High / Low Timing Patterns

Just few days ago stock.nu published a post with charts showing that empirical distribution of daily lows follows a U-shaped pattern, i.e. daily low is not equally likely to happen at any time during the trading day, rather low is more likely to occur near the open or the close. Similar (symmetric) U pattern exists for distribution of daily highs - they also more likely to happen near the open or the close.

This particular distributions of daily extremes is actually not surprising - it is one of the properties of random walk process. To re-phrase: if we were to simulate a random walk process, we would not see a uniform distribution of highs and lows throughout the day, rather we would see this U-shape pattern of highs and lows.

This phenomenon is known as arcsine law (law in this context means probability distribution) and has several manifestations. The one displayed in the charts is described in Wikipedia as Third Arcsine Law

This is theoretical distribution of lows (or highs) that we would expect according to the formula. If you compare it to the original (empirical distribution) chart, the numbers are quit similar. 

However I must note that I cheated a little bit: both the empirical chart from stock.nu and mine below have seven columns, but time intervals are not the same - in the top chart the first bar corresponds to the first half-hour, while the rest are one hour. In theoretical chart all time intervals are the same. 

So how much do these distributions really differ? If we recalculate the theoretical chart, we still see the U-shaped pattern, but it would look somewhat different from empirical. 

What explains the differences? Well - really one thing - there is a difference in trading activity in different parts of the day. I speculate that if were to adjust the time intervals for equal trading volume, or equal number of trades, the resulting empirical distribution will be very similar.

Check out my follow-up post.