In this study I have attempted to go back to the very basic elements of price – open, close, high, low, daily range, daily change – and attempted to analyse these phenomena statistically.

Latter I discuss my conclusions and possible ways in which they can be used in trading.

The study was undertaken using the EUR/USD pair, using daily data only.

#### Types of days

In total 2594 days were included.

Initially these were categorised into 3 classifications: up days, down days and unchanged days. The results are below:

Total up days = 1319

Total down days = 1260

Total unchanged days = 15

This leads to a percentage breakdown as follows:

Up days as a percentage of all days = 50.85%

Down days as a percentage of all days = 48.57%

Unchanged days as a percentage of all days = 0.58%

#### Sequences of days

Another phenomenon which was studied was sequences of the same type of day occurring uninterrupted in a row. Below is a list of the sequences for up days, the number of times they occurred in the sample and the total number of days in brackets.

2 in a row occurred 178 times (356 days)

3 in a row occurred 89 times (267 days)

4 in a row occurred 40 times (160 days)

5 in a row occurred 13 times (65 days)

6 in a row occurred 9 times (54 days)

7 in a row occurred 4 times (28 days)

8 in a row occurred 2 times (16 days)

9 in a row occurred 1 time (9 days).

**The table below compares clustering in the data to what might be expected from a random sample:**

The first column shows the length of the sequences studied.

The second column shows the ‘Discrete Frequency’ of the clustering, which is the number of separate individual incidences of the sequence occurring.

The ‘Total Frequency’ column is a sum of the discrete incidences and also the number of times the sequence occurred within a larger sequence – for example when 2 up days in a row repeated twice, once after each other within a longer sequence of 4 up days in a row.

‘Probability’ represents the random chance of the sequences occurring using the base chance of an up day happening, which was 50.85%.

Finally ‘Probable Frequency’ uses the empirical probability of the previous column to calculate the number of times the sequences should have occurred randomly within the data.

#### From single days to 2 or more.

The number of single up days which extended to 2 or more up days was also analyzed.

Below are the calculations:

362 up days occurred on their own and were not followed by a further up day.

Another 337 days – that is one for each cluster longer than 1 day were also not followed by a further up day as they completed the sequences ie they constituted the final day of a sequence.

In total this gives 699 up days which were not followed by another up day.

This leaves 1319 – 699 = 620 up days which were followed by another up day.

This means that 47% of up days were followed by one or more further up days in a continuous sequence, and 53% were not.

#### 2 day sequences which led to 3 or more days in a row

The data was also analysed to check how many of the 2 in a row up days were followed by a further up day.

Total number of 2 in a row or more days = 337

The 2 in a row days which failed to become 3 in a row days = 178

Therefore those which extended = 337 – 178 = 159

In percentage terms = 47.2% of 2 in a row up days led to 3 or more up days and,

52.8% of 2 in a row up days did not extend to 3 or more up days in a row.

#### Distribution according to Change

The change in price during a day was analysed. This was the difference between the opening price and the closing price. The results were rounded up and down in order to allow them to be categorised and analysed more easily. This process is described as ‘binning’. In this case the percentage change was rounded up or down to the nearest first decimal place. The results are shown in the study below. The x-axis at the bottom shows the percentage change in the day and the y axis the number of occurrences of days with that level of change.

The table below shows the raw data used in the graph above. The “95%” column shows where 95% of the days fell. This is bounded by the light blue row. It occurs between the -1.4% and +1.4% change levels. This means that the majority of days have a 1.4% or less level of change. Furthermore the data showed that 80% of days had less than a 0.8% change. Overall the data showed a tendency to low volatility or reversion.

#### Distribution according to Range

The percentage change in price or range was also analysed. The range is the difference between the high and the low. The graph below shows the results:

The raw data is also shown in a table below. The table shows that roughly 95% of days had a range of between 0.2% and 1.9%. 80% of days had a range of between 0.4% and 1.3% and 57% (It wasn’t possible to get closer to 50% than this without going below 50%) had a range of between 0.5% and 1.0%.

#### Conclusion

The distribution of up and down days was roughly equal ie 50:50. This implied a purely random distribution.

The frequency of uninterrupted sequences of days was slightly higher than would have been expected randomly using the chance of an up day occurring at all (50.85%). This was particularly true for 2, 3 and 4 sequence clusters.

The 2 cluster occurred 427 times when it would have been expected to occur only 335 times randomly. The 3 sequence happened 177 times versus 113 expected randomly and finally the 4 happened 73 versus 43 times. These are quite large differences and maight point to a bias in market data to clustering.

Despite the higher frequency of clustering the chances of sequences lengthening based on statistical evidence was less than 50%. This seems to imply a bias to discontinuation of sequences of up days although the sample was probably too small to prove this given most of the results were near to 50%.

Regarding Range, 80% of days in EUR/USD showed a daily range of between 0.3% and 1.3%, and this data could be useful to construct probable ranges in which future price movements might be expected to probably range between.

Likewise the fact that 80% of daily changes occur within a 0.8% band around the opening price is also a useful statistic. These two stats could even be used together possibly to trade range extremes, given the probability prices will pull back as the day progresses; however, this may require further study to confirm.

The increased frequency of days with a low percentage change may explain the preponderance of Option’s strategies such as the ‘Butterfly’, ‘Condor’ and ‘Iron Condor’ which profit from low volatility non-directional movement.

Indeed it may be that strategies based on the same principles are best placed to take advantage of the daily changes in price in EUR/USD, particularly on an intraday basis.