Repeating Christmas jump in LIBOR

Background: London Interbank Offered Rate (LIBOR) exists since 1986 as a benchmark interest rate. Methods: Using two-layer linear regression method, we found a pattern of shortterm nature in LIBOR behaviour. Results: To wit, 2-month LIBOR experiences a jump after Xmas for the last two decades. The direction and size of the jump depend on the data trend on 21 days before Xmas. Conclusions: The obtained results can be used to build a winning strategy on the Swap Market.

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Introduction
In 1986, a new benchmark interest rate was introduced, named the London Interbank Offered Rate (LIBOR). At LIBOR, major banks of the world lend to one another in the international interbank market for short-term loans. From a mathematical point of view, LIBOR is a sequence of daily changing real values. LIBOR data is in open access and can be found on multiple web-sites, for example, here 1 .
In Figure 1, one can see a large scale sample of 2-month LIBOR for loans in USD. In Figure 2, the values of 2-month LIBOR 21 days before Xmas and 6 days after from 2004-2019 years were put together.
LIBOR has a crucial role in the Swap Market, where people exchange their loan interests and can win or lose money depending on their right or wrong predictions of LIBOR dynamics. For example, person P got a one-million-dollar loan with 5% interest and person E borrowed the same amount but with the interest 2%+LIBOR. After some time, they decide to exchange their interest rates because P thinks that LIBOR will go lower than 3% but E believes that it will go higher than 3%. Both their opinions are based on some prediction methods, even if it is just an intuition. We intend to bring another prediction tool into the game. A curious reader may find more complex models and measures on LIBOR for different problems [2][3][4][5] .
Thus, here we are not interested to LIBOR nature per se but in its volatility. More precisely, we study the behavior of LIBOR after Christmas from December 26 to December 31. Examples of such data are in Figure 3 and Figure 4.
Although LIBOR itself is going to disappear in 2021 [6][7][8][9][10][11] , and one can apply our results only for Xmas 2020, we think that the model we introduce here might be useful for short-term analysis in other problems.
So, how does Christmas affect dynamics of LIBOR until the next holiday?

Definition and models
The research we conducted indicates convincingly that a jump does exist. But what is a jump in a discrete sequence of numbers? The following seems to be the most acceptable: There are a real number x and a real discrete function given tabularly for some natural k ∈ [2,n − 2]. One chooses approximant A : R r × R 1 → R 1 among functions having continuous derivative by second argument notated as A′ 2 and chooses some quality criterion Q(z,A(p, ·) to minimize by parameter vector p ∈ R r . Then we appoint  and consider the approximation problem with the criterion Q(z 0 ,A(p,·)) → min p∈R r . Let its solution be denoted as 0 ( , ). A z x Then we consider the next problem Let its solution be denoted as In other words, the jump at x of a discrete function given by a tabular is the difference at x between the obtained solutions of two approximation problems of the same type, the first problem is formulated on the left part of the table, the second problem is formulated on the right part of the table and must keep at x trend (i.e. derivative) of the first problem solution. The left part corresponds to the nodes less than x, the right part corresponds to the nodes greater than x.
In our case, x =Dec 25).
It is easy to see that the so-defined jump depends on the type of approximation and on the amount of the input data. On top of that, we have to decide the amount of input data in z 0 . Notice that the amount of the data in z 1 is only three pairs (date, LIBOR of this date) because there are exactly three working banking days between Xmas and New Year's Eve (NYE). The data source is available in 1 or in many other sources.
Variability of the data due to random factors leads to the choice of the simplest approximation. We use linear approximating functions, which coefficients may be found by linear regression with its own quality criterion. We restrict ourselves to LIBOR data for the last 22 years, because it is natural to expect the evolution of LIBOR behavior over the years.   or decrease) does not change a lot over a short time interval, it is almost the same before and after Xmas. Therefore, we seek an approximation after Xmas in the following form: Thus, for each selected year j there is a relationship ( , ) .
According to Definition 1 the difference : is the jump we have been looking for. Having such connections over 23 years, one can try to find a pattern. To do that, we turn to linear-quadratic regression on two-dimensional nodes. This time the approximating function has the form: with an approximation table F J (a j ,b j ) ≈ Δ j , j ∈ J ⊂ {1995,...,2019}. Sub index J at F points at which subset of years over the past 23 has been chosen to construct the regression. The remaining years will be used to verify the statistical reliability of the result.
Why 15 week days before and 3 week days after Xmas? On one hand, we want as much data for our approximation as possible. On the other hand, the longer the time interval, the less  accurate the trend on the end of the interval. Someone could say: "Why don't you take a more complex approximant to capture more complicated futures of the time series?" Well, that would require even more data for statistical power of such approximant. Since we want to detect a short-term pattern, we should avoid such approach. It made sense to take a number of days before Xmas divisible by 5, so each week day would appear evenly. After trying 25, 20, 15, 10, 5, we found the model worked best with 15 days.
Regarding 3 LIBOR days after Xmas, the same logic explained above is applicable here too. Again, empirically we have found that 3 days work the best. Notice every year has exactly 3 LIBOR days between Xmas and NYE. It is possible that NYE plays a big role in that.

Results
We conducted the process above for several different numbers of years for F regression (from 5 to 20 years), different LIBOR data (overnight, 1 month, 2 months, etc.). The most convincing results have been obtained with the following setups: 21 calendar day regression for each year from 15-year intervals; 2-month-loan values of LIBOR.
Observe the results in the Table 1.
So, our prediction for the jump formula after Xmas 2020 are: 2020 2020 2020 2020 2020 0.004770 9.265766 0.002346 2.018758 The 95%-confidence intervals for the coefficients β 0 ,β 1 ,β 2 ,β 3 in (3)  Put them into any program to find linear regression, for example, into our code in R, which is available as Extended data 12 . The result of the regression is two numbers: that corresponding to free term is b 2020 , the other is a 2020 . Substitution of them to (3) yields the jump.
The prediction of the jump can be used to predict the mean LIBOR after Xmas and before NYE ( ). y a x Hence Notice that the last term in (4)  In the pre-print 13 of this paper, one can find our predictions for the jump after Xmas of 2019 and see that later data from the event confirmed it.

Data availability
Source data All data used in this paper can be found at IBORate (http:// iborate.com/usd-libor/) 1 .

Extended data
The code used to develop the model is available at: https:// github.com/keshmish/Chistmas-Jump-in-LIBOR/.

Open Peer Review
help us a lot.
2. The articles you mentioned are talking about LIBOR transition, which is a little unrelated to our topic. Unfortunately, adding data after 2020 is hard since the LIBOR got mostly shut down in 2021. There above, you mentioned similar to LIBOR data where the proposed here method could be tried on. This is a very useful hint. Thanks!