Can the famous Monty Hall Problem be used in a stock index context? We think so.
For those of you unfamiliar with the Monty Hall Problem, the name originates from ‘Let’s Make A Deal’, a popular 60s American TV show hosted by – that’s right – Monty Hall. The show’s premise was simple: contestants were offered something of value, and could then either keep it, or made a ‘trade’ with the host.
At one point in the show, contestants are given a choice of three doors. One of them has a prize behind it, and the remaining two doors open to show nothing.
They pick Door #1, and the host (who knows what’s behind the doors), opens Door #3, which turns out to be empty. Monty then turns to them, grins, and says, “Do you want to pick the other door instead?”
The question is:
Is it to the contestant’s advantage to switch their choice?
Put yourself in their shoes. Your intuition will probably say of course I shouldn’t switch! But sometimes, intuition can be misleading. Mathematically speaking. While Door #1 has a 1/3 chance of having something behind it, since you now know that Door #3 is empty, switching to Door #2 still gives you a 2/3 chance to win something. That’s double the odds!
Take a look at the video below, which takes you through the whole process.
So, there you have it. The best chance you have is in changing your original choice and selecting Door #2.
The key is to remember that Monty already knows which door hides the prize. This makes the chance to change your door a conditioned action. Once Door #3 is opened, the remaining doors no longer have the same probability they did before.
For a more formal solution, we recommend reviewing Bayes Theorum, and this link to a more step-by-step solution, in particular.
Very curious but… how can we apply this in a financial environment?
I have designed a “metaphor” between this problem and the problem of identify trends in the Euro Stoxx 50 Price Index. In my last post I presented a method to predict trends based on the Naïve Bayesian classifier. (If you want to refresh your memory, go directly to Sir Bayes: all but not naïve!). It was a fair method (in the sense that it didn’t play with future information). It also drove results in stocks, with a bias in favour of success in predictions (>50%). I have tested this in the indexes S&P 500 and Euro Stoxx 50 and the results for S&P 500 show a bias in favour of success, correctly predicting 53% of the days. They were not as good in Euro Stoxx 50, however, where the success is an insufficient 42%.
To solve this weak result in Euro Stoxx 50, we want a more rewarding prediction of this index. To gain this, we are going to use the following relationship of causality:
We start then from the hypothesis that S&P 500 has an impact on Euro Stoxx 50
(with a lag of one day due to the physical location of time zones) in a “direct” mode:
So the simile between Monty Hall problem and the task we are going to test are:
- Which mix will be Monty Hall, the TV showman?
For us, our version of Monty is the Naïve Bayesian Classifier (from now on, referred as NB) applied to S&P 500. I have assumed a big simplification with a difference within the MH Problem. In the famous problem, Monty has absolute certainty about where the prize is; not the case in our NB method.
- Which situation is the simile of Monty Hall opening a door which has no prize? It has to be a situation that makes the participant think that he could have selected the correct door. In our case this will be:“ B Method applied to S&P 500 indicates the trend is ranged market and NB Method applied to Euro Stoxx 50 indicates that yesterday it was in trend. This means bullish or bearish so… we have to make the choice of not continuing to bet today for bullish or bearish markets for the Euro Stoxx 50, and going out of the market (this means, betting for ranged market).
We have also rejected the possibility of:
NB Method applied to S&P 500 indicates bullish/bearish and NB Method applied to Euro Stoxx 50 indicates that the yesterday was bullish/bearish (positive relationship). I therefore decide to change my bet in an extreme way to the opposite trend, and I estimate that today Euro Stoxx is in a bearish/ bullish trend:
We have not contemplated this option because we would be changing the bet in favour of something that goes against our initial hypothesis of a positive relationship between both indexes. This means that we are only applying the Monty Hall solution when the bet is not “clear” for Monty. That is, when Monty shows that S&P 500 is not in trend, and I’m not sure if I’ve chosen the correct trend (= the correct door) for Euro Stoxx 50.
Well, after our tests, the sad result of 42% success in Euro Stoxx prediction, has become 52%. This is not a result we should throw away.
If you want to know more interesting applications of the MH Problem, I suggest you read about “The principle of restricted choice“.
See you next time!