In The Multivariate Ornstein-Uhlenbeck Model as an Alternative to Regressions, we considered the relative valuation of the CADUSD exchange rate and the price of front-month Brent futures, from the perspective of statistical arbitrage. And rather than a linear regression to characterize the relation between the two series, we used a multivariate Ornstein-Uhlenbeck process.
The MVOU process is well-suited to modeling the dynamic behavior of variables over time, particularly when there’s a possibility that the trajectory of the variables is influenced not only by a tendency to revert toward their long-run means but also a possibility that the variables are influenced by each other.
For example, we found that the CADUSD exchange rate exhibited a tendency to revert toward a long-run mean of 90 US cents per Canadian dollar with a half-life of roughly three years. In contrast, the Brent futures price exhibited almost no tendency to revert toward a long-run mean – but it did display a tendency to be attracted to the CADUSD exchange rate, with the latter scaled by a factor of 2.25.
As a result, the correlation and covariance between these two series were found to be dependent on the horizon – in this case with the correlation increasing as the horizon increased from daily to weekly to monthly to quarterly and eventually using the levels of the two series.
The ability of the MVOU model to reflect these dynamics is quite useful, as it more accurately characterizes the observed behavior of the series than do more static models, such as regressions. But the price we pay for this flexibility is that there are more issues we need to consider when constructing spread trades.
For example, we need to decide whether we want to use dynamic hedge ratios in order to capture the dynamic correlation and covariance of the two series – or whether we’d prefer to use a static hedge ratio, with a focus on a single trading horizon. If we planned to enter a spread trade with a fixed horizon of, say, three months, we could choose a static hedge ratio that optimizes the ex ante risk-adjusted performance of the trade over this three-month horizon, avoiding the need to rebalance dynamically.
When considering these two alternatives, we should keep in mind that a strategy that is practical yet suboptimal may still be preferable to a strategy this is optimal yet impractical. In general, I believe the dynamic hedge ratio will most often lead to better performance. But if trading costs are high, or if the person calculating and executing the dynamic hedges is especially pressed for time, it may be preferable to settle for a static hedge ratio.
With this in mind, it’s also useful to perform some sensitivity analysis to gauge the extent to which the optimal hedge ratio varies with the trading horizon. For example, the graph below shows the optimal static hedge ratio and the ex ante annualized Sharpe ratio of the trade, as a function of the trading horizon. In this case, the optimal hedge ratio is taken to be the one that maximizes the ex ante Sharpe ratio to that particular horizon, given the expected price changes and covariances calculated by our calibrated model for that horizon.
Note that the optimal hedge ratio does depend on the trading horizon but that it doesn’t appear especially sensitive to the trading horizon. In particular, if we expected our trade to perform over a period of, say, three months, we might choose a static hedge ratio of 1.75, with the understanding that we may well have a bit larger position in CADUSD than would be optimal over horizons of a few weeks and that we may have a bit smaller position in CADUSD than would be optimal should we hold the trade over a longer horizon.
To gain another perspective on this issue, the path of expected values for the spread is shown in the next graph, along with the historical values for the spread using this static weighting.
Our sense is that this path of expected values is attractive despite the use of a static hedge ratio, in absolute terms and relative to the conditional standard deviation of the spread over a three-month horizon. In other words, the static hedge ratio may not be strictly optimal, but it looks pretty good. And depending on circumstances, the difference between ‘pretty good’ and ‘optimal’ may not be worth the additional effort to dynamically hedge the position.
At this point, it would seem natural to quantify the difference between ‘pretty good’ and ‘optimal’ – eg, by comparing the ex ante Sharpe ratio of the static hedging strategy to that of the optimal, dynamic strategy. But while it’s straightforward to undertake each of the steps required to determine the optimal portfolio for any particular horizon at any particular date, it’s not straightforward to derive a general expression for this optimal weight, in the limit, as a function of the prices of the respective assets, and of the parameters of the multivariate Ornstein-Uhlenbeck process. We could approach this problem algorithmically rather than analytically .... but this is a complicated topic on its own – and one we'll save for another post.
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