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The statistical arbitrage strategy is one of the most traditional investment strategies. There are many theoretical and empirical studies until now. However, almost all of the statistical arbitrage strategies focus on the price difference (spread) between two similar assets in the same asset class and exploit the mean reversion of spreads,
* i.e.* pairs trading. In this study, we extend the strategy to multiple assets in the multi-asset market. Although mean-reverting portfolios were derived based on a single criterion in related researches, we derive a mean-reverting portfolio by optimizing multiple mean-reversion criteria. We expect that a mean-reverting portfolio based on multiple indicators leads to a higher return/risk. We perform an empirical analysis in multi-asset market and show the profitability of our strategy.

Portfolio selection is one of the most important topics in mathematical finance. Modern portfolio theory has its genesis in the seminal works of Markowitz [

Also, deriving a mean reverting portfolio is one of the most popular methods in portfolio selection [

The remaining sections of this paper are organized as follows. In Section 2, we briefly describe the related studies of the mean reverting portfolio using the time-series model. In Section 3, we introduce multiple indicators denoting the “goodness” of the mean reversion and a method of integrating the indicators called PGP. In Section 4, we describe the pairs trading strategy in the multi-asset market and in Section 5, we verify its effectiveness through empirical analysis with the actual financial market data. Finally, we conclude.

Quantitative indicators of the mean reversion have been proposed in various forms. Here, we use three types of indicators describing the mean reversion: Predictability, Portmanteau Statistics and Crossing Statistics. Predictability indicates how close to the white noise in terms of the variance of time series [

When N assets exist at the time of t, y t = { y 1 , t , ⋯ , y N , t } denotes the log prices at the time. When w = { w 1 , ⋯ , w N } denotes the weight vector of each assets, the portfolio can be described as below.

z t = w T y t (1)

Logarithmic return of the portfolio r t can be described as below.

r t = z t − z t − 1 = w T ( y t − y t − 1 ) (2)

The problem in this study is to determine the weight w in which the portfolio z t in the Equation (1) is mean reverting. In other words, the object is to calculate the weight as much mean reverting as possible between multiple assets.

We introduce multiple indicators which show the goodness in terms of the mean reversion of the portfolio z t . Specifically, we introduce (1) Predictability, (2) Portmanteau Statistics, (3) Crossing Statistics to quantify the mean reversion. We start by defining the ith order (lag-i) autocovariance matrix for a stochastic process y t as

M i : = Cov ( y t , y t + i ) = E [ ( y t − E [ y t ] ) ( y t + i − E [ y t + i ] ) T ] . (3)

Note that M 0 represents the covariance matrix.

Predictability shows how the time series is close to the white noise in terms of

Paper | Criteria | Investment assets |
---|---|---|

Cuturi and d’Aspremont [ | Pred, Port, Cross | Implied volatility of U.S. stocks (single asset) |

Zhao and Palomar [ | Pred, Port, Cross | U.S. stocks (single asset) |

Our research | Pred, Port, Cross Pred + Port + Cross | Global futures (multi-asset) |

a. Pred, Port and Cross represent Predictability, Portmanteau Statistics, and Crossing Statistics respectively.

the variance. We consider the following stationary time-series.

y t = y ^ t − 1 + ε t (4)

y ^ t − 1 denotes the predicted value of y based on the information of the time of t − 1 . The simplest example of Equation (4) is AR (1) model representing y ^ t − 1 = α y t − 1 . ε t denotes the white noise which is independent from y ^ t − 1 and whose variance is σ ε 2 . Taking the variance of Equation (4), σ y 2 = σ y ^ 2 + σ ε 2 . σ y 2 denotes the variance of y t and σ y ^ 2 denotes the variance of y ^ t − 1 . Predictability is defined as follows.

predictability = σ y ^ 2 σ y 2 (5)

From the definition of predictability, predictability means that the smaller, the more mean revering and vice versa. Here, we assume y ^ t − 1 can be modeled by the following VAR (1) model. Notice that we can extend to VARMA model because VARMA (p, q) model can be reduced to VAR (1) model [

y t = A y t − 1 + e t (6)

where e t is the white noise.

Multiplying VAR (1) of Equation (6) by w T , we can get w T y t = w T A y t − 1 + w T e t . Taking its variance, the term on the left hand side is w T M 0 w and the first term on the right hand side w T A y t − 1 is w T A M 0 A T w . Since A = M 1 T M 0 − 1 according to the property of VAR (1) model, the first term on the right hand side is w T A M 0 A T w = w T M 1 T M 0 − 1 M 0 ( M 1 T M 0 − 1 ) T w = w T M 1 T M 0 − 1 M 1 w .

Therefore, predictability of VAR (1) model is as follows.

predictability ( w ) = w T M 1 T M 0 − 1 M 1 w w T M 0 w (7)

Portmanteau Statistics indicator shows how the time series is close to the white noise in terms of the correlation. We consider stationary time-series with lag-p.

y t = y ^ t − 1 + ⋯ + y ^ t − p + ε t (8)

Portmanteau statistics are defined as follows.

portmanteau = ∑ i = 1 p ρ i 2 (9)

where ρ i is the ith order autocorrelation, and defined as E [ z t z t + i ] / E [ z t 2 ] .

Autocorrelation of the white noise is zero and portmanteau ≥ 0 by its definition. The time series is close to the white noise when portmanteau is close to 0. Therefore, we can get a mean reverting portfolio by minimizing Portmanteau statistics. For a mean reverting portfolio z t = w T y t , we can get the expression for Portmanteau statistics as

portmanteau ( w ) = ∑ i = 1 p ( w T M i w w T M 0 w ) 2 (10)

Crossing Statistics indicator counts how many times the time series crosses the average level in the time interval T.

For a stationary Gaussian process, the crossing statistics is defined as follows.

crossing = 1 T − 1 ∑ t = 2 T 1 E ( y t ) (11)

where 1 E ( y t ) denotes an indicator function that returns 1 when E = { y t y t − 1 ≤ 0 } , or 0 otherwise.

For a centered stationary Gaussian process, we can get the expression for Crossing Statistics as

crossing = 1 π arccos ( ρ 1 ) . (12)

In order to get a spread having many zero-crossing, we minimize ρ 1 . Therefore, for mean reverting portfolio z t = w T y t , we define the crossing statistics as

crossing ( w ) = w T M 1 w w T M 0 w (13)

In this section, we determine the optimal weights of the portfolio, integrating three mean reversion indicators introduced in the previous section by PGP. Goal programming (GP) was first proposed in [

The GP has many extensions and applications. For example, Liu and Chen proposed an uncertain goal programming and [

PGP method is originally proposed in [

The first step in PGP is to get the optimal values pred * , port * , cross * by solving independent minimizing problems whose objective functions are Equation (7), Equation (10), and Equation (13). We can get the optimal weights of the portfolio by solving the problem as follows.

arg min w | d 1 pred * | λ 1 + | d 2 port * | λ 2 + | d 3 cross * | λ 3 (14)

where d 1 = predictability ( w ) − pred *

d 2 = portmanteau ( w ) − port *

d 3 = crossing ( w ) − cross *

where λ 1 , λ 2 , λ 3 are variables of investor preferences for predictability, portmanteau statistics, and crossing statistics. We show the conceptual figure of three indicators of the mean reversion and PGP in

In this study, we propose the pairs trading strategy by deriving the mean reverting portfolio based on the three mean reversion indicators introduced in the previous section.

Specifically, the pairs trading strategy includes following three steps.

Step 1

For investing on multi-assets, we select the order p that minimize AIC of VAR (p) model under the condition that p is equal to or less than the full-order selected in advance.

Step 2

We derive the mean reverting portfolios by minimizing the functions, which are 1) Predictability, 2) Portmanteau Statistics, 3) Crossing Statistics, 4) the multi-objective function integrated with Predictability, Portmanteau Statistics, and Crossing Statistics.

Step 3

We calculate the spread from the moving average of past return of the portfolio derived in Step 2. We get the position when the spread is ±1 standard deviation farther from the average. As a loss cut, we unwind the position if the spread is ±2 standard deviations farther from the average.

This section describes the empirical study with real market data.

We test our method using real market data from global futures. We show the investment universe in ^{th}, 2007 to August 30^{th}, 2019. Sample size during the period is 650.

Investment assets | ||||||||
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Equity future (16 assets) | S & P500 (SP) | NAS DAQ (NQ) | CA (PT) | GB (Z) | FR (CF) | DE (GX) | EU (VG) | ES (IB) |

NL (EO) | NO (OI) | CH (SM) | NIKKEI (NK) | TOPIX (TP) | HK (HI) | AU (XP) | SG (QZ) | |

Bond future (13 assets) | US2Y (TU) | US5Y (FV) | US10Y (TY) | US20Y (US) | AU3Y (YM) | AU10Y (XM) | CA10Y (CN) | DE2Y (DU) |

DE5Y (OE) | DE10Y (RX) | DE30Y (UB) | GB10Y (G) | JP10Y (JB) |

a. words in parentheses denote tickers.

Performance statistics | SP | NQ | PT | Z | CF | GX | VG | IB |
---|---|---|---|---|---|---|---|---|

Return(%, Ann) | 8.4 | 15.1 | 5.2 | 5.8 | 5.9 | 6.9 | 5.2 | 2.8 |

Risk(%, Ann) | 18.0 | 19.2 | 17.1 | 17.8 | 21.1 | 22.2 | 21.9 | 23.4 |

Return/risk | 0.46 | 0.79 | 0.30 | 0.33 | 0.24 | 0.31 | 0.24 | 0.12 |

Maximum drawdown (%) | −56.8 | −51.6 | −49.1 | −49.3 | −58.8 | −57.8 | −61.6 | −54.5 |

Performance statistics | EO | OI | SM | NK | TP | HI | XP | QZ |
---|---|---|---|---|---|---|---|---|

Return(%, Ann) | 6.4 | 7.5 | 5.8 | 5.8 | 3.4 | 7.9 | 3.7 | 4.4 |

Risk(%, Ann) | 21.0 | 23.1 | 18.3 | 22.7 | 21.5 | 22.7 | 16.8 | 18.9 |

Return/risk | 0.30 | 0.32 | 0.32 | 0.26 | 0.16 | 0.35 | 0.22 | 0.23 |

Maximum drawdown (%) | −64.2 | −64.4 | −57.6 | −60.4 | −58.5 | −59.5 | −55.6 | −60.2 |

Performance statistics | TU | FV | TY | US | YM | XM | CN | DU |
---|---|---|---|---|---|---|---|---|

Return(%, Ann) | 1.2 | 2.9 | 4.3 | 5.8 | 1.3 | 3.7 | 3.8 | 1.0 |

Risk(%, Ann) | 1.2 | 3.5 | 5.6 | 10.0 | 2.5 | 6.7 | 5.4 | 1.1 |

Return/risk | 1.01 | 0.84 | 0.77 | 0.58 | 0.51 | 0.55 | 0.71 | 0.91 |

Maximum drawdown (%) | −2.4 | −5.9 | −8.8 | −17.7 | −5.6 | −13.0 | −10.0 | −3.1 |

Performance statistics | OE | RX | UB | G | JB |
---|---|---|---|---|---|

Return(%, Ann) | 3.2 | 5.6 | 9.2 | 5.0 | 2.0 |

Risk(%, Ann) | 3.0 | 5.5 | 12.6 | 6.4 | 2.4 |

Return/risk | 1.03 | 1.02 | 0.72 | 0.78 | 0.85 |

Maximum drawdown (%) | −6.0 | −7.2 | −18.3 | −10.0 | −5.8 |

We determine the weights of the optimal portfolio on condition that the full-order equals to 5 and data in the past 52 weeks are used for the model selection. Portfolio leverage is determined so that ex-ante risk of the portfolio equals to 5% calculated by the covariance matrix based on the data in the past 52 weeks. Note that model of predictability is VAR (1) model for the integration of indicators. It is decided to unwind the position every quarter even though we have the position. If we have a loss cut, we don’t have the position in the quarter. All preferences of predictability, portmanteau statistics, and crossing statistics in PGP equal to 1. Moving average of portfolio return is calculated based on data in past 13 weeks and average and standard deviation of the spread are calculated based on data in past 52 weeks. We call the period from getting the position to unwinding the position a strategy.

We show the performance summary of the portfolio based on the indicators of the mean reversion in

The performance statistics in

The return/risk and winning percentage of the strategy in

In this study, we propose pairs trading strategy where we derive the mean reverting portfolio in the multi-asset market by using the time series model. We

PGP | Predictability | Portmanteau | Crossing | |
---|---|---|---|---|

Return in investment period (%, Ann) | 11.9 | 9.9 | 3.1 | 4.3 |

Risk in investment period (%, Ann) | 6.9 | 8.9 | 4.5 | 4.8 |

Return/risk in investment period | 1.73 | 1.11 | 0.70 | 0.90 |

Winning percentage of the strategy (%) | 59.8 | 57.3 | 50.6 | 54.8 |

Maximum drawdown of the strategy (%) | −2.5 | −4.4 | −1.7 | −2.0 |

Proportion of the investment period (%) | 28.0 | 24.5 | 30.0 | 27.7 |

Average investment period (weeks) | 1.9 | 1.8 | 2.4 | 2.1 |

derive the portfolios based on predictability, which is measured in terms of the variance, portmanteau statistics, which is measured in terms of the correlation, crossing statistics, which represents how many times the time series crosses the average level, and the indicator integrated by a method called PGP. We get the empirical results that the return/risk and winning percentage of the strategy are best in the case of PGP, and it suggests that it is effective to combine multiple indicators of the mean reversion for deriving the mean reverting portfolio in the multi-asset market.

The authors acknowledge Taku Imahase and Akio Ito for discussions and insights that helped clarify the ideas in this paper. The authors also thank all the reviewers for insightful comments.

The authors declare no conflicts of interest regarding the publication of this paper.

Imai, T. and Nakagawa, K. (2020) Statistical Arbitrage Strategy in Multi-Asset Market Using Time Series Analysis. Journal of Mathematical Finance, 10, 334-344. https://doi.org/10.4236/jmf.2020.102020