# Statistical Arbitrage in Cryptocurrency Markets

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## Abstract

**:**

## 1. Introduction

- Development of an advanced, machine-learning-based statistical arbitrage approach for the cryptocurrency space: we build our approach on the ideas of Fischer and Krauss (2018); Huck (2009, 2010); Krauss et al. (2017); Moritz and Zimmermann (2014); Takeuchi and Lee (2013), who have developed similar methods for U.S. cash equities, but on much lower frequencies (days to months). With the present manuscript, we successfully show that relative-value arbitrage opportunities exist in this young and aspiring market, given that a random forest is able to produce daily returns of 7.1 bps after transaction costs.
- Consideration of microstructural effects: advancing to higher frequencies, e.g., minute-binned data, brings along substantial challenges. First, trading volume needs to be taken into account. In cash equities, many strategies are backtested on the closing price, which captures 7 percent of daily liquidity for NYSE listed stocks—see Intercontinental Exchange (2018). In stark contrast, liquidity needs to be carefully assessed for every minute bar in the cryptocurrency space, especially in case of smaller coins. We incorporate this effect in our study and only execute trades in case liquidity is present. Second, micro-structural effects, and especially the bid-ask bounce, need to be considered. We therefore introduce a lag between the price on which the prediction is generated, and the subsequent price on which execution is taking place. Hence, we eliminate the bid-ask bounce see, e.g., (Gatev et al. 2006) and we render the strategy realistic in the digital age, given that there is sufficient time for signal generation, order routing, and order execution.
- Shining light into the black box: machine learning models often have the downside of being intransparent and opaque. Hence, we analyze feature importances, and we compare the random forest to the transparent logistic regression. We find that both methods capture short-term characteristics in the data, with past returns over the past 60 min contributing most when explaining future returns over the subsequent 120 min.

## 2. Data and Software

#### 2.1. Data

#### 2.2. Software

## 3. Methodology

#### 3.1. Generation of Training and Trading Set

#### 3.2. Feature and Target Generation

#### 3.2.1. Features—Multiperiod Returns

#### 3.2.2. Targets

#### 3.3. Models

#### 3.3.1. Logistic regression

#### 3.3.2. Random forest

#### 3.4. Forecasting, Ranking and Trading

- Execution gap: We create the trading signal at the end of minute t and place the order for execution at the closing price of the following minute $t+1$. In other words, we introduce a one period gap between signal generation and execution to account for the time frame required for data processing, prediction making, and order management.
- Volume constraint (opening of position): A position is only opened when at least one unit of the currency pair is traded at the respective point in time—otherwise, the order is canceled and the amount of capital foreseen for the position is kept in cash for the two hours period.
- Volume constraint (closing of position): Once the position has reached its two hours lifetime, a closing order is triggered and executed at the first bar with sufficient volume.
- Elimination of starting point bias: To avoid any bias related to the starting point (point in time at which the first portfolio is opened), we open a new portfolio at every minute $t\in \{1,2,\dots ,120\}$ and average the results across the 120 portfolios that are opened at each time t.
- Transaction costs: We assume 15 bps per half turn, based on analyses on transaction costs and liquidity costs provided in Schnaubelt et al. (2019) on cryptocurrency limit order book data.

## 4. Results

#### 4.1. Trade-Level Results

- Positive mean returns: Both models yield positive and statistically significant mean returns with the RF (3.8 bps) clearly outperforming the LR (2.0 bps) by a factor of almost two. Looking at the contribution from long trades and short trades, we find that the latter are more profitable (−2.1 bps. vs. 5.6 bps (LR) and 0.2 bps. vs. 6.4 bps. (RF))—a finding that is likely driven by the overall decline of the cryptocurrency market during this period.
- Extreme price movements: Looking at the minimum (−42.8 percent) and maximum returns (34.4 percent), we find astonishingly high values given the two hour holding period. However, these observations can be attributed to the extreme price movements in cryptocurrency markets—see Osterrieder and Lorenz (2017). The 25 percent and 75 percent quartiles are less extreme with values between −1.2 and 1.3 percent for both models.
- Negative median: We further notice that both, the RF and the LR model, have negative median returns. In other words, more trades lead to a loss than to a profit. However, taking into account the magnitude of the profits and losses, we find that the profits surpass the losses by approximately 5 bps (LR) and 10 bps (RF) on average (simply speaking, more money is made when the model is right than lost when it is wrong). In result, the mean trade of the RF is positive, i.e., $0.49587\times 0.01774+0.50413\times (-0.01669)=0.00038>0$.
- Skewness and Kurtosis: Both, LR and RF exhibit positive skewness, which is a favorable property for investors, given that the right tail tends to be more pronounced than the left tail. By contrast, kurtosis values above 9 indicate leptokurtic behavior, and that significant risk lies in the extremes—see Osterrieder and Lorenz (2017).
- Differing number of trades: Finally, we observe that the number of executed trades differs between the two models as well as the long and short leg. As described in the previous section, our backtesting engine cancels orders in case no volume is available to execute the respective trade. We may therefore cautiously conclude that the RF model selects a larger share of less liquid coins (119,829 executed trades) compared to the LR model (158,408 trades). Note: the overall high number of trades results from the backtesting logic in which we open a new portfolio with three long orders and three short orders by the end of each minute to avoid starting point bias.

#### 4.2. Return Development over Time

- Panel A—daily return characteristics: With regard to mean return, the random forest surpasses the logistic regression by 2.2 bps per day (7.1 bps vs. 4.9 bps). We further observe that both, the maximum and minimum daily returns, are within reasonable levels of −2.6 percent (LR) and +2.1 percent (RF), respectively. The underlying reason is the large number of active positions at each point in time (see Section 3.4) which also explains the low standard deviation of 66 bps (LR) and 53 bps (RF). Looking at Bitcoin (BTC) and the general market (MKT), we find mean returns of −0.5 bps per day and −28.1 bps, respectively.
- Panel B—risk metrics: Panel B reveals favorable risk metrics for the random forest with a 1-percent value at risk of −1.0 percent compared to −1.5 percent for the logistic regression. Moreover, we find a significantly lower maximum drawdown of −2.4 percent for the RF and -5.9 percent for the LR compared to −26.7 percent for Bitcoin and −32.9 percent for the general market. The difference is caused by the short leg of the portfolio, i.e., the investment in the flop-3 coins which helps in eliminating market risk.
- Panel C—annualized risk-return metrics: Finally, panel C depicts annualized risk-return metrics. We observe annualized returns of 29.0 percent for the random forest and 18.8 percent for the logistic regression, compared to vastly negative results for the buy-and-hold benchmarks. Given the low volatility, these results translate into a Sharpe ratio of 1.4 (LR) and 2.5 (RF) respectively—hereby outperforming both Bitcoin and the general market by a clear margin.

#### 4.3. Beyond Returns—Shedding Light Into the Patterns Exploited for Trading

- Feature importance analysis: The upper half of the figure shows the features (explanatory variables) used by the random forest, sorted by feature importance in descending order. The most important features are the returns over the past 20, 40 and 60 min. In other words, the random forest pays most attention to the price development over the past hour. By contrast, the longer term price development (past 12–24 h) does not seem to have a substantial contribution to predicting the price change over the next two hours.
- Coefficient analysis: Looking at the lower part of the figure, we take advantage of the high transparency and explanatory value of the logistic regression model. The highest regression coefficient of approximately −6.5 belongs to the return over the past 20 min, followed by the coefficients for the 40 and 60 min returns. Moreover, we find that almost all regression coefficients exhibit a negative sign—in other words, the model likely produces a positive forecast (long), in case the respective coin has experienced a decline in the recent past (negative feature values which are multiplied with negative regression coefficients) and vice versa. We may therefore cautiously conclude that the model capitalizes on short-term mean-reversion—see Jegadeesh (1990); Lehmann (1990).

## 5. Discussion—Limits to Arbitrage

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BTC | Bitcoin |

LR | logistic regression |

MKT | market, i.e., an equal investment in all coins at the beginning of the trading period |

RF | random forest |

VaR | value at risk |

## Appendix A

**Table A1.**Overview of coins and corresponding exchanges used throughout this study. Note: All coins are denominated in USD prices as provided by www.cryptocompare.com.

No | Coin | Exchange | No | Coin | Exchange | |
---|---|---|---|---|---|---|

1 | ADA | BitTrex | 21 | QTUM | Bitfinex | |

2 | BCH | Bitfinex | 22 | RDD | Yobit | |

3 | BCN | HitBTC | 23 | SAN | Bitfinex | |

4 | BTC | Bitfinex | 24 | SNT | Bitfinex | |

5 | BTG | Bitfinex | 25 | STRAT | HitBTC | |

6 | CND | HitBTC | 26 | TNB | Bitfinex | |

7 | CVC | HitBTC | 27 | TNT | HitBTC | |

8 | DASH | Bitfinex | 28 | TRX | Bitfinex | |

9 | DATA | Bitfinex | 29 | USDT | Kraken | |

10 | EOS | Bitfinex | 30 | VIB | HitBTC | |

11 | ETC | Bitfinex | 31 | WAVES | Yobit | |

12 | ETH | Bitfinex | 32 | XDN | HitBTC | |

13 | ETP | Bitfinex | 33 | XEM | Yobit | |

14 | GNT | Bitfinex | 34 | XLM | Poloniex | |

15 | LTC | Bitfinex | 35 | XMR | Bitfinex | |

16 | MANA | Bitfinex | 36 | XRP | Bitfinex | |

17 | NEO | Bitfinex | 37 | XVG | BitTrex | |

18 | NXT | Poloniex | 38 | YOYOW | Bitfinex | |

19 | OMG | Bitfinex | 39 | ZEC | Bitfinex | |

20 | QASH | Bitfinex | 40 | ZRX | Bitfinex |

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1 | The emotional valence and opinion polarization are computed on a daily basis as proposed by Warriner et al. (2013). |

2 | |

3 | Not all time-series examined are complete in the sense that they cover the whole period from January to September 2018. This could be due to several reasons such as the delisting of a coin. It is noteworthy that such time-series are not eliminated but traded according to the available data. |

4 | More precisely, by executing at the opening price of minute $t+1$, we still leave a small gap compared to an execution at the closing price of minute t (which is used to make the prediction). |

**Figure 1.**Development of financial performance of random forest model (RF) when investing in the top-3 and flop-3 coins vs. Bitcoin (BTC) and general market (MKT), i.e., an equal investment in all coins at the beginning of the trading period.

**Figure 2.**Feature importance extracted from the random forest and regression coefficients for the logistic regression model. The features (explanatory variables) are sorted in descending order based on their importance extracted from the random forest model. The coefficients of the logistic regression model are plotted following the same order.

**Table 1.**Key return characteristics on the level of individual round trip trades for the logistic regression (LR) and the random forest model (RF) when investing in the top-3 and flop-3 coins, after transaction costs of 30 bps for the round trip trade.

LR | RF | |||||
---|---|---|---|---|---|---|

Long | Short | Total | Long | Short | Total | |

No. trades | 73319 | 85089 | 158408 | 49689 | 70140 | 119829 |

Mean return | −0.00021 | 0.00056 | 0.00020 | 0.00002 | 0.00064 | 0.00038 |

Standard error | 0.00009 | 0.00009 | 0.00006 | 0.00011 | 0.00010 | 0.00008 |

t-Statistic | −2.35284 | 6.19182 | 3.17475 | 0.19865 | 6.39796 | 5.14330 |

Minimum | −0.17736 | −0.42764 | −0.42764 | −0.17649 | −0.42764 | −0.42764 |

25% Quantile | −0.01169 | −0.01086 | −0.01127 | −0.01140 | −0.01064 | −0.01094 |

Median | −0.00141 | 0.00109 | −0.00004 | −0.00192 | 0.00095 | −0.00015 |

75% Quantile | 0.00993 | 0.01313 | 0.01172 | 0.00990 | 0.01299 | 0.01183 |

Maximum | 0.29043 | 0.34424 | 0.34424 | 0.26296 | 0.34424 | 0.34424 |

Share > 0 | 0.46677 | 0.52671 | 0.49897 | 0.45622 | 0.52395 | 0.49587 |

Standard dev. | 0.02449 | 0.02656 | 0.02563 | 0.02490 | 0.02653 | 0.02587 |

Skewness | 1.00453 | −0.44146 | 0.14509 | 1.03417 | −0.38629 | 0.14070 |

Kurtosis | 9.26031 | 9.46260 | 9.41992 | 8.98506 | 9.55134 | 9.36387 |

Mean return positive trade | 0.01726 | 0.01750 | 0.01739 | 0.01802 | 0.01757 | 0.01774 |

Mean return negative trade | −0.01551 | −0.01828 | −0.01691 | −0.01508 | −0.01799 | −0.01669 |

**Table 2.**Daily and annualized risk-return metrics for the logistic regression (LR) and the random forest model (RF) model when investing in the top-3 and flop-3 coins, versus Bitcoin (BTC) and the general market (MKT), i.e., an equal investment in all coins at the beginning of the trading period. Panel A depicts daily return characteristics, panel B depicts risk and panel C annualized risk-return metrics.

LR | RF | BTC | MKT | ||
---|---|---|---|---|---|

A | Mean return | 0.00049 | 0.00071 | −0.00005 | −0.00281 |

Standard dev. | 0.00661 | 0.00534 | 0.03260 | 0.03680 | |

Minimum | −0.02583 | −0.01027 | −0.10016 | −0.10805 | |

25% Quantile | −0.00323 | −0.00212 | −0.01598 | −0.02270 | |

Median | 0.00025 | 0.00020 | 0.00111 | 0.00069 | |

75% Quantile | 0.00388 | 0.00324 | 0.01458 | 0.01829 | |

Maximum | 0.01920 | 0.02115 | 0.08777 | 0.11555 | |

Share > 0 | 0.51807 | 0.53012 | 0.50602 | 0.50602 | |

B | Historic VaR 1% | −0.01523 | −0.01025 | −0.09112 | −0.10461 |

Historic VaR 5% | −0.00809 | −0.00756 | −0.05482 | −0.05978 | |

Maximum drawdown | −0.05892 | −0.02432 | −0.26738 | −0.32908 | |

C | Annual return | 0.18762 | 0.29012 | −0.18754 | −0.71640 |

Annual volatility | 0.12632 | 0.10203 | 0.62284 | 0.70310 | |

Sharpe ratio | 1.42394 | 2.54785 | −0.02755 | −1.46060 | |

Sortino ratio | 2.16255 | 4.51777 | −0.03787 | −1.90273 |

**Table 3.**Key return characteristics on the level of individual round trip trades for the random forest (RF) model when investing in the top-3 and flop-3 coins, after transaction costs of 30 bps. Each column represents the gap between signal generation and signal execution, i.e., gap 0 refers to signal generation at the closing price of bar t and execution at the opening price of bar $t+1$. Gap 1 refers to a delayed execution at the closing price of bar $t+1$, gap 2 to a delayed execution at the closing price of bar $t+2$, and so forth.

Gap 0 | Gap 1 | Gap 2 | Gap 3 | Gap 4 | Gap 5 | |
---|---|---|---|---|---|---|

No. trades | 119829 | 119829 | 118948 | 118424 | 118055 | 117630 |

Mean return | 0.00205 | 0.00038 | 0.00024 | 0.00016 | 0.00009 | −0.00001 |

Standard error | 0.00008 | 0.00008 | 0.00008 | 0.00008 | 0.00008 | 0.00007 |

t−Statistic | 26.97626 | 5.14330 | 3.24117 | 2.15184 | 1.15309 | −0.09429 |

Minimum | −0.42764 | −0.42764 | −0.40397 | −0.43317 | −0.40940 | −0.37498 |

25% Quantile | −0.00974 | −0.01094 | −0.01104 | −0.01113 | −0.01120 | −0.01126 |

Median | 0.00097 | −0.00015 | −0.00031 | −0.00043 | −0.00053 | −0.00061 |

75% Quantile | 0.01330 | 0.01183 | 0.01163 | 0.01146 | 0.01135 | 0.01124 |

Maximum | 0.34424 | 0.34424 | 0.34424 | 0.34424 | 0.34424 | 0.34424 |

Share > 0 | 0.52342 | 0.49587 | 0.49294 | 0.49070 | 0.48810 | 0.48605 |

Standard dev. | 0.02626 | 0.02587 | 0.02574 | 0.02566 | 0.02566 | 0.02552 |

Skewness | 0.32786 | 0.14070 | 0.11708 | 0.09076 | 0.04651 | 0.09360 |

Kurtosis | 9.19105 | 9.36387 | 9.14659 | 9.35075 | 9.76571 | 9.47677 |

Mean return positive trade | 0.01869 | 0.01774 | 0.01763 | 0.01756 | 0.01755 | 0.01745 |

Mean return negative trade | −0.01623 | −0.01669 | −0.01666 | −0.01660 | −0.01656 | −0.01651 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fischer, T.G.; Krauss, C.; Deinert, A. Statistical Arbitrage in Cryptocurrency Markets. *J. Risk Financial Manag.* **2019**, *12*, 31.
https://doi.org/10.3390/jrfm12010031

**AMA Style**

Fischer TG, Krauss C, Deinert A. Statistical Arbitrage in Cryptocurrency Markets. *Journal of Risk and Financial Management*. 2019; 12(1):31.
https://doi.org/10.3390/jrfm12010031

**Chicago/Turabian Style**

Fischer, Thomas Günter, Christopher Krauss, and Alexander Deinert. 2019. "Statistical Arbitrage in Cryptocurrency Markets" *Journal of Risk and Financial Management* 12, no. 1: 31.
https://doi.org/10.3390/jrfm12010031