This article is the final project submitted by the author as a part of his coursework in Executive Programme in Algorithmic Trading (EPAT™) at QuantInsti. Do check our Projects page and have a look at what our students are building**.**

**About the Author**

Harish Maranani is an EPATian. His educational qualifications include:

- Bachelors in Technology in Electronics and Communications Engineering from Acharya Nagarjuna University,
- MBA Finance from Staffordshire University (UK),
- Certificate in Quantitative Finance (CQF), and
- Master of Science in Mathematical and Computational Finance from New Jersey Institute of Technology, Newark, USA.

**Aim**: To implement pairs trading/statistical arbitrage strategy in currencies.

**Pairs Chosen: **EURINR, USDINR, GBPINR, AUDINR, CADINR, JPYINR

**Frequency: **Daily

**Time Period: **2011/4/21 to 2013/5/22

**Implemented using: **Python.

**Pair Selection Criteria for FX Markets:**

- The time series data for the above-chosen currency pairs is imported from quandl.
- Co-integration Test is carried out on all possible pair combinations viz. EURINR-USDINR, EURINR-GBPINR etc.
- Selecting Co-integrated pairs whose t-static value is less than 5% critical value of -2.8.
- Slicing the pairs which meet the co-integration condition for further analysis.
- To further test for confirmation of co-integration, CADF test is carried out on the sliced pairs from the pool.
- Z-score is calculated for each selected pair combination and the strategy is applied.
- Profit/loss, equity curve, maximum drawdown, are calculated/tabulated/plotted.
- Consider two currency pairs EUR/INR and USD/INR. Here the base currencies are EUR and USD respectively and the counter currency is INR.

**Preliminary Test:**

- In order to find the pairs of currencies that are co-integrated, a preliminary test through coint(x,y) from statsmodels.tsa.stattools is carried out and their respective pvalues, tstatic are plotted below.
- The t-static values that are displayed below are the ones that passed the co-integration test. i.e the t-static values smaller than the 5% critical value of -2.86.

**Below is the list of pairs whose T-static values are less than the 5% critical value of -2.86:**

- ['EURINR/USDINR: -3.89372142826',
- 'EURINR/GBPINR: -3.04457063111',
- 'EURINR/CADINR: -3.16044058632',
- 'USDINR/AUDINR: -3.14784526027',
- 'USDINR/CADINR: -3.19434173492',
- 'GBPINR/CADINR: -3.86588509209',
- 'AUDINR/CADINR: -3.10827352646']

**Below is the plot of p-values of the co-integrated pairs:**

Before rejecting null hypothesis to confirm the prices are mean-reverting, we shall conduct Co-Integrated Augmented Dickey-Fuller (CADF) test to confirm the same for the above sliced pairs out from the whole set of currencies. Below are the Results and plots.

**We shall consider the 4 co-integrated pairs based on T-Static Values for CADF testing.**

**The following are the 4 Co-integrated pairs:**

EURINR/USDINR: -3.89372142826

GBPINR/CADINR: -3.86588509209

USDINR/CADINR: -3.19434173492

EURINR/CADINR: -3.16044058632

#### EURINR/USDINR

**TIME SERIES PLOTS OF EURINR/USDINR**

From the above graph, it is visibly evident that the prices are co-integrated, however, to statistically confirm the same, the below set of tests/procedures are implemented.

Creating a scatter plot of the prices, to see the relationship is broadly linear.Given the above residual plot, it is relatively stationary.

**Co-integrated Augmented Dickey-Fuller Test Results**

Co-integrated Augmented Dickey-Fuller (CADF) test determines the optimal hedge ratio by performing a linear regression against the two-time series and then tests for stationarity under the linear combination.
Implementing in python gives the following result:

(-3.0420602182962395, 0.03114885626164075, 1L, 652L, {'1%': -3.440419374623044, '10%': -2.5691361169972526, '5%': -2.8659830798370352}, 852.99818965061797)

Given the above results, the t-static to be -3.04 less than 5% critical value of -2.8, we can reject the null hypothesis and can confirm that the prices are mean-reverting.

**GBPINR/CADINR**

Below are the time series, scatter and residual plots of GBPINR/CADINR
**CADF Test results:**

(-3.3637522231183872, 0.012258395060108089, 2L, 651L, {'1%': -3.440434903803665, '10%': -2.569139761751388, '5%': -2.865989920612213}, -179.04749802146216)

Given the above results, the t-static to be -3.36 smaller than 5% critical value of -2.8, we can reject the null hypothesis and can confirm that the prices are mean-reverting.

** ****USDINR/CADINR**

**CADF Test results**

Given the above results, the t-static to be -2.93 smaller than 5% critical value of -2.8, we can reject the null hypothesis and can confirm that the prices are mean-reverting.

(-2.9344605252608607, 0.041484961304201866, 1L, 652L, {'1%': -3.440419374623044, '10%': -2.5691361169972526, '5%': -2.8659830798370352}, -99.577663481220952)

**USDINR/AUDINR**

Below are the results from CADF test:

(-3.2595055880757768, 0.016788501512565262, 4L, 649L, {'1%': -3.440466106307706, '10%': -2.5691470850496558, '5%': -2.8660036655537744}, 381.77145926378489)

With the t-static value of -3.25 smaller than the 5% critical value of -2.86, we can reject the null hypothesis and can confirm that the pair is co-integrating.

Now that we have found the co-integrated pairs in the form of following pairs with t-static values:

- EURINR/USDINR: -3.04
- GBPINR/CADINR: -3.363
- USDINR/CADINR: -2.934
- USDINR/AUDINR: -3.259

- Calculating price ratios and creating a new column ratio in the data frames (df, df1, df2, df4) of the above currency pairs respectively.

df:

Df1:

**Calculation of Z-score of the price ratio for the 30-day window of moving average and standard deviation:**

- Below are the plots of z-scores for the above co-integrated pairs with their respective price ratios:

**From the above Z-Score plots of the selected pairs, Z-score is exhibiting mean reverting behavior within 2 standard deviations. **

**Building a Trading Strategy:**

- When z-score touches +2 short the pair and close the position when it reverts back to +1
- When z-score touches -2 long the pair and close the position when it reverts back to -1.
- Only one position is held at a single instance of time.

**Equity Curve:**

Plotting the equity curve with the starting capital of 100 INR equally divided among 4 pairs.

With 100 INR initial Capital, equity ended at 114.05.

Cumulative profit to be 14% without any leverage. With 10 times leverage (ideal for FX trading), the profits can be seen at 140%. Below are the important performance metrics of the strategy.

Profit percentage Without Leverage | 14.0514144897 % |

Profit percentage with 10 times leverage | 140.514144897 % |

Number of Positive Trades | 59 |

Number of Negative Trades | 23 |

Hit Ratio | 71.9512195122 % |

Average Positive Trade | 0.46886657456220338 |

Average Negative Trade | -0.59181362649660851 |

Average Profit/Average Loss | 0.792253766338 |

Maximum Drawdown | -5.1832506579 % |

The above graph shows the maximum drawdown points marked with red dots and the value is added in the above table.

**Instructions for Implementation**

- Please run the IPython notebook named harish_stat_arb.ipynb for the confirmation of results and plots.
- Another option is to run the python script harish_quantinsti_final_project_code.py on any python IDE to confirm the results and graph.
- Use the below code for exporting the final dataframe to an excel file.

writer = pd.ExcelWriter('pairs_final.xlsx',engine = 'xlsxwriter')pairs.to_excel(writer,'Sheet5')

writer.save()

**Conclusion**

Though the strategy has generated 140% returns over the backtest period of 2 years, the following factors should be considered in order to evaluate a more accurate performance of the strategy.
- The model has ignored the slippage and commissions
- The model ignored the bid-ask spread while placing buy or sell orders

**Bibliography**

- Statistical Arbitrage lecture Quantinsti, Nitesh Khandelwal
- Pairs Trading, Ganapathy Vidyamurthy, Wiley Finance
- Successful Algorithmic trading, Michael Halls-Moore