An Application of the Ornstein-Uhlenbeck Process to Pairs Trading

Pairs trading is a widely used market-neutral strategy in quantitative finance that capitalizes on the price convergence of two historically correlated securities. By simultaneously shorting the overpriced asset and longing the underpriced one, the strategy aims to profit from the mean-reverting behavior of their price spread. Unlike directional trading strategies, pairs trading does not depend on the overall market direction, allowing it to be robust in volatile or stagnant market conditions.

This project seeks to improve the traditional pairs trading framework by leveraging statistical and econometric techniques. Specifically, we explore the application of cointegration tests, and modelling the spread as a mean-reverting processes using the Ornstein-Uhlenbeck (OU) model. The methodology is tested on historical market data to evaluate its performance and robustness under varying market conditions.

The selection of pairs is critical. Pairs chosen should exhibit high correlation between the percentage change of the prices of each security, and the price spread should exhibit a mean reverting behavior. Once pairs are chosen, an algorithm can be developed to model the spread and to forecast future pricing trends to take a directional view on the spread. In our case, we employ the Ornstein-Uhlenbeck process to do so, and we compare our results to a basic computation of the z score using the rolling mean and standard deviation by backtesting our results on historical pricing data.

Additionally, we retrieved metadata from the wrds_ds_names table, which includes key identifiers like company names, tickers, and other relevant securities information. After obtaining this data, we removed duplicates based on the combination of infocode and dscode, ensuring that each security was represented uniquely.

The next step was merging the filtered stock data (the universe) with the metadata. By performing an inner join on both infocode and dscode, we ensured that only the securities present in both datasets were retained. This merge allowed us to combine financial data with corresponding company names and other identifiers, creating a comprehensive dataset. Finally, an additional merge was performed to include the general industry descriptions and specific industry group descriptions. The final dataset contains 3,986,371 rows and 26 columns.

After merging, we selected a subset of important columns that were relevant to our analysis. These columns included essential identifiers such as infocode, dscode, ticker, and isin, as well as financial metrics like market capitalization, returns, and closing prices and industry descriptions. An example of the data is shown in Table 1 and Table 2.

To quantify the similarity between the returns of two companies, the squared mean squared distance (MSD) was computed for each pair. This was calculated only on data that was from 2018 to 2020 to ensure point in time analysis. The data which the pairs trading strategy was analyzed on was after the 2018-2020 period. The mean squared distance between two companies $i$ and $j$, based on their returns $d_{ij}$, is defined as:

$$d_{ij}=\frac{1}{N}\sum_{t=1}^{N}\left(R_{i,t}-R_{j,t}\right)^{2}$$

(1)

where $R_{i,t}$ and $R_{j,t}$ represent the returns of company $i$ and company $j$ at time $t$, respectively.

The pairs with the smallest pairwise MSD values were selected, as we assume they demonstrated the highest degree of return similarity. To ensure independence and avoid redundancy, once a company was included in a pair, it was excluded from further pairings. This process resulted in a list of unique pairs with minimal return deviations, serving as strong candidates for further validation.

For instance, the pair United Airlines and Delta Airlines returned a cointegration test statistic of $-12.0389$ with a p-value of $0.000$, demonstrating a statistically significant relationship suitable for trading. In contrast, the pair Credit Corp LTD and Farmers & Merchants Bank produced a cointegration test statistic of $-0.252617$ with a p-value of $0.9779$, indicating no significant cointegration and, therefore, making it unsuitable for further analysis.

Our preliminary results are shown in Table 3 and Table 4. We find that the standard portfolio has an expected return of 7.21% with a Sharpe ratio of 0.778. The frequency of the daily returns were also plotted and the results were shown in (4) and (9).

We find that the portfolio using the Ornstein-Uhlenbeck process has an expected return of 4.92% with a Sharpe ratio of 0.468. Both strategies displayed a positive return. Examining the correlation matrix for both the standard and OU pairs trading strategy in Figures (5) and 10 shows that there is very little inter-pair correlation, which suggests that the returns are reasonably diversified and orthogonal to each other.

Our findings show that the standard pairs trading strategy delivered a positive daily return 54.63 % of the time and the OU pairs trading strategy deliver a positive daily return 53.3 % of the time. While this is a strong win-loss ratio, further work must be done to test the robustness of the strategies on out of sample data to verify the robustness of the win-loss results.

Although the OU Model performed worse than the standard model, the cumulative returns plot indicates that the model captures the correct signals and overall trend, as shown in (3) and (8). This suggests that the model has strong predictive power but may have underperformed due to factors such as suboptimal parameter tuning or an inappropriate Z-score threshold. We suspect that this is likely due to potential non-stationarity of some pairs which led to poor mean reversion. To mitigate this, the Augmented Dickey-Fuller Test could have been used to test for stationarity of the pairs, which gives more reliability for the z-score calculation because the mean and standard deviation are constant. Nevertheless, further testing is required to ensure the robustness and validity of our results. Additional out of sample data and tests will need to be performed. Additionally, the cointegration test was performed in the period from 2018-2020 which was prior to the COVID-19 crisis. A change in regime may have led to a change in the behavior of the spread post-pandemic as well, and further work should be done to explore whether the price correlations were largely affected by major market regimes and events.

Our findings suggest that the pairs trading with Ornstein-Uhlenbeck process underperformed the naive pairs trading algorithm. The underperformance is likely due to the limited predictive power of the model and the fact that we are generalizing all the results to all companies. Future work can consider conditioning the pairs trading strategy on different industries and fine tuning parameters in the OU parameter estimation.

Future work should also explore different statistical methodologies to select pairs. Our pairs were chosen by choosing pairs that have the lowest mean squared distance between percentage price changes. Future work should explore different metrics or a combination such as Percentage Absolute difference, Correlation based distance, and Cosine Similarity to choose pairs. Testing the same pairs trading algorithm with fixed parameters while varying pair selection metrics can help evaluate the effectiveness of each metric. Incorporating additional autocorrelation tests may further refine pair selection.

Additional work on portfolio optimization to maintain factor and/or sector neutrality could also be helpful. This could be done by finding additional time series of the returns of various factors or sectors, running a multivariate regression or principal component analysis (to further eliminate multicollinearity between factors) to solve for betas, and a quadratic program can be solved to optimize the weight of each spread in the portfolio to ensure that the portfolio respects factor and sector neutral constraints.

Finally, hyperparameter tuning should also be explored to enhance the model’s performance. Specifically, the Z-score thresholds, that is currently set arbitrarily at the 25th and 75th percentiles should be fine-tuned to account for variations in performances across different industries.

The simulation did not incorporate transaction costs and if this strategy were to be implemented in real life the returns could be even lower. Slippage was also not accounted for. We assume that we transact on the next day and that the size of trades is negligible and does not move the market. The strategy also relies on the assumption that all the selected pairs exhibits mean-reverting behavior when in reality this assumption may not hold. Non-Stationarity pairs could lead to poor performance if the pairs fail to revert. Finally, the strategy was tested on a limited one-year time frame, which may not fully capture the effects of varying economic conditions such as booms, recessions, leadership changes, or global events like pandemics.