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Journal of Mathematical Finance, 2020, 10, 334-344

https://www.scirp.org/journal/jmf

ISSN Online: 2162-2442

ISSN Print: 2162-2434

DOI:

10.4236/jmf.2020.102020 May 21, 2020 334

Journal of Mathematical Finance

Statistical Arbitrage Strategy in Multi-Asset

Market Using Time Series Analysis

Takahiro Imai, Kei Nakagawa

Innovation Lab Department, Nomura Asset Management Co. Ltd., Tokyo, Japan

Abstract

The statistical arbitrage strategy is one of the most traditional investment

strategies. There are many theoretical and empirical studies until now. How-

ever, almost all of the statistical arbitrage strategies focus on the price differ-

ence (spread) between two similar assets in the same asset class and exploit

the mean reversion of spreads,

i.e.

pairs tradin

g. 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 cri-

teria. 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.

Keywords

Statistical Arbitrage Strategy, Asset Allocation, Mean Reverting Portfolio

1. Introduction

Portfolio selection is one of the most important topics in mathematical finance.

Modern portfolio theory has its genesis in the seminal works of Markowitz [1].

In Markowitz analysis, the investment return should be maximized for a given

level of risk. Therefore, the main problem of portfolio selection is how to derive

a portfolio with a higher return/risk. Several researchers have been built some

models to maximize return/risk of the portfolio. For example, there are many

studies based on methods such as machine learning [2] and uncertainty theory

[3] in recent years. In addition to maximizing return/risk, there have been pro-

posed methods for constructing portfolios based on various criteria. Risk-based

portfolio that focuses only on risk such as risk parity [4], factor risk parity [5]

How to cite this paper:

Imai, T. and Na-

kagawa

, K. (2020) Statistical Arbitrage Strat-

egy in Multi

Asset Market Using Time Series

Analysis

Journal of Mathematical Finance

, 334-344.

https://doi.org/10.4236/jmf.2020.102020

Received:

March 3, 2020

Accepted:

May 18, 2020

Published:

May 21, 2020

20 by author(s) and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution

International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

T. Imai, K. Nakagawa

DOI:

10.4236/jmf.2020.102020 335

Journal of Mathematical Finance

and complex valued risk parity [6] is a typical example.

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

in portfolio selection [7] [8]. Traditionally, a mean reverting portfolio originated

from pairs trading. There are many studies on the mean reversion of the price

difference between two similar assets,

i.e.

spread until now. A broad range of in-

vestors from individual investors to institutional investors invest in pairs trading

strategy exploiting the mean reversion of spread [9]. According to [10], there are

many approaches to the pairs trading strategy such as stocks distance, time series

model e.g. co-integration and stochastic control. However, since many related

works of the pairs trading strategy focused on the spread only between similar

stocks, the investment universe was stocks in a single asset class. In this study,

we propose pairs trading strategy which invest on assets in different asset classes,

by deriving the mean-reverting portfolio in not a single asset market but the

multi-asset market. When the portfolio is far away to a certain extent from the

average level, we take a position in the direction of mean reversion. Specifically,

we construct the portfolio based on multiple criteria for the mean reversion de-

fined based on different perspectives. By using the technique of a multi-objective

optimization problem called Polynomial Goal Programming (PGP), we propose

a fair approach to combine the quantitative criteria of the mean reversion. We

aim to obtain the arbitrage opportunity between global asset classes in the mul-

ti-asset market.

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.

2. Related Work

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 in-

dicates how close to the white noise in terms of the variance of time series [11]

[12]. Portmanteau Statistics indicates how close to the white noise in terms of

the correlation of time series [13]. Crossing Statistics indicates how many times

the time series crosses the average level in the unit time interval [14]. As pairs

trading strategies using predictability, portmanteau statistics, crossing statistics

alone respectively, there are related researches investing on the implied volatility

of the U.S. stocks [7] and U.S. stocks [8]. Although these researches are useful in

that they evaluate the effectiveness of the single quantitative indicator of the

mean reversion, they do not construct a mean reverting portfolio based on mul-

tiple perspectives. Our method combines Predictability, Portmanteau Statistics

T. Imai, K. Nakagawa

DOI:

10.4236/jmf.2020.102020 336 Journal of

Mathematical Finance

and Crossing Statistics by solving the multi-objective optimization problem. We

expect that a mean-reverting portfolio based on multiple indicators leads to a

higher return/risk. However, there remains a problem that we have no idea how

to combine the multiple indicators fairly. As a method of solving such mul-

ti-objective optimization problems fairly, a method called PGP is often used for

portfolio optimization problems with higher-order moments [15]. In this study,

we apply PGP to solve the multi-objective optimization problem. Furthermore,

we extend the strategy to multiple assets in the multi-asset market. The compar-

ison of our research and related work is summarized in

Table 1.

3. Mean Reverting Portfolio

When

assets exist at the time of

{ }

1, ,

t t Nt

yy=y 

denotes the log prices

at the time. When

{ }

ww=w 

denotes the weight vector of each assets, the

portfolio can be described as below.

z = wy

(1)

Logarithmic return of the portfolio

can be described as below.

( )

11t tt t t

rzz yy

−−

=−= −w

(2)

The problem in this study is to determine the weight

in which the portfo-

lio

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.

3.1. Indicators of the Mean Reversion

We introduce multiple indicators which show the goodness in terms of the mean

reversion of the portfolio

. Specifically, we introduce (1) Predictability, (2)

Portmanteau Statistics, (3) Crossing Statistics to quantify the mean reversion.

We start by defining the

th order (lag-

) autocovariance matrix for a stochastic

process

( )

[ ]

( )

[ ]

( )

: Cov , .

i ti t t ti ti+ ++



= =−−





M yy y y y y 

(3)

Note that

represents the covariance matrix.

3.1.1. Predictability

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

Table 1. Comparison of our research and related work.

Paper

Criteria

Investment assets

Cuturi and d’Aspremont [2017] Pred, Port, Cross

Implied volatility of

U.S. stocks (single asset)

Zhao and Palomar [2018]

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.

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