# More on Waiting Times, from Vic Niederhoffer

April 27, 2013 |

It is interesting to contemplate the distribution of waiting times to the next 10 day max's and 10 day min's in S&P futures since 1 1 1999.

[Ed.: an n day max is defined by {for 1 = 1 to n, c[t] > c[t-i] } i.e. the close is higher than the previous n closes]

`Col A: days since last 10 day minimum`
`Col B: Number of occurrences`
```Col C: Waiting time to next one
```
`Col D: days since last 10 day maximum`
`Col E: Number of occurrences`
`Col F: Waiting time to next one`

.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

```Col A  Col B     Col C            Col D     Col E       Col F

0        524                          0         789
1        297        10.5              1         373        8
2        219        13                2         276        9
3        188        14                3         226        10
4        165        15                4         185        11
5        146        15                5         161        11
6        140        15                6         148        11
7        130        15                7         131        11
8        124        15                8         119       11.5
9        119        14                9         115        11
10       110        14                10        108        11
11-15      426        15              11-15      375        11
16-20      289        15              16-20      226        12
21-30      339        16             21-30       257        11
31-50      288        13             31-50       257        11
51-100     54         06             51-100       11         05
```

There were 3575 trading days during this period with no drift per day, i.e. the adjusted  futures started and ended at the same level.

Many assymmetries are evident.

The maxima occur with much greater frequency. The average waiting time to the next 10 day min or 10 day max is constant until 51 days has elapsed. The waiting time to the next 10 day minima is greater than the wait to the next day maxima  after more than 10 days have elapsed.

I studied this because I am brushing up on my knowledge of stochastic models, reading many good books on the subject. I find the subject very refreshing and useful and it is good to keep the paper and pencil less feeble at counting.

One would point out that Mr. Tom Downing did very good work on writing this statistics program while he was employed by my firm.   Vic, chair.