

10-2

10-1

100

101

CPU time (seconds)

0.6

0.8

1

1.2

1.4

1.6

1.8

obje

ctive v

alu

e

fmincon

SCA + CVX

SCA + CVX (acc.)

SCA + MOSEK

SCA + MOSEK (acc.)

SCA + ADMM

SCA + ADMM (acc.)

Fig. 1. Convergence comparison for objective function value.

The SCA-MRP algorithm is first compared with the gen-eral purpose constrained optimization solver fmincon [25] inMATLAB. From Figure 1, it is easy to see that SCA-MRPobtains a faster convergence and converges to a better solu-tion than fmincon. We further compare the SCA-MRP withthe inner problem solved by CVX, MOSEK, and ADMM. Theinner problem solved by ADMM can uniformly get a fasterconvergence either using acceleration or not than the others.

In Figure 2, we show that by tuning the parameter µ in prob-lem (3), our formulation is able to get a trade-off between MRand variance of the portfolio. However, this desirable prop-erty cannot be attained with existing methods in the literature.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

var 10-3

0

0.01

0.02

0.03

0.04

0.05

0.06

mean r

evers

ion

Fig. 2. Trade-off between MR and variance in MRP design.We also test the performance on market data from the

Standard & Poor’s 500 (S&P 500) Index, which are retrievedfrom Google Finance1. We apply the designed MRP to amean reversion trading based on the trading framework andperformance measure introduced in [16]. We choose stockcandidates into one asset pool as {APA, AXP, CAT, COF,FCX, IBM, MMM}, where they are denoted by their tickersymbols in Figure 3. Three spreads are constructed from thispool based on the Johansen method [9]. The methods pro-posed in this paper are employed for optimal MRP design.In Figure 3, we compare the performance of our designedMRP with spread s2. The performance metrics like returnon investment (ROI), Sharpe ratio, and cumulative P&Ls arereported. It is shown that the designed MRP can achieve ahigher Sharpe ratio and a better final cumulative return.

3

3.5

4

4.5

5

5.5

Lo

g-p

rice

s

APA

AXP

CAT

COF

FCX

IBM

MMM

-202

-202

2010-02-01 2010-12-15 2011-11-01 2012-09-19 2013-08-09 2014-06-27

-202

-1

0

1

Spre

ad

-0.02

-0.01

0

0.01

RO

I

-505

10

RO

I

10-3

2013-03-05 2013-06-06 2013-09-11 2013-12-13 2014-03-24 2014-06-270

10

20

Cum

. P

&L

Fig. 3. A mean-reversion trading based on real data.

5. CONCLUSIONSThe optimal mean-reverting portfolio design problem arisingfrom statistical arbitrage has been considered in this paper.We first proposed a general model for MRP design where atrade-off can be attained between the MR and variance of anMRP and the investment leverage constraint is considered. Tosolve the problem, a SCA-based algorithm is used with theinner convex subproblem efficiently solved by ADMM. Nu-merical results show that our proposed method can generateconsistent profits and outperform the benchmark methods.

1https://www.google.com/finance

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