Nov

8

I read an old Spec List post of mine about Mathematical Constants, by Stephen R. Finch, Cambridge University Press on constants like Pi, e. One is Feller's constant. Feller's constant measures the probability of a run. Suppose we toss a fair coin 100 times. What is the probability that we encounter at least one run of five consecutive heads? In a simplified stock market model where each day is independently up or down with probability 50%, what is the probability of experiencing five consecutive down days during 100 trading sessions? Feller's formula approximates the probability of no such run occurring:

Probability of at least one run of consecutive down days:

In 252 trading sessions

Run length - Probability

3 days - 100.0%
4 days - 100.0%
5 days - 98.6%
6 days - 87.0%
7 days - 63.0%
8 days - 38.6%
10 days - 11.3%

This provides a baseline.

Yet over the last 5 years here are the actual count of runs of down days non overlapping:

Exact streak length - Occurrences

Exactly 3 days - 43
Exactly 4 days - 19
Exactly 5 days - 7
Exactly 6 days - 3
7+ days - 0

Nothing that randomness can't explain but interesting to me. Probably drift accounts for fewer down runs.

Larry Williams asks:

How about consecutive up days would drift make that a stronger bias to continue the sequence?

Zubin Al Genubi responds:

Larry, Yes. Here are up day streak counts last 5 years. More than down.

Exact up-streak length - Occurrences

3 days - 41
4 days - 25
5 days - 13
6 days - 4
7+ days - 8

The eight streaks of 7+ days broke down into five 7-day streaks, two 8-day streaks, and one 9-day streak. Each streak counts once in its own length category.

Adam Grimes writes:

I think drift is the explanation. Try modeling as random walk with the mean = market drift (optionally with distribution drawn from actual market returns) and I believe you see the same thing even under iid. Haven't done it in years so I could be mistaken, but that's my recollection. (Maybe autocorrelation plays a part here too. Stock returns do show some serial dependency in a departure from randomness.)

Kora Reddy adds:

When I first read "Some A Posteriori Probabilities in Stock Market Action" by Alfred Cowles III and Herbert E. Jones, I didn't explore the rest of Cowles' publications. Today, I finally took the time to look into his broader research and stumbled upon this goldmine: Cowles Foundation for Research in Economics


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