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Statistical Arbitrage (SA) is a common financial term. However, there is no common definition in the literature while investors use the expression SA for a variety of different strategies. So, what is SA? In order to answer this question, we investigate SA strategies across equity, fixed income and commodity. The analysis of strategies’ key features indicates that no existing definition fully describes them. To bridge this gap, we identify a general definition and propose a classification system that encompasses the current forms of SA strategies while facilitating the inclusion of new types as they emerge.

The concept of arbitrage is fundamental in financial literature and has been used in classical analysis of market efficiency [

This paper addresses this question with an in-depth investigation of SA. We begin by reviewing existing definitions of arbitrage, which are reduced to a common framework to analyze and compare them. We survey statistically determined arbitrage strategies analyzing both the academic and financial industry research. In total, we review 165 articles on the subject, published between 1995 and 2016. Particular attention is paid to hedge funds techniques, market neutral investment strategies and algorithmic trading. The strategies are discussed in a standardized way analyzing equity, fixed income and, for the first time, commodity. We find that these strategies show significant similarities and common features that define them. The comparison of theoretical definitions and strategies’ key features indicates that no available definition appropriately describes SA strategies. To bridge this gap, we propose a general definition, which more closely reflects investors’ strategies. In addition, we suggest that, instead of searching for a definitive theoretical definition of SA, scholars should instead agree on a classification system that encompasses the current forms of SA while facilitating the inclusion of new types as they emerge. We propose a simple system for classifying strategies that takes into account the strategies’ risk and return profile. We illustrate the advantages of this approach by demonstrating how it can guide theoretical development and empirical testing. We also provide examples of potential future research directions.

We make several contributions to the existing literature. We identify a general definition, which encompasses all SA strategies and introduce a classification system that facilitates their study. This is achieved through an innovative investigation of SA both in academic and financial industry research. In our review, for the first time, we analyze SA across all asset classes (equity, fixed income and commodity) to identify common features and defining elements. Our analysis brings clarity in SA investing and allows investors to have a common framework to assess different investment opportunities.

The paper is organized as follows. In Section 2, we review existing definitions of SA producing a comprehensive mapping. In Section 3, we report a survey of statistically determined arbitrage strategies. In Section 4, we identify the key features which are common to the various strategies. We combine the findings of the previous sections and propose a general definition and classification system. Section 5 concludes the paper.

It is commonly accepted that Statistical Arbitrage (SA) started with Nunzio Tartaglia who, in the mid-1980s, assembled a team of quantitative analysts at Morgan Stanley to uncover statistical mispricing in equity markets [

The literature on the limits of arbitrage is quite broad and provides some insights on why SA opportunities exist. Mou [

In this section, we review all definitions of arbitrage available in literature which may be suitable to define SA. Our analysis encompasses both alternative definitions of arbitrage as well as definitions of statistical arbitrage. Before reviewing the various definitions, we briefly recall the four types of definitions that are commonly used: 1) lexical, 2) conceptual, 3) abstract and 4) operational [

Some lexical definitions tend to be vague and lack formalism because traders, for good commercial reasons, tend to be obscure about their investment methods. Pole [

A general definition of SA strategy should describe what SA is and its objectives. We find instead that some definitions focus on specific implementations and techniques. In particular, in a broad range of papers, SA is associated with pairs trading [

Another set of definitions can be classified as conceptual as they can be associated with specific measures. In reviewing Hedge Funds (HFs) strategies, Connor and Lasarte [

We next discuss the various extensions of arbitrage available in the literature that are used mainly in asset pricing. All definitions can be classified as operational and are mathematically formulated. Here, we provide a description of the various arbitrages while we refer to the relative papers for a more rigorous formulation.

We first introduce the classical definition of arbitrage, defined as a zero-cost trading strategy with positive expected payoff and no possibility of a loss. The absence of arbitrage is a necessary condition for equilibrium models, however this condition alone is often too weak to be practically useful for certain applications such as option pricing [

A first attempt to provide a new definition of arbitrage is made by Ledoit [

Bernardo and Ledoit [

In the literature, there are two definitions of Statistical Arbitrage (SA) which differ significantly from each other. Bondarenko’s SA [

As a summary, we provide a high-level description of all the reviewed arbitrage definitions in

The existing literature on SA includes a small number of reviews of arbitrage strategies which cover only single asset classes. In fixed income, Duarte,

Author/Name | Definition |
---|---|

Panel A: Lexical definitions | |

Burgess (2000) | SA is a framework for identifying, modelling and exploiting small but consistent regularities in asset price dynamics |

Zapart (2003) | SA is an investment opportunity arising from the choice of models for hedging |

Do et al. (2006) | SA is an equity trading strategy that employs time series methods to identify relative mispricing between stocks |

Thomaidis and Kondakis (2006) | SA is an attempt to profit from pricing discrepancies that appear in a group of assets |

Pole (2007) | SA uses mathematical models to generate returns from systematic movements in securities prices |

Avellaneda and Lee (2008) | SA encompasses a variety of strategies characterized by: i) systematic trading signals, ii) market neutral trades and iii) statistical methods |

Montana et al. (2008) | SA is an investment strategy that exploits patterns detected in financial data streams |

Panel B: Conceptual definitions | |

Connor and Lasarte (2003) | SA is a zero-cost portfolio where the probability of a negative payoff is very small but not exactly zero |

Stefanini (2006) | SA seeks to capture imbalances in expected value of financial instruments, while trying to be market neutral |

Saks and Maringer (2008) | SA 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 |

Focardi, Fabozzi and Mitov (2016) | SA strategies aim at producing positive, low-volatility returns that are uncorrelated with market returns |

Panel C: Operational definitions | |

δ-Arbitrage | Is a strategy with a Sharpe ratio above a constant and positive δ |

Good Deal | Consists in buying (selling) securities whose market price lies outside a range of plausible prices |

Approximate Arbitrage | Is a strategy whose gain-loss ratio is above a predefined constant value greater than 1 |

Acceptable Opportunity | Is a strategy with a non-negative expected value under each valuation measure and losses capped under the set of stress measures |

ε-Arbitrage | Consists in buying (selling) those derivatives strategies whose price significantly differs from the least costly optimal replication strategy |

SA (Bondarenko) | Is a strategy with expected positive payoff and expected non-negative payoff conditional on the augmented information set |

SA (Hogan et al.) | With time the strategy has positive expected payoff, probability of a loss which tends to zero and time averaged variance which converges to zero |

Longstaff and Yu [

^{1}Ornstein-Uhlenbeck is a model used to describe the multivariate dynamics of financial variables [

In our review, for the first time, we look at SA across all asset classes to identify common features and defining elements. We review the existing literature on statistically determined arbitrage strategies and, particularly, on those labelled as SA. We identify 165 articles in literature discussing SA strategies spanning from 1995 to 2016 (see

We categorize the various strategies based on the classification proposed by Duarte, Longstaff and Yu [^{1} stochastic process (10) and, more recently, high frequency trading (9). Pairs trading is predominantly an equity strategy (103). Capital structure arbitrage is the second most documented strategy (30) which includes primarily convertible arbitrage strategies (19). Term structure strategies are documented only in eight studies of which four analyze bonds. Swap spread arbitrage and mortgage arbitrage are discussed in three studies each.

SA strategy | Equities | Bonds | Commodities | Volatility | FX | Mix | Total |
---|---|---|---|---|---|---|---|

Pairs trading | 103 | 6 | 1 | 2 | 112 | ||

Capital structure arbitrage | 30 | 30 | |||||

Volatility arbitrage | 9 | 9 | |||||

Term structure arbitrage | 1 | 4 | 3 | 8 | |||

Swap spread arbitrage | 3 | 3 | |||||

Mortgage arbitrage | 3 | 3 | |||||

Total | 104 | 40 | 9 | 9 | 1 | 2 | 165 |

We next describe the six identified trading strategies. Pairs trading is a SA strategy which is particularly popular in equity [

Term structure arbitrage is a common SA strategy which typically involves taking market-neutral long-short positions at different points of a term structure as suggested by a relative value analysis [

Volatility arbitrage is a popular and widely used strategy [

Swap spread arbitrage is another popular fixed income strategy which bets on the difference between a fixed and a floating yield [

Mortgage arbitrage consists of buying mortgage-backed securities (MBSs) while hedging their interest rate exposure primarily through derivatives [

Capital structure arbitrage involves taking long and short positions in the various instruments of a company’s capital structure [

Convertible Arbitrage is one of the most popular capital structure strategies and involves buying a portfolio of convertible bonds while selling short the underlying stocks [

This review allows us to identify the defining features of the different strategies across asset classes. They are summarized in

In this section, we define SA strategies. We identify those features which are common to the surveyed arbitrage strategies. We compare them with the available definitions and provide a new definition in conjunction with a classification scheme. The new definition incorporates all strategies’ key elements and the classification scheme encompasses the important dimensions of SA while being flexible and easy to use.

All strategies aim to exploit relative value opportunities through the implementation of long-short positions. Pairs trading invests in the spread between two stocks. Term structure models the spread between yields or future prices.

Strategy | Descriptions |
---|---|

Pairs trading | Plays mean reversion in the spreads of two securities |

Term structure arbitrage | Takes long-short positions across the term structure |

Volatility arbitrage | Plays the spread of implied vs. realized volatility of the same security or implied vs. implied volatility of the same or different securities |

Swap spread arbitrage | Profits from the spread between a fix and a floating leg by entering a short (long) Treasury position and simultaneously buying (selling) an IRS |

Mortgage arbitrage | Buys MBS hedging the interest rates exposure |

Capital structure arbitrage | Takes long-short positions on different instruments of a company (credit arbitrage and convertible arbitrage) |

Volatility arbitrage identifies relative value opportunities between volatilities. Swap spread plays a fixed spread versus a floating spread. Mortgage arbitrage models the spread of MBS over treasury. Capital structure arbitrage profits from the spread between various instruments of the same company. Spreads trading involves taking long-short positions in order to profit from spreads or simply to bet on a security while being market-neutral.

However, not all strategies need mean reversion. Pairs trading and term structure arbitrage need spreads to revert to their mean to be profitable. Other strategies instead need a persistent positive spread-carry: between implied and realized volatility (volatility arbitrage), between the fixed and the floating spread (swap spread arbitrage), in the MBS spread over treasury (mortgage arbitrage) and between various instruments of the same company (capital structure arbitrage). If spreads narrow these strategies are less profitable and can turn into a loss. In addition, not all strategies are zero-cost. This is not only due to market frictions or trading costs but it is true by construction. For example, pairs trading (in the market-neutral form) may require a net payment and mortgage arbitrage requires the purchase of MBSs.

It is not possible to clearly define whether SA strategies are market-neutral. All strategies invest in some risk factors while hedging others. For example, term structure arbitrage may hedge only against parallel shifts of the term structure. Volatility arbitrage hedges against movements of the underlying but not of the underlying volatility. Swap spread arbitrage hedges against changes in treasury and swap rates but not against credit risk. Mortgage arbitrage hedges against movements in treasury rates but not mortgage spreads.

Not all strategies guarantee gains but rather offer positive expected excess returns with an acceptably small potential loss. Arbitrageurs require a positive expected excess return over the risk free to compensate for risk. The potential loss must be acceptably small in order to qualify the strategy as arbitrage rather than simple investment. Although not all the academic literature reports it, trades always have take profit and stop loss features. The take profit identifies when a trade no longer offers positive expected excess returns. A take profit is triggered in case there is reversion to the mean (pairs trading, term structure arbitrage, volatility arbitrage and capital structure arbitrage) or when the positive carry disappears (swap spread arbitrage and mortgage arbitrage). The stop loss quantifies when a loss is no longer acceptably small and results from investors’ risk tolerance.

From the previous analysis, it is possible to conclude that three key factors define statistically determined arbitrage opportunities: 1) relative value, 2) positive expected excess returns and 3) acceptably small potential loss. Take profit and stop loss are features which enable to operationalize SA strategies (see

From the review of strategies and definitions, we find that both in the definitions and strategies, statistics are used to explain securities mispricing. In particular,

Main features by strategy | Pairs trading | Term structure arbitrage | Volatility arbitrage | Swap spread arbitrage | Mortgage arbitrage | Capital structure arbitrage |
---|---|---|---|---|---|---|

Relative value | Yes | Yes | Yes | Yes | Yes | Yes |

Mean reversion | Yes | Yes | - | No | No | - |

Market neutral | - | - | - | - | - | - |

Zero cost | - | - | - | Yes | No | - |

Expected positive excess return | Yes | Yes | Yes | Yes | Yes | Yes |

Acceptably small potential loss | Yes | Yes | Yes | Yes | Yes | Yes |

Take profit | Yes | Yes | Yes | Yes | Yes | Yes |

Stop loss | Yes | Yes | Yes | Yes | Yes | Yes |

they focus on the same observable phenomenon but from different perspectives. Definitions focus primarily in strengthening the concept of arbitrage introducing additional constraints that can make theory more consistent with financial markets. In some cases, they use tools common to practitioners, such as the Sharpe ratio in δA. In other cases, instead the focus is more on the theoretical framework, such as in the augmented information set in Bondarenko’s definition [

We aim to create a definition which is measurable. That rules out lexical definitions which focus generically on systematic strategies [

The available conceptual definitions do not capture all key features: Connor and Lasarte [

Key feature | Relative value | Expected positive excess return | Acceptably small potential loss | Take profit | Stop loss |
---|---|---|---|---|---|

Panel A: Conceptual definitions | |||||

Connor and Lasarte (2003) | No | - | Yes | Yes | Yes |

Stefanini (2006) | Yes | Yes | No | Yes | No |

Saks and Maringer (2008) | No | Yes | Yes | Yes | Yes |

Focardi et al. (2016) | - | Yes | - | Yes | No |

Panel A: Conceptual definitions | |||||

δ?Arbitrage (Ledoit, 1995) | No | Yes | No | Yes | No |

Good Deal (Cochrane and Saa-Requejo, 1998) | No | Yes | No | Yes | No |

Approximate Arbitrage (Bernardo and Ledoit, 2000) | No | Yes | Yes | Yes | - |

Acceptable Opportunity (Carr et al., 2001) | No | - | Yes | Yes | Yes |

ε-Arbitrage (Bertsimas et al., 2001) | Yes | - | No | Yes | No |

SA (Bondarenko, 2003) | No | - | No | Yes | No |

SA (Hogan et al., 2004) | No | - | Yes | Yes | Yes |

Only δA, GD and AA incorporate the feature of positive excess returns while the other definitions generically refer to positive expected returns as there is no initial cost involved. AA embeds the feature of acceptably small potential loss but this is limited to a specific measure (gain-loss ratio). AO limits losses through the use of generic stress measures. Hogan’s SA partially requires acceptably small potential losses as the probability of a loss converges to zero with time. All definitions embed the concept of take profit as long as it is assumed that a strategy is closed at maturity or when the expected returns are no longer positive. AOs can be closed in stop loss if the realized loss is higher than what is acceptable according to the stress measures. Hogan’s SA has the concept of stop loss if it is assumed that a strategy is closed when the constraints on the probability of a loss are no longer satisfied. AA trades are closed in stop loss only if the gain-loss ratio is lower than one. According to the other definitions instead a trade is closed only when the defining criteria are no longer met and this does not necessarily involve a stop loss. In conclusion, there are some differences across definitions. Although some definitions are compatible with various strategies’ common features, nevertheless they fail to incorporate all of them as defining elements.

As no available definition fully captures what is done in practice, we identify a conceptual definition that incorporates all strategies’ key elements. We choose to use a conceptual definition as it clearly defines SA while leaving each analyst to select the most appropriate measure as explained below.

We define a SA strategy as a relative value strategy with a positive expected excess return and an acceptably small potential loss. We note the following in relation to our proposed new definition. First, SA is a relative value strategy. This reflects the fact that all the reviewed strategies play the spread of a security against another one. It should be noted that, while the concept of relative value is universally accepted, its boundaries are not clearly defined. A priori a total return strategy can be considered a relative value strategy of an investment against the overnight rate (which is close to zero). It is using the common understanding that we refer to relative value strategies as strategies aiming to find mispricing using historical relationships. As a relative value strategy, SA requires that the underlying securities are combined in a long-short portfolio. This allows to more accurately isolate some sources of risk (expected to deliver positive excess returns) while hedging others. The underlying securities may or may not belong to the same asset class.

Another element is given by the expected positive excess return. This part of the definition incorporates two features. The first one is given by the fact that the strategy focuses on the expected return. This differs from the definition of arbitrage where the strategy has no admissible possible negative outcomes. Losses are allowed in our definition of SA. The second one is given by the excess return. This reflects the fact that every arbitrageur embarks on a strategy involving some risk only if there are expectations of returns higher than the risk free whenever an initial investment is required.

The last requirement is given by the acceptably small potential loss. This element is fundamental in order to differentiate SA from a simple investment strategy. To be called arbitrage, a strategy needs to have a constrained loss profile. A strategy is closed whenever the defining criteria are no longer satisfied: 1) in stop loss, if the loss is no longer acceptably small or 2) in take profit, if the performance is positive and the expected excess return is no longer positive.

This definition cannot be operational unless we define how to measure a positive expected excess return and an acceptably small potential loss. The need for clarity on this issue is critical. However, the complex and dynamic landscape of financial markets suggests that no definitive theoretical or operational definition of SA is likely to be agreed. Because of this we propose to use the definition in conjunction with a classification scheme.

A positive expected excess returns requires defining the risk free and a probability measure. The risk free can be the cost of financing (for unfunded strategies) or the cash rate (for funded strategies). In the case of a zero-cost trading strategy, the risk free is equal to zero. Defining an acceptably small potential loss requires identifying a set of suitable risk measures and criteria to establish what is acceptably small. Examples of risk measures are the probability of a loss, the Value at Risk (VaR) and the Conditional Value at Risk (CVaR), see [

This classification scheme aims to be sufficiently detailed to encompass the important dimensions of SA while at the same time being intuitive and easy to use. To be widely accepted, a definition should also appeal to practitioners and other stakeholders by reflecting the world as it is perceived. Our definition, with annexed classification scheme, satisfies the four canons of a good definition: adequacy, differentiation, impartiality and completeness [

Our definition and classification system could guide future research. For example, the use of a common classification system allows investigating the profitability and riskiness of SA strategies across asset classes and time. This enables mapping pricing anomalies and can provide directions on how to improve pricing models. The existence of persistent SA opportunities in selected strategies can be used as an indicator to direct future research to less studied asset classes and instruments. Having a framework brings transparency to the term SA, helping investors in making investment decisions. For example, our definition of SA can be used in the hedge funds industry where there is no agreement on a standardized classification system of strategies [

In this paper, we investigate the concept of statistical arbitrage (SA). As there is no agreement in literature on a common definition, we review both the theoretical and empirical works on SA since its introduction. In particular, we look at all those definitions, which may be suitable to identify this class of strategies. We produce a review of all strategies which may be associated with the concept of statistically determined arbitrage opportunities. We identify those common features which define the concept embedded in investors thinking. As no definition is suitable to describe this type of strategies, we introduce a general definition and propose a classification system that encompasses the current forms of SA strategies while facilitating the inclusion of new types as they emerge.

Our study makes several contributions to the existing literature. We bridge the gap existing between the literature on arbitrage definitions and SA strategies. We perform an innovative investigation of SA both in academic and financial industry research analyzing, for the first time, SA across all asset classes (equity, fixed income and commodity). We find a general definition, which includes all SA strategies and propose a classification system measuring the strategies’ risk and return profile. This facilitates the inclusion of new strategies and measures as they emerge. Our analysis allows investors to have a common framework to evaluate investment opportunities and brings clarity in SA investing, guiding theoretical development and empirical testing. We also provide examples of potential future research directions.

Lazzarino, M., Berrill, J. and Šević, A. (2018) What Is Statistical Arbitrage? Theoretical Economics Letters, 8, 888-908. https://doi.org/10.4236/tel.2018.85063