# The simplest strategy of paired trading in the bond market - static arbitrage in action

*This article presents the simplest, but very interesting strategy of paired trading on the example of trading securities of the German bond market.*

## What is pair trading

Pair trading is a so-called market-neutral trading strategy, which essentially allows you to profit regardless of the phase of the market (uptrend/downtrend or flat). Alternatively, pair trading is called static arbitrage or convergence trading technique.

Two well-correlated financial instruments are used in trading. When correlation temporarily weaken, and the charts move in different directions, one instrument makes a short trade, and the other - a long one. The main calculation is made on restoration of correlation.

## The essence of the strategy

The trading strategy was tested on the German bond market. The main attention was paid to Euro-bund and Bobl instruments (Euro-bund - 10-year treasury bond futures; Bobl - futures on 5-year Treasury bonds). The correlation between these two instruments is quite high and amounts to more than 90%.

Testing was done on Euro-bund and Bobl closing prices between December 1, 2009 and October 17, 2014 (but there are two dips in quotes between February 28, 2013 and April 29, 2013).

**Components of the formula:**

- Signal (t) is a divergence function.
- R (A,t) is the yield of the instrument A for period i (in our case i = 1).
- R (B,t) is the yield of instrument B for period I (in our case i = 1).
- Sigma (A,20) is the standard deviation of the return of the instrument A for 20 periods.
- Sigma (B,20) is the standard deviation of the return of instrument B for 20 periods.

Example of calculation of the Euro-bund yield as of October 31, 2014 for 1 period. The prices of the futures contract are shown in the table below.

To calculate the yield, we take the values from the right column.

R(Euro-bund, 31.10.2014) = (151.05-150.87)/150.87 = 151.05/150.87 - 1 = 0.001193

To calculate the yield for 2 periods, you need to take the prices on 10/29/2014 (instead of 10/30/2014).

## Rules of the trade

- Stop-losses are not used. It is assumed that the amount of capital is sufficient to withstand short-term drawdowns.
- On the Euro-bund 1/3 of the total position volume, on the Bobl - 2/3 (because the Euro-bund is more volatile instrument).
- Positions are opened every time the Signal(t) function crosses the +1 level from bottom to top or the -1 level from top to bottom.
- Positions are closed when the Signal(t) function crosses the +1 level from top to bottom or the -1 level from bottom to top.
- Trading period: the first trading day is the first day of the futures end month (the same month); the last trading day is the last trading day of the month that comes before the futures end month (for example, the Bobl futures trading period of September 2014 would begin on June 2 and end on August 29, 2014).
- To calculate the Signal (t) function, the data of the previous 20 periods from the trading period of the same futures are used.

## Trading results

- The total number of transactions is 27.
- The percentage of profitable trades is 74.07%.
- The Sharpe Ratio is 0.25.
- The first drawdown is about 10.6% (for the period from November 29, 2010 to July 30, 2012).
- The second drawdown is about 3.3% (from June 24, 2013 to June 9, 2014).

## Statistics

- Average: 267.59
- Error type: 71,4
- Median: 270
- Fashion: 280
- Standard deviation: 371.02
- Sampling variance: 137660.32
- Excess: 4.51
- Asymmetry coefficient: 1.56
- Range: 1910
- Minimum: -410
- Maximum: 1500
- Amount: 7225
- Number of samples: 27
- Reliability level (95.0%): 146,77

## Conclusion

The results are quite interesting, given parameter optimization due to missing data and curve fitting (which means over-optimizing model parameters based on transient noise in the historical data).

The histogram of profits/losses shows us a positive asymmetry coefficient, which indicates the profitability of the given pair trading strategy (which can be confirmed by bootstrapping or Monte Carlo simulation). However, due to the small number of trades and two failures in the data, the results should be treated with caution. In real trading there may be deeper drawdowns.

More than 66% trades are closed in 1 day or in the next 2 trading days. The percentage of profitable trades is 74.04%, which overlap the losses by a significant margin. The short-term trades on the German bond market are interesting in that you can use additional identical pair trading strategies on the same trading account. In doing so, the profits will increase (but so will the losses).