*4.3. Trading Period*

The top 10 stocks with the highest *z*-statistic were considered in the one-day trading period. For every top stock, we applied the following trading rules:


Motivated by Section 3, the trade was reversed 120 min later. Our strategy was based on a two-stage logic. First, we identified significant overnight price changes that had a substantial impact on future stock prices. Second, the top stocks possessed mean-reverting price dynamics, so that we could take advantage of these temporary market inefficiencies. If our assumption was correct, we were in a position to capture transient mispricings and generate profits. Concluding, we created a statistical arbitrage strategy based on a mean-reverting jump-diffusion model, the individual jump threshold depends on the underlying volatility.

As we aim for a classic long-short investment strategy in the sense of Gatev et al. (2006), we followed the principles of Avellaneda and Lee (2010) and Stübinger et al. (2018) and secured the market exposure with appropriate capital investments in the S&P 500 index. Every activity carried out on the market involves transaction costs. Therefore, it would be naive to ignore these fees as our high-frequency framework is based on permanent trading. According to Prager et al. (2012) and Stübinger and Bredthauer (2017), estimating exact values is not possible, but the bid-ask spread had abated to lower than one percent for stocks of the S&P 500 index, i.e., two basis points for an average stock price of 50 USD. In the same vein, Voya Investment Management (2016) accounted for a bid-ask spread of 3.5 basis points for the S&P 500, which was caused by increased use of algorithmic trading, decimalization, and changes in the stock market landscape. To be in line with Stübinger and Endres (2018), we assumed transaction costs of five basis points per share per half-turn. Consequently, transaction costs per complete round-trip corresponded to 20 basis points. This assumption appears realistic in light of our high-turnover strategy in a highly-liquid equity market.

In order to evaluate the value-add of our strategy, we benchmarked it against strategies from the same research field, but less flexible. More specifically, we considered the S&P 500 buy-and-hold strategy (BHS), fixed threshold strategy (FTS), general volatility strategy (GVS), and reverting volatility strategy (RVS) (see Table 2). The characteristic "individual" implies that the trading behavior depends on the underlying variable. If the model captured the behavior of fluctuations of stock price dynamics, we assigned the "volatility" property. The feature "mean-reverting" was fulfilled for statistical approaches that were able to model convergence to equilibrium after divergence. Finally, the explicit inclusion of a jump term led to the characteristic "jump-diffusion". Data and the general frame were set identically to the JDS in order to ensure a fair comparison. Especially, we transferred the top 10 stocks to the trading period for each day across all strategies. Details of the four benchmark strategies are presented in the following paragraphs.

**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).


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

First, we compared JDS to a naive S&P 500 buy-and-hold strategy (BHS). To be more specific, the index was bought in January 1998 and held during the complete time period. This passive investment neglected all the characteristics required for a successful strategy, namely, "individual", "volatility", "mean-reverting", and "jump-diffusion".
