# EUR/USD, by Craig Mee

January 18, 2007 |

A quick observation …

Since the start of 1995 through 2006, the opening week of the year in eur/usd has been the extreme (HIGH OR LOW) for the year nine out of 12 years … Will ‘07 follow this suit?

This looks pretty nonrandom to me notwithstanding the arcsine effect.

Define S as the number of years (out of 12) in which the min or max falls within the first week … In 10,000 simulated 12 year periods, here is the distribution of S when price changes follow a standard normal distribution: (mean 0, standard deviation 1):

S       N        Prob       Odds

0     988     0.0988     10.12
1     2504   0.2504      3.99
2     2984   0.2984      3.35
3     2145   0.2145      4.66
4     951     0.0951     10.52
5     324     0.0324     30.86
6     86       0.0086     116.28
7     16       0.0016     625.00
8     1         0.0001    10000.00
9     1         0.0001    10000.00
10   0         0.0000      NA
11   0         0.0000      NA
12   0         0.0000      NA

In only 1 of the 10000 simulations did at least 9 years of the 12 have a min or max within the first week.

If you assume some sort of drift (for example, since 2002 euro/\$ mean = 3.3 pips with standard deviation of 68 pips/day), the probability of having at least one first week min or max increases, but the probability rapidly drops off after S=7:

S       N         Prob      Odds

0     579      0.0579    17.27
1     1814    0.1814     5.51
2     2789    0.2789     3.59
3     2460    0.2460     4.07
4     1473    0.1473     6.79
5     628      0.0628    15.92
6     210      0.0210    47.62
7     44       0.0044     227.27
8      3        0.0003    3333.33
9      0        0.0000      NA
10    0        0.0000      NA
11    0        0.0000      NA
12    0        0.0000      NA

Another approach would be to estimate the probability of observing a first week min or max in any given year (conditional on a price change distribution), and then calculate the probability of having at least 9 successes out of 12 trials under binomial distribution.

EUUS_W.DAT : column = OPEN 02/01/1995-25/12/2006

WEEK_1 WK_MIN WK_MAX   DIFF
1995 1.2040  1.2040   1.3422   0.0000
1996 1.2740  1.2250   1.2837   0.0097
1997 1.2400  1.0556   1.2406   0.0006
1998 1.1091  1.0762   1.2085   0.0329
1999 1.1756  1.0098   1.1830   0.0074
2000 1.0133  0.8352   1.0256   0.0123
2001 0.8956  0.8437   0.9472   0.0516
2002 0.9016  0.8613   1.0100   0.0403
2003 1.0225  1.0225   1.2184   0.0000
2004 1.2352  1.1790   1.3444   0.0562
2005 1.3313  1.1709   1.3576   0.0263
2006 1.1854  1.1834   1.3353   0.0020

I took a quick look at this as a finger-exercise. Below is R code with some user-tweakable parameters, currently set to roughly mimic Tom's work (though I took a clean-room approach; didn't use Tom's code as a base). The idea, as suggested by Tom, is to find the "probability of observing a first week min or max in any given year," which is "Prop" in this R script, and turns out to be .177 (I'm sure Dr. Phil or others could find a closed-form solution) and plug this into the binomial, thus chopping out an order of magnitude of computing. The results I get are almost exactly Tom's, so either his work is correct (as usual) or he/I made the same mistakes.

Days<- 252 # Biz days in a year
Year<- 12 # Number of years
Week<- 5 # Biz days in a week
Sims<- 10000 # Number of sims
Data<- apply(matrix(rnorm(Days*Sims),Days),2,cumsum)
Prop<-sum(pmin(apply(Data,2,which.min),apply(Data,2,which.max))<=Week)/Sims
Prob<- round(diff(pbinom(Year:0,Year,Prop,F)),4); Prob<- c(Prob,1-sum(Prob))
Odds<- round(1/Prob,2)
data.frame(S=Year:0,Prob,Odds)

Days<- 252
Year<- 12
Week<- 5
Sims<- 10000
Data<- apply(matrix(rnorm(Days*Sims),Days),2,cumsum)
Prop<-sum(pmin(apply(Data,2,which.min),apply(Data,2,which.max))<=Week)/Sims
Prob<- round(diff(pbinom(Year:0,Year,Prop,F)),4); Prob<-
c(Prob,1-sum(Prob))
Odds<- round(1/Prob,2)
data.frame(S=Year:0,Prob,Odds)
S          Prob      Odds
1      12    0.0000     Inf
2      11    0.0000     Inf
3      10    0.0000     Inf
4      9      0.0000     Inf
5      8      0.0002     5000.00
6      7      0.0016     625.00
7      6      0.0088     113.64
8      5      0.0352     28.41
9      4      0.1023     9.78
10    3      0.2113     4.73
11    2      0.2948     3.39
12    1      0.2492     4.01
13    0      0.0966     10.35