I recently analyzed a butterfly spread involving the yields of three Italian government bonds:
BTP 4.5% Mar-24
BTP 1.85% May-24
BTP 3.75% Sep-24.
At one point in the analysis, I needed to assess the directionality of the fly – ie, the extent to which this butterfly spread was statistically independent of a proxy for the overall market, taken here to be the average of the three bond yields.
And as part of that analysis, I dutifully calculated two correlations:
The correlation between the daily changes in the butterfly spread and the daily changes in the average bond yield
The correlation between the level of the butterfly spread and the level of the average bond yield.
A scatterplot of the daily changes is shown below. From this perspective, these two series appear largely uncorrelated, with an estimated correlation coefficient of only -0.05.
In contrast, a scatterplot of the levels of the two series is shown in the next graph.
And a time-series plot of the two series appears in the next graph, with the average bond yield shown on the inverted, right-hand axis.
From this perspective, the correlation between these two series appears strongly negative. And in fact, the estimated correlation coefficient in this case is -0.90
So are we dealing with uncorrelated data or with strongly (negatively) correlated data? Well, as we’ve seen in recent posts (eg, When Correlations Depend on Frequency of Observation and Interactions Between 5Y and 10Y Treasury Yields), it’s possible for the correct answer to be both. In particular, it’s not uncommon to for financial data to exhibit horizon-dependent correlation – a characteristic that can readily be modeled using the multivariate Ornstein-Uhlenbeck process.
With that in mind, I calibrated a MVOU model to the data. And with the calibrated model, I was able to calculate a vector of conditional expected values and a matrix of conditional covariances for any trading horizon date. The corresponding correlation coefficient between the fly and the market proxy is shown as a function of the trading horizon in the graph below.
Given the estimated strengths of the attractions between the three yields, I was particularly interested in a trading horizon of three months. But as seen in the graph above, this butterfly spread would be expected to have a correlation with the market proxy of -0.63 on a trading horizon of three months. I was looking for a trade with less directional exposure to the overall market.
With this in mind, I decided to perform a constrained optimization. More specifically, I choose weights so as to maximize the ratio of the expected yield change of the fly divided by the standard deviation of the fly, subject to the constraint that the covariance between the fly and my market proxy, on a trading horizon of three months, was equal to zero.
In particular, if we define
w as a 3x 1 vector containing the weights defining this butterfly spread
r as a 3 x 1 vector containing the conditional expected yield changes of these three yields on a three-month horizon
i as a 3 x 1 vector of 1s
V as a 3 x 3 matrix of conditional covariances for the three yields on a three-month horizon
then the problem is to choose w so as to minimize w'r/sqrt(w'Vw), subject to w'Vi=0, where w' denotes the transpose of w. I also normalized the problem so that the second element of w (ie, the weight corresponding to the yield of the May-24 issue) would be equal to 2, to make the result more readily comparable to the standard weights of -1:2:-1.
As it happened, the solution corresponded to weights of -0.975:2:-0.975, meaning we’d be buying a bit less of the Mar-24 and Sep-24 issues in this weighted fly than would be the case had we used the standard weights. And the ratio of the expected change in the fly divided by the standard deviation in the change in the fly was 2.8 on a three-month horizon.
Note that this weighted fly still has a correlation with the market proxy that depends on the trading horizon, as shown in the graph below. It’s just that we were able to adjust the weights of the fly so that the correlation at a three-month horizon was equal to zero, by construction. We should expect this fly and the market proxy to have a positive correlation over shorter horizons and to have a negative correlation over longer horizons.
The three yields comprising this butterfly spread constitute a particularly pronounced example of horizon-dependent correlations. But horizon-dependent correlations are not at all uncommon in financial data sets, and the multivariate Ornstein-Uhlenbeck model is especially useful in modeling these dynamics. In fact, in situations like this, it's an excellent choice, as it not only allows us to quantify the effect of horizon-dependent correlations on trades, it also provides a framework for optimizing trades under various risk constraints, such as the constraint that the trade have a correlation of zero with our market proxy on a horizon of three months.
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