Roulette & the Stock Market, Part 2

Slicing through the red+even win probability surface

In this followup to our prior article showing the parallels between a particular roulette wager and a bearish but volatile stock market asset, we now do for the roulette model the same thing we allow in our stock market risk modelers MCarloRisk3D. That is, we make a slice plane at a given number of roulette spins forward in time (analogous to the number of days forward in our stock market modeler), slice thru the time/win/probability surface, and analyze the win probability distribution of that slice.

Figure 1: Chart generated by our example web site for a constant red+even roulette bet. Hand-drawn bright green slice shown at spins = 39

Recall that at each given spin (time step) forward, we computed the win probability distribution. This is the source of the red, black, and dark green curves on the above chart. Now we pull that whole probability distribution out separately at a given spin to analyze it.

For this example, we will use spin 39 because it crosses the middle of a plateau of the rapidly descending (bearish) 5th percentile constant probability curve.

To our eyes just looking at the downward tilting envelope in the above chart, it looks like the distribution that we had cut through the win/probability/spin surface should come out to have a negative skew to it. However, this does not prove to be the case, as we will show with a little bit of analysis in the Gretl econometrics software.

Win distribution (on X axis) histogram of constant red+even wagers at spin = 39

Summary statistics, using the observations 1 - 100000
for the variable 'redeven' (100000 valid observations)

  Mean                        -4.0914
  Median                      -4.0000
  Minimum                     -40.000
  Maximum                      30.000
  Standard deviation           8.5628
  C.V.                         2.0929
  Skewness                   0.021222
  Ex. kurtosis              -0.012843
  5% percentile               -18.000
  95% percentile               10.000
  Interquartile range          12.000
  Missing obs.                      0

Table 1

5th, 50th (median) and 95th percentiles agree with our web site’s analysis at -18, -4, and 10, respectively. Whew. Good thing.

 (Output from Gretl)

 Test for normality of redeven:

 Doornik-Hansen test = 8.31592, with p-value 0.0156394

 Shapiro-Wilk W = 0.995466, with p-value 9.80852e-44

 Lilliefors test = 0.0485127, with p-value ~= 0

 Jarque-Bera test = 8.1933, with p-value 0.0166283

Table 2

While formal the tests for normality —

the scope of which is far beyond this article… we started to type something about this, but hoo boy, it’s just too much… look up the above keywords in the context of “tests for normality” for more info

— in Gretl show that this distribution is not precisely normal for our monte carlo count of 100k, and small but non trivial skewness and kurtosis (noted above) indicate this as well, the distribution seems approximately symmetric and close to normal by eye.

Even though Gretl’s outputs indicate a slightly non-normal distribution, we were expecting very not normal, this being a traditional house-wins betting game (e.g. a big skew bearish and maybe a fat tail on the bearish side). But this is not the case. The tilt toward loss seems almost entirely caused by the mean of the distribution being << 0 (about -4 wagering chips). “So we got that going for us,” type of thing.

Let’s cut one more slice at spin = 9. This is the highest spin value where the median black curve is still flatline (neutral), which we can see if we examine Figure 1 above.

In this case, the data was so close to normal that Gretl plots both the histogram of the raw data and also the closest normal distribution it could find to it (the N( ) annotation at the above right, mean -0.95, stdev 4.11)

Summary statistics, using the observations 1 - 100000
for the variable 'redevenSpin9' (100000 valid observations)

  Mean                       -0.95316
  Median                       0.0000
  Minimum                     -18.000
  Maximum                      16.000
  Standard deviation           4.1149
  C.V.                         4.3171
  Skewness                   0.012723
  Ex. kurtosis               -0.10262
  5% percentile               -8.0000
  95% percentile               6.0000
  Interquartile range          6.0000
  Missing obs.                      0

Table 3

Test for normality of redevenSpin9:

 Doornik-Hansen test = 48.3354, with p-value 3.19227e-11

 Shapiro-Wilk W = 0.978967, with p-value 1.12773e-73

 Lilliefors test = 0.0967992, with p-value ~= 0

 Jarque-Bera test = 46.5752, with p-value 7.69685e-11

Table 4

The formal tests say “not normal” again, but positive or negative skew and kurtosis (tail fatness) values indicate close to normal, in fact maybe within the tolerance of our 100k run monte carlo results. After all, a monte carlo solution like this is only approximate. It would be interesting to see if our model output becomes more normal according to skew and kurtosis the more monte carlo runs we make.

Check of the square root of time rule from the stock markets

There is a well know approximation in the stock markets that the volatility of an asset scales as the square root of time. Let’s check this for our two slices we made here. We’ll use the first slice at spin (discrete time) 9 as the first value, along with the future value at spin 39. These two form a ratio of 39/9 = 4.333. The square root of this is 2.08. This is assuming a unit time of 9 spins, so then spin 39 is this time unit * 4.333.

Now let’s look at the standard deviation of our wins (volatility) from above Gretl outputs at those two time steps.

stdev of winnings at 9 spins = 4.1149

stdev of winnings at 39 spins = 8.5628

The ratio of these is also about 2.08. Hence, our win volatility in this red+even wager seems to also scale with root t, at least for these two data points. This was surprising to the present author. Maybe it will be to readers, too! Budding high rollers, please “hold your horses” and note that we are only studying one particular wager here.

I suppose we should go and make a plot of this root t phenomenon over various time steps. It could be that we just got lucky picking 9 and 39 as slice points, and their volatilities just happen to scale like root t. Again, in due time.

Implications

Future incarnations of Alexis Ivanovitch only have to be lucky in the mean and not worry about also being lucky in higher order statistical moments with this red+even wager. There are those who may scoff at Alexis’ potential knowledge of statistics, but his author does reference logarithms in his Notes From The Underground (1864):

All human actions will then, of course, be tabulated according to these laws, mathematically, like tables of logarithms up to 108,000, and entered in an index; or, better still, there would be published certain edifying works of the nature of encyclopaedic lexicons, in which everything will be so clearly calculated and explained that there will be no more incidents or adventures in the world.

That this resembles statements from the latest hucksters of A.I. is also surprising, yet purely coincidental. I think.

Example site

The example site that generates our red+even forward in time win surface is located here. You can view the source code to see how it works, make a copy, modify it, etc. If you copy the source code locally, just edit it as you see fit and “open file” it from a browser. It should run from a local file.

Further reading

Part 3 of this series is now online:

https://medium.com/@nttp/roulette-and-the-stock-market-part-3-9b9370becc20