# Statistical Arbitrage with Mean-Reverting Overnight Price Gaps on High-Frequency Data of the S&P 500

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Barndorff–Nielsen and Shephard Jump Test

^{th}period. Based on this case, the variation of the jump term can be isolated by subtracting $BPV$ from $RV$:

#### 2.2. Jump Detection Scheme

## 3. Event Study of the S&P 500 Index

## 4. Back-Testing Framework

#### 4.1. Data and Software

`R`(R Core Team 2019). For computation-intensive calculations, we used both the general-purpose programming language

`C++`and on-demand cloud computing platforms with virtual computer clusters that are available 24/7 via the Internet.

#### 4.2. Formation Period

#### 4.3. Trading Period

- We observe a negative price gap during the night, i.e., the stock is undervalued. Consequently, we go long in the stock.
- We observe a positive price gap during the night, i.e., the stock is overvalued. Consequently, we go short in the stock.

#### S&P 500 Buy-and-Hold Strategy (BHS)

#### Fixed Threshold Strategy (FTS)

#### General Volatility Strategy (GVS)

#### Reverting Volatility Strategy (RVS)

## 5. Results

#### 5.1. Risk-Return Characteristics

#### 5.2. Sub-Period Analysis

#### 5.3. Robustness Check

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Detailed development of the $ACR$ with p-values of the two-sided t-test from January 1998–December 2015. $ACR$ denotes the average cumulative returns.

Positive Gap | Negative Gap | |||
---|---|---|---|---|

Target Time | $\mathit{ACR}$ in % | $\mathit{p}$-Value | $\mathit{ACR}$ in % | $\mathit{p}$-Value |

5 min | 0.0056 | 0.1180 | 0.0013 | 0.8140 |

10 min | −0.0037 | 0.4690 | 0.0178 | 0.0180 |

15 min | −0.0155 | 0.0160 | 0.0334 | 0.0000 |

20 min | −0.0229 | 0.0020 | 0.0293 | 0.0060 |

25 min | −0.0248 | 0.0030 | 0.0355 | 0.0030 |

30 min | −0.0207 | 0.0260 | 0.0310 | 0.0160 |

35 min | −0.0150 | 0.1580 | 0.0327 | 0.0320 |

40 min | −0.0193 | 0.0850 | 0.0284 | 0.0670 |

45 min | −0.0245 | 0.0400 | 0.0314 | 0.0480 |

50 min | −0.0227 | 0.0670 | 0.0318 | 0.0560 |

55 min | −0.0236 | 0.0610 | 0.0415 | 0.0160 |

60 min | −0.0255 | 0.0490 | 0.0420 | 0.0150 |

65 min | −0.0209 | 0.1230 | 0.0346 | 0.0490 |

70 min | −0.0231 | 0.1000 | 0.0333 | 0.0690 |

75 min | −0.0257 | 0.0810 | 0.0387 | 0.0470 |

80 min | −0.0287 | 0.0600 | 0.0417 | 0.0310 |

85 min | −0.0301 | 0.0490 | 0.0432 | 0.0280 |

90 min | −0.0256 | 0.1030 | 0.0415 | 0.0360 |

95 min | −0.0230 | 0.1540 | 0.0373 | 0.0690 |

100 min | −0.0286 | 0.0810 | 0.0338 | 0.0990 |

105 min | −0.0316 | 0.0550 | 0.0334 | 0.1060 |

110 min | −0.0288 | 0.0820 | 0.0334 | 0.1070 |

115 min | −0.0294 | 0.0780 | 0.0382 | 0.0740 |

120 min | −0.0248 | 0.1360 | 0.0328 | 0.1260 |

130 min | −0.0189 | 0.2620 | 0.0307 | 0.1670 |

140 min | −0.0207 | 0.2290 | 0.0256 | 0.2660 |

150 min | −0.0224 | 0.2150 | 0.0343 | 0.1430 |

160 min | −0.0177 | 0.3300 | 0.0313 | 0.1920 |

170 min | −0.0156 | 0.3940 | 0.0295 | 0.2230 |

180 min | −0.0091 | 0.6250 | 0.0248 | 0.3160 |

190 min | −0.0066 | 0.7270 | 0.0276 | 0.2620 |

200 min | −0.0069 | 0.7180 | 0.0296 | 0.2320 |

210 min | −0.0111 | 0.5700 | 0.0340 | 0.1770 |

220 min | −0.0070 | 0.7210 | 0.0304 | 0.2370 |

230 min | −0.0027 | 0.8880 | 0.0258 | 0.3260 |

240 min | −0.0038 | 0.8450 | 0.0288 | 0.2650 |

250 min | −0.0068 | 0.7300 | 0.0254 | 0.3340 |

260 min | −0.0121 | 0.5460 | 0.0302 | 0.2640 |

270 min | −0.0175 | 0.3850 | 0.0314 | 0.2470 |

280 min | −0.0218 | 0.2890 | 0.0275 | 0.3260 |

290 min | −0.0212 | 0.3130 | 0.0241 | 0.3960 |

310 min | −0.0189 | 0.3880 | 0.0347 | 0.2390 |

330 min | −0.0131 | 0.5740 | 0.0309 | 0.3140 |

350 min | −0.0115 | 0.6360 | 0.0344 | 0.2820 |

370 min | −0.0073 | 0.7720 | 0.0276 | 0.4320 |

390 min | 0.0229 | 0.4070 | −0.0052 | 0.8900 |

391 min | 0.0236 | 0.3900 | −0.0052 | 0.8870 |

**Table A2.**Annualized risk-return measures for BHS, FTS, GVS, RVS, and JDS for sub-periods of 3 years from January 1998–December 2015.

Before Transaction Costs | After Transaction Costs | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

BHS | FTS | GVS | RVS | JDS | FTS | GVS | RVS | JDS | ||

1998–2000 | Mean return | 0.0624 | 1.1054 | 1.3520 | 1.8718 | 0.1674 | 0.2731 | 0.4226 | 0.7376 | −0.2956 |

Mean excess return | 0.0106 | 1.0030 | 1.2378 | 1.7324 | 0.1105 | 0.2111 | 0.3534 | 0.6531 | −0.3300 | |

Standard deviation | 0.2055 | 0.1037 | 0.1172 | 0.1193 | 0.1439 | 0.1037 | 0.1172 | 0.1193 | 0.1435 | |

Downside deviation | 0.1442 | 0.0449 | 0.0546 | 0.0515 | 0.1035 | 0.0591 | 0.0687 | 0.0641 | 0.1191 | |

Sharpe ratio | 0.0516 | 9.6750 | 10.5590 | 14.5236 | 0.7682 | 2.0364 | 3.0143 | 5.4756 | −2.2991 | |

Sortino ratio | 0.4324 | 24.6231 | 24.7455 | 36.3203 | 1.6165 | 4.6206 | 6.1472 | 11.5012 | −2.4817 | |

2001–2003 | Mean return | −0.0781 | 0.5919 | 0.7374 | 1.7218 | 1.5412 | −0.0379 | 0.0502 | 0.6467 | 0.6408 |

Mean excess return | −0.0978 | 0.5580 | 0.7005 | 1.6640 | 1.4872 | −0.0584 | 0.0278 | 0.6116 | 0.6058 | |

Standard deviation | 0.2184 | 0.1214 | 0.1463 | 0.1636 | 0.2037 | 0.1214 | 0.1463 | 0.1636 | 0.2024 | |

Downside deviation | 0.1538 | 0.0685 | 0.0827 | 0.0823 | 0.1057 | 0.0849 | 0.0981 | 0.0966 | 0.1187 | |

Sharpe ratio | −0.4478 | 4.5978 | 4.7880 | 10.1681 | 7.2995 | −0.4816 | 0.1902 | 3.7375 | 2.9939 | |

Sortino ratio | −0.5080 | 8.6396 | 8.9161 | 20.9326 | 14.5766 | −0.4467 | 0.5116 | 6.6958 | 5.3993 | |

2004–2006 | Mean return | 0.0787 | 0.4021 | 0.2554 | 0.9144 | 1.3252 | −0.1529 | −0.2417 | 0.1574 | 0.4035 |

Mean excess return | 0.0475 | 0.3616 | 0.2191 | 0.8592 | 1.2582 | −0.1774 | −0.2636 | 0.1240 | 0.3630 | |

Standard deviation | 0.1046 | 0.0601 | 0.0656 | 0.1019 | 0.1278 | 0.0601 | 0.0656 | 0.1019 | 0.1276 | |

Downside deviation | 0.0720 | 0.0323 | 0.0396 | 0.0499 | 0.0613 | 0.0483 | 0.0562 | 0.0650 | 0.0758 | |

Sharpe ratio | 0.4542 | 6.0149 | 3.3407 | 8.4349 | 9.8430 | −2.9506 | −4.0187 | 1.2171 | 2.8453 | |

Sortino ratio | 1.0935 | 12.4580 | 6.4486 | 18.3127 | 21.6319 | −3.1672 | −4.2977 | 2.4206 | 5.3199 | |

2007–2009 | Mean return | −0.1177 | 1.1060 | 1.2654 | 2.5881 | 5.8734 | 0.2735 | 0.3701 | 1.1720 | 3.1502 |

Mean excess return | −0.1358 | 1.0628 | 1.2189 | 2.5147 | 5.7331 | 0.2473 | 0.3419 | 1.1274 | 3.0653 | |

Standard deviation | 0.2995 | 0.1500 | 0.1991 | 0.2193 | 0.3477 | 0.1500 | 0.1991 | 0.2193 | 0.3470 | |

Downside deviation | 0.2209 | 0.0687 | 0.0977 | 0.1009 | 0.1323 | 0.0823 | 0.1111 | 0.1134 | 0.1437 | |

Sharpe ratio | −0.4534 | 7.0874 | 6.1215 | 11.4693 | 16.4905 | 1.6491 | 1.7170 | 5.1421 | 8.8346 | |

Sortino ratio | −0.5328 | 16.0953 | 12.9488 | 25.6392 | 44.4096 | 3.3231 | 3.3318 | 10.3376 | 21.9222 | |

2010–2012 | Mean return | 0.0671 | 0.1413 | 0.1445 | 0.3591 | 0.6918 | −0.3107 | −0.3088 | −0.1789 | 0.0215 |

Mean excess return | 0.0663 | 0.1404 | 0.1436 | 0.3581 | 0.6905 | −0.3112 | −0.3093 | −0.1795 | 0.0207 | |

Standard deviation | 0.1856 | 0.0538 | 0.0651 | 0.1017 | 0.1543 | 0.0538 | 0.0651 | 0.1017 | 0.1540 | |

Downside deviation | 0.1341 | 0.0328 | 0.0415 | 0.0658 | 0.0905 | 0.0512 | 0.0590 | 0.0815 | 0.1061 | |

Sharpe ratio | 0.3572 | 2.6115 | 2.2080 | 3.5202 | 4.4759 | −5.7880 | −4.7542 | −1.7647 | 0.1348 | |

Sortino ratio | 0.5004 | 4.3142 | 3.4861 | 5.4583 | 7.6420 | −6.0655 | −5.2322 | −2.1942 | 0.2029 | |

2013–2015 | Mean return | 0.1219 | 0.2262 | 0.2275 | 1.0296 | 1.5703 | −0.2593 | −0.2585 | 0.2272 | 0.6838 |

Mean excess return | 0.1219 | 0.2262 | 0.2275 | 1.0296 | 1.5703 | −0.2593 | −0.2585 | 0.2272 | 0.6838 | |

Standard deviation | 0.1281 | 0.0487 | 0.0611 | 0.1022 | 0.1392 | 0.0487 | 0.0611 | 0.1022 | 0.1372 | |

Downside deviation | 0.0904 | 0.0289 | 0.0367 | 0.0506 | 0.0535 | 0.0455 | 0.0535 | 0.0647 | 0.0662 | |

Sharpe ratio | 0.9516 | 4.6472 | 3.7216 | 10.0702 | 11.2826 | −5.3274 | −4.2295 | 2.2221 | 4.9853 | |

Sortino ratio | 1.3484 | 7.8150 | 6.2034 | 20.3644 | 29.3617 | −5.7035 | −4.8294 | 3.5130 | 10.3278 |

## References

- Andersen, Torben G., and Tim Bollerslev. 1998. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39: 885. [Google Scholar] [CrossRef]
- Andersen, Torben G., Tim Bollerslev, Francis X. Diebold, and Paul Labys. 2001. The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96: 42–55. [Google Scholar] [CrossRef]
- Andersen, Torben G., Tim Bollerslev, Francis X. Diebold, and Paul Labys. 2003. Modeling and forecasting realized volatility. Econometrica 71: 579–625. [Google Scholar] [CrossRef]
- Andersen, Torben G., Tim Bollerslev, Per Frederiksen, and Morten Ørregaard Nielsen. 2010. Continuous-time models, realized volatilities, and testable distributional implications for daily stock returns. Journal of Applied Econometrics 25: 233–61. [Google Scholar] [CrossRef]
- Avellaneda, Marco, and Jeong-Hyun Lee. 2010. Statistical arbitrage in the US equities market. Quantitative Finance 10: 761–82. [Google Scholar] [CrossRef]
- Balance. 2019. Make Money Personal. Available online: https://www.thebalance.com (accessed on 27 February 2019).
- Banerjee, Prithviraj S., James S. Doran, and David R. Peterson. 2007. Implied volatility and future portfolio returns. Journal of Banking & Finance 31: 3183–99. [Google Scholar]
- Bariviera, Aurelio F. 2017. The inefficiency of bitcoin revisited: A dynamic approach. Economics Letters 161: 1–4. [Google Scholar] [CrossRef]
- Barndorff-Nielsen, Ole E., and Neil Shephard. 2002. Estimating quadratic variation using realized variance. Journal of Applied Econometrics 17: 457–77. [Google Scholar] [CrossRef]
- Barndorff-Nielsen, Ole E., and Neil Shephard. 2004. Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2: 1–37. [Google Scholar] [CrossRef]
- Barndorff-Nielsen, Ole E., and Neil Shephard. 2006. Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4: 1–30. [Google Scholar] [CrossRef]
- Bedowska-Sojka, Barbara. 2013. Macroeconomic news effects on the stock markets in intraday data. Central European Journal of Economic Modelling and Econometrics 5: 249–69. [Google Scholar]
- Bertram, William K. 2010. Analytic solutions for optimal statistical arbitrage trading. Physica A: Statistical Mechanics and Its Applications 389: 2234–43. [Google Scholar] [CrossRef]
- Business Insider. 2015. Markets Insider by Intelligence. Available online: https://www.businessinsider.com/ (accessed on 27 February 2019).
- Caporale, Guglielmo Maria, and Alex Plastun. 2017. Price gaps: Another market anomaly? Investment Analysts Journal 46: 279–93. [Google Scholar] [CrossRef]
- Cartea, Álvaro, Sebastian Jaimungal, and José Penalva. 2015. Algorithmic and High-Frequency Trading. Cambridge: Cambridge University Press. [Google Scholar]
- Chen, Huafeng, Shaojun Chen, Zhuo Chen, and Feng Li. 2017. Empirical investigation of an equity pairs trading strategy. Management Science 65: 370–89. [Google Scholar] [CrossRef]
- Cont, Rama. 2001. Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1: 223–36. [Google Scholar] [CrossRef]
- Do, Binh, and Robert Faff. 2010. Does simple pairs trading still work? Financial Analysts Journal 66: 83–95. [Google Scholar] [CrossRef]
- Do, Binh, and Robert Faff. 2012. Are pairs trading profits robust to trading costs? Journal of Financial Research 35: 261–87. [Google Scholar] [CrossRef]
- Endres, Sylvia, and Johannes Stübinger. 2019a. Optimal trading strategies for Lévy-driven Ornstein-Uhlenbeck processes. Applied Economics. forthcoming. [Google Scholar] [CrossRef]
- Endres, Sylvia, and Johannes Stübinger. 2019b. Regime-switching modeling of high-frequency stock returns with Lévy jumps. Quantitative Finance. forthcoming. [Google Scholar] [CrossRef]
- Evans, Kevin P. 2011. Intraday jumps and us macroeconomic news announcements. Journal of Banking & Finance 35: 2511–27. [Google Scholar]
- Frömmel, Michael, Xing Han, and Frederick van Gysegem. 2015. Further evidence on foreign exchange jumps and news announcements. Emerging Markets Finance and Trade 51: 774–87. [Google Scholar] [CrossRef]
- Fung, Alexander Kwok-Wah, Debby MY Mok, and Kin Lam. 2000. Intraday price reversals for index futures in the US and Hong Kong. Journal of Banking & Finance 24: 1179–201. [Google Scholar]
- Gatev, Evan, William N. Goetzmann, and K. Geert Rouwenhorst. 2006. Pairs trading: Performance of a relative-value arbitrage rule. Review of Financial Studies 19: 797–827. [Google Scholar] [CrossRef]
- Göncü, Ahmet, and Erdinc Akyildirim. 2016. A stochastic model for commodity pairs trading. Quantitative Finance 16: 1843–57. [Google Scholar] [CrossRef]
- Grant, James L, Avner Wolf, and Susana Yu. 2005. Intraday price reversals in the US stock index futures market: A 15-year study. Journal of Banking & Finance 29: 1311–27. [Google Scholar]
- Hogan, Steve, Robert Jarrow, Melvyn Teo, and Mitch Warachka. 2004. Testing market efficiency using statistical arbitrage with applications to momentum and value strategies. Journal of Financial Economics 73: 525–65. [Google Scholar] [CrossRef]
- Huang, Xin, and George Tauchen. 2005. The relative contribution of jumps to total price variance. Journal of Financial Econometrics 3: 456–99. [Google Scholar] [CrossRef]
- Knoll, Julian, Johannes Stübinger, and Michael Grottke. 2018. Exploiting social media with higher-order factorization machines: Statistical arbitrage on high-frequency data of the S&P 500. Quantitative Finance. forthcoming. [Google Scholar]
- Larsson, Stig, Carl Lindberg, and Marcus Warfheimer. 2013. Optimal closing of a pair trade with a model containing jumps. Applications of Mathematics 58: 249–68. [Google Scholar] [CrossRef]
- Leung, Tim, and Xin Li. 2015. Optimal mean reversion trading with transaction costs and stop-loss exit. International Journal of Theoretical and Applied Finance 18: 1550020. [Google Scholar] [CrossRef]
- Liu, Bo, Lo-Bin Chang, and Hélyette Geman. 2017. Intraday pairs trading strategies on high frequency data: The case of oil companies. Quantitative Finance 17: 87–100. [Google Scholar] [CrossRef]
- Lubnau, Thorben, and Neda Todorova. 2015. Trading on mean-reversion in energy futures markets. Energy Economics 51: 312–19. [Google Scholar] [CrossRef]
- Miao, George J. 2014. High frequency and dynamic pairs trading based on statistical arbitrage using a two-stage correlation and cointegration approach. International Journal of Economics and Finance 6: 96–110. [Google Scholar] [CrossRef]
- Mitchell, John B. 2010. Soybean futures crush spread arbitrage: Trading strategies and market efficiency. Journal of Risk and Financial Management 3: 63–96. [Google Scholar] [CrossRef]
- Nakajima, Tadahiro. 2019. Expectations for statistical arbitrage in energy futures markets. Journal of Risk and Financial Management 12: 14. [Google Scholar] [CrossRef]
- Pole, Andrew. 2011. Statistical Arbitrage: Algorithmic Trading Insights and Techniques. Hoboken: John Wiley & Sons. [Google Scholar]
- Poterba, James M., and Lawrence H. Summers. 1988. Mean reversion in stock prices: Evidence and implications. Journal of Financial Economics 22: 27–59. [Google Scholar] [CrossRef]
- Prager, Richard, Supurna Vedbrat, Chris Vogel, and Even Cameron Watt. 2012. Got Liquidity? New York: BlackRock Investment Institute. [Google Scholar]
- QuantQuote. 2016. QuantQuote Market Data and Software. Available online: https://quantquote.com (accessed on 27 February 2019).
- R Core Team. 2019. Stats: A Language and Environment for Statistical Computing. R package. Wien: R Core Team. [Google Scholar]
- Rad, Hossein, Rand Kwong Yew Low, and Robert Faff. 2016. The profitability of pairs trading strategies: Distance, cointegration and copula methods. Quantitative Finance 16: 1541–58. [Google Scholar] [CrossRef]
- Rombouts, Jeroen V. K., and Lars Stentoft. 2011. Multivariate option pricing with time varying volatility and correlations. Journal of Banking & Finance 35: 2267–81. [Google Scholar]
- S&P Dow Jones Indices. 2015. S&P Global—Equity S&P 500 Index. Available online: https://us.spindices.com/indices/equity/sp-500 (accessed on 27 February 2019).
- Stübinger, Johannes. 2018. Statistical arbitrage with optimal causal paths on high-frequency data of the S&P 500. Quantitative Finance. forthcoming. [Google Scholar]
- Stübinger, Johannes, and Jens Bredthauer. 2017. Statistical arbitrage pairs trading with high-frequency data. International Journal of Economics and Financial Issues 7: 650–62. [Google Scholar]
- Stübinger, Johannes, and Sylvia Endres. 2018. Pairs trading with a mean-reverting jump-diffusion model on high-frequency data. Quantitative Finance 18: 1735–51. [Google Scholar] [CrossRef]
- Stübinger, Johannes, and Julian Knoll. 2018. Beat the bookmaker—Winning football bets with machine learning (Best Application Paper). Paper presented at 38th SGAI International Conference on Artificial Intelligence, Cambridge, UK, December 11–13; pp. 219–33. [Google Scholar]
- Stübinger, Johannes, Benedikt Mangold, and Christopher Krauss. 2018. Statistical arbitrage with vine copulas. Quantitative Finance 18: 1831–49. [Google Scholar] [CrossRef]
- Suleman, Muhammad Tahir. 2012. Stock market reaction to good and bad political news. Asian Journal of Finance & Accounting 4: 299–312. [Google Scholar]
- Vidyamurthy, Ganapathy. 2004. Pairs Trading: Quantitative Methods and Analysis. Hoboken: John Wiley & Sons. [Google Scholar]
- Voya Investment Management. 2016. The Impact of Equity Market Fragmentation and Dark Pools on Trading and Alpha Generation. Available online: https://investments.voya.com (accessed on 27 February 2019).

1 | If less than 10 shares satisfied the condition of Andersen et al. (2010), we traded accordingly less. However, this case is extremely rare. |

**Figure 1.**Histogram of positive and negative overnight gaps, which were identified by the BNS jump test, from January 1998–December 2015.

**Figure 2.**Development of positive and negative overnight gaps, which were identified by the BNS jump test, from 1998–2015.

**Figure 3.**Average cumulative returns (%) after positive and negative overnight gaps, which were identified by the BNS jump test, from January 1998–December 2015.

**Figure 4.**Development of an investment of 1 USD after transaction costs for FTS, GVS, RVS, and JDS (first column) compared to the S&P 500 buy-and-hold-strategy (BHS) (second column). The time period from January 1998–December 2015 is divided into three sub-periods (March 1998/December 2006, January 2007/December 2009, January 2010/December 2015).

**Table 1.**Characteristics of positive and negative overnight gaps, which are identified by the Barndorff–Nielsen and Shephard (BNS) jump test, from January 1998–December 2015.

Positive Gap | Negative Gap | |
---|---|---|

Number of gaps | 1154 | 974 |

Mean | 0.0060 | −0.0067 |

Minimum | 0.0003 | −0.0764 |

Quartile 1 | 0.0029 | −0.0085 |

Median | 0.0045 | −0.0049 |

Quartile 3 | 0.0072 | −0.0029 |

Maximum | 0.0602 | −0.0005 |

Standard deviation | 0.0053 | 0.0063 |

Skewness | 3.2771 | −3.8289 |

Kurtosis | 20.8453 | 29.3100 |

**Table 2.**Overview of the characteristics of the S&P 500 buy-and-hold strategy (BHS), fixed threshold strategy (FTS), generalized volatility strategy (GVS), reverting volatility strategy (RVS), and jump-diffusion strategy (JDS).

Characteristic | BHS | FTS | GVS | RVS | JDS |
---|---|---|---|---|---|

Individual | No | Yes | Yes | Yes | Yes |

Volatility | No | No | Yes | Yes | Yes |

Mean-reverting | No | No | No | Yes | Yes |

Jump-diffusion | No | No | No | No | Yes |

**Table 3.**Daily return characteristics and risk metrics for BHS, FTS, GVS, RVS, and JDS from January 1998–December 2015. NW denotes Newey–West standard errors with 1-lag correction and CVaR the conditional value at risk.

Before Transaction Costs | After Transaction Costs | ||||||||
---|---|---|---|---|---|---|---|---|---|

BHS | FTS | GVS | RVS | JDS | FTS | GVS | RVS | JDS | |

Mean return | 0.0002 | 0.0017 | 0.0019 | 0.0033 | 0.0036 | −0.0003 | −0.0001 | 0.0013 | 0.0017 |

Standard error (NW) | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0002 |

t-Statistic (NW) | 0.8617 | 17.9433 | 15.8454 | 23.0251 | 16.4912 | −2.5816 | −1.1504 | 9.2534 | 7.8870 |

Minimum | −0.0947 | −0.0410 | −0.0521 | −0.0544 | −0.1169 | −0.0430 | −0.0541 | −0.0564 | −0.1187 |

Quartile 1 | −0.0056 | −0.0012 | −0.0016 | −0.0013 | −0.0021 | −0.0032 | −0.0036 | −0.0033 | −0.0041 |

Median | 0.0005 | 0.0012 | 0.0013 | 0.0030 | 0.0028 | −0.0008 | −0.0007 | 0.0010 | 0.0008 |

Quartile 3 | 0.0061 | 0.0040 | 0.0046 | 0.0076 | 0.0085 | 0.0020 | 0.0026 | 0.0056 | 0.0065 |

Maximum | 0.1096 | 0.0604 | 0.0776 | 0.0889 | 0.1947 | 0.0584 | 0.0756 | 0.0869 | 0.1923 |

Standard deviation | 0.0126 | 0.0062 | 0.0077 | 0.0090 | 0.0129 | 0.0062 | 0.0077 | 0.0090 | 0.0128 |

Skewness | −0.1987 | 1.2552 | 1.2987 | 0.9082 | 2.7078 | 1.2552 | 1.2987 | 0.9082 | 2.6990 |

Kurtosis | 7.5278 | 9.5525 | 11.7337 | 8.3119 | 29.7136 | 9.5525 | 11.7337 | 8.3119 | 29.8425 |

Historical VaR 1% | −0.0350 | −0.0136 | −0.0178 | −0.0187 | −0.0255 | −0.0156 | −0.0198 | −0.0207 | −0.0275 |

Historical CVaR 1% | −0.0506 | −0.0186 | −0.0249 | −0.0263 | −0.0346 | −0.0206 | −0.0269 | −0.0283 | −0.0365 |

Historical VaR 5% | −0.0197 | −0.0068 | −0.0078 | −0.0093 | −0.0129 | −0.0088 | −0.0098 | −0.0113 | −0.0149 |

Historical CVaR 5% | −0.0302 | −0.0110 | −0.0141 | −0.0155 | −0.0209 | −0.0130 | −0.0161 | −0.0175 | −0.0228 |

Maximum drawdown | 0.6433 | 0.0667 | 0.0860 | 0.1012 | 0.2707 | 0.8784 | 0.8947 | 0.5991 | 0.6817 |

Share with return ≥ 0 | 0.5313 | 0.6327 | 0.6200 | 0.6782 | 0.6715 | 0.4179 | 0.4288 | 0.5592 | 0.5841 |

**Table 4.**Annualized risk-return measures for BHS, FTS, GVS, RVS, and JDS from January 1998–December 2015.

Before Transaction Costs | After Transaction Costs | ||||||||
---|---|---|---|---|---|---|---|---|---|

BHS | FTS | GVS | RVS | JDS | FTS | GVS | RVS | JDS | |

Mean return | 0.0181 | 0.5456 | 0.5874 | 1.2959 | 1.4472 | −0.0659 | −0.0407 | 0.3885 | 0.5147 |

Mean excess return | −0.0022 | 0.5149 | 0.5558 | 1.2503 | 1.3985 | −0.0846 | −0.0598 | 0.3609 | 0.4845 |

Standard deviation | 0.2005 | 0.0984 | 0.1219 | 0.1432 | 0.2045 | 0.0984 | 0.1219 | 0.1432 | 0.2037 |

Downside deviation | 0.1441 | 0.0490 | 0.0633 | 0.0696 | 0.0950 | 0.0639 | 0.0777 | 0.0832 | 0.1082 |

Sharpe ratio | −0.0110 | 5.2312 | 4.5598 | 8.7339 | 6.8392 | −0.8592 | −0.4904 | 2.5211 | 2.3781 |

Sortino ratio | 0.1256 | 11.1380 | 9.2757 | 18.6058 | 15.2388 | −1.0315 | −0.5229 | 4.6719 | 4.7587 |

**Table 5.**Yearly returns for BHS, FTS, GVS, RVS, and JDS for a varying target time from January 1998–December 2015.

Before Transaction Costs | After Transaction Costs | ||||||||
---|---|---|---|---|---|---|---|---|---|

Target Time | BHS | FTS | GVS | RVS | JDS | FTS | GVS | RVS | JDS |

20 min | 0.0181 | 0.4997 | 0.4944 | 1.4201 | 1.6941 | −0.0937 | −0.0970 | 0.4638 | 0.6261 |

40 min | 0.0181 | 0.5030 | 0.5382 | 1.4120 | 1.6685 | −0.0918 | −0.0704 | 0.4589 | 0.6104 |

60 min | 0.0181 | 0.5214 | 0.5525 | 1.3706 | 1.6624 | −0.0806 | −0.0618 | 0.4338 | 0.6068 |

80 min | 0.0181 | 0.5088 | 0.5483 | 1.3132 | 1.5883 | −0.0882 | −0.0643 | 0.3990 | 0.5628 |

100 min | 0.0181 | 0.5065 | 0.5583 | 1.3107 | 1.5893 | −0.0896 | −0.0583 | 0.3975 | 0.5634 |

120 min | 0.0181 | 0.5456 | 0.5874 | 1.2959 | 1.4472 | −0.0659 | −0.0407 | 0.3885 | 0.5147 |

140 min | 0.0181 | 0.5346 | 0.5748 | 1.2748 | 1.5233 | −0.0726 | −0.0483 | 0.3757 | 0.5241 |

160 min | 0.0181 | 0.5384 | 0.5897 | 1.2653 | 1.5226 | −0.0703 | −0.0392 | 0.3700 | 0.5229 |

180 min | 0.0181 | 0.5699 | 0.5848 | 1.2510 | 1.4946 | −0.0512 | −0.0422 | 0.3613 | 0.5061 |

200 min | 0.0181 | 0.5268 | 0.5764 | 1.2255 | 1.4783 | −0.0773 | −0.0473 | 0.3459 | 0.4965 |

220 min | 0.0181 | 0.5165 | 0.5838 | 1.2358 | 1.4865 | −0.0836 | −0.0428 | 0.3521 | 0.5014 |

**Table 6.**Yearly returns for BHS, FTS, GVS, RVS, and JDS for a target time of 5, 35, 65, and 95 min from January 1998–December 2015.

Before Transaction Costs | After Transaction Costs | ||||||||
---|---|---|---|---|---|---|---|---|---|

Target Time | BHS | FTS | GVS | RVS | JDS | FTS | GVS | RVS | JDS |

5 min | 0.0181 | 0.3217 | 0.2529 | 1.0387 | 1.1842 | −0.2015 | −0.2432 | 0.2327 | 0.3174 |

35 min | 0.0181 | 0.5233 | 0.5349 | 1.4023 | 1.6719 | −0.0795 | −0.0725 | 0.4531 | 0.6134 |

65 min | 0.0181 | 0.5232 | 0.5515 | 1.3793 | 1.6431 | −0.0795 | −0.0624 | 0.4391 | 0.5956 |

95 min | 0.0181 | 0.5118 | 0.5601 | 1.3102 | 1.5824 | −0.0864 | −0.0572 | 0.3972 | 0.5589 |

© 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**

Stübinger, J.; Schneider, L.
Statistical Arbitrage with Mean-Reverting Overnight Price Gaps on High-Frequency Data of the S&P 500. *J. Risk Financial Manag.* **2019**, *12*, 51.
https://doi.org/10.3390/jrfm12020051

**AMA Style**

Stübinger J, Schneider L.
Statistical Arbitrage with Mean-Reverting Overnight Price Gaps on High-Frequency Data of the S&P 500. *Journal of Risk and Financial Management*. 2019; 12(2):51.
https://doi.org/10.3390/jrfm12020051

**Chicago/Turabian Style**

Stübinger, Johannes, and Lucas Schneider.
2019. "Statistical Arbitrage with Mean-Reverting Overnight Price Gaps on High-Frequency Data of the S&P 500" *Journal of Risk and Financial Management* 12, no. 2: 51.
https://doi.org/10.3390/jrfm12020051