With this article we are starting a series devoted to the theory of probability, in particular about its most unexpected application in the financial sphere.
For hundreds of years mathematicians sum, divide, multiply and compare different infinities with each other. Infinite values are the most abstract matters of mathematics, but sometimes they influence our everyday lives. In this series we will discuss how they affect the financial sphere.
There are some paradoxes in the probability theory that imply endless profits and flourishing in case of their implementation. Such hypotheses are never taken seriously, but in some cases they reflect the reality, at least partly. These cases will be discussed in our next articles.
Special focus in the probability theory is made on the random motion. In mathematical terms this phenomenon is described in a rather abstract way: it’s when a point moves randomly in different directions. In the real life, however, this activity can be applied to a variety of situations, including some conventional states of the financial sphere. Let’s start our voyage of discovery, however, from a comic hypothetic situation: a drunk person moves randomly after leaving a pub.
Wandering pub crawlers
Imagine a person leaving a pub and going somewhere. He is so drunk that each step he makes is random: he does not remember where he is going, and he does not even control the direction of his movement. Luckily the bar is situated in the dead end of a narrow street, so he can only make steps in two directions: away from the bar or back. The first step is always made outside the bar, but afterwards he is always moving randomly: step forward, step forward, step back, step forward, step forward, step forward, step back… In mathematics, such process is called one-dimension random motion.
In our case, if a person goes back to the bar, he will be met by the guards who will take him back inside and get him to sleep in the nearest armchair.
There are a lot of drunk people in the bar we are watching. They are leaving the institution, and we are recording their pathways.
Our three questions are:
The answer to the first question is quite simple, it is taught to the students at universities: the average distance is equal to root of N. If a person makes 100 steps, he will most likely be 10 steps away from the bar.
The second question is easily solved with the help of logic. It’s given that the first step is always made away from the bar; the second step leads him back inside with the probability of ½. It means that the half of wanderers will return after two steps, which means it is the median.
The third question is not that evident. Median and average values can differ a lot, and in our case the average time of return will be… infinite. At least this is derived from the mathematical formula.
How does it happen?
Wandering pub crawlers in imaginary experiments
To find an average position of a drunk person after N steps, we should conduct the experiment multiple times and find an arithmetic average of the number of steps made by all of them.
Say, at first we watched five people and get simple results. Two people returned to the bar after two steps, the third came back after four steps, and the rest came back after eight steps.The median is thus 3 steps, while the average is (2*2+1*4+2*8)/5=24/5=4.8 steps. Well, it’s not that much, and is not even close to infinity.
Five experiments is obviously not enough for good statistics. Next time we will watch 20 people and get the following picture:
The half of rally participants again came back after 2-4 steps. 6 crawlers made several dozens of steps and came back, too. But the other 4 people went so far that their adventures did not even fit into the chart. They returned in 140, 198, 202 and 298 steps.
The mean deviation of the wanderers corresponds with the theory. For example, in the theory the average distance from the bar after 20 steps should be between 4 and 5 steps (root of 20). When N=20, we are still observing the movement of 7 people, while the rest are already sleeping in the bar. Two of them have just come back, one is two steps away, another one’s distance is four steps, two more have made six ‘pure’ steps and the last one — 10 steps. The average distance is (2*0+1*2+1*4+2*6+1*10)/7=28/7=4, exactly as prescribed by the theory.
The median time of return is twice bigger than the theoretical estimate, four steps. This deviation is allowable taking into account the small number of cases.
It is much more difficult when speaking about the average return time. In our case it is (7*2+3*4+1*8+1*10+2*20+1*30+1*34+1*140+1*198+1*202+1*298)/20 = 986/20 = 49.3 steps (!), and it has nothing to do with the previous case when we had only five people. This is all caused by the four ‘adventurers’ that went have made too many steps. Why do they appear? Will it be possible to find some real wandering time after some more experiments?
The point is: the more experiments we make during our observation and the more wanderers we watch, the bigger will be the average number of steps made. The half of all drinkers will still return after 2-4 steps, but the rest will be able to go very far away.
These charts illustrate the probability of pub crawler’s wandering in the street after N steps and the probability of his returning back exactly at the N-th step. Both charts are exponential.
Mathematicians have derived a formula that allows counting the probability of a person’s return to the bar at the N-th step and the probability of his continuous walk. They are rather complex, so we will just take a look at the corresponding charts (above). A pub crawler can come back only at an even step. For the second step, the probability of comeback is ½. For the fourth step this probability will be ⅛, while the probability of remaining in the street will be ½-⅛=⅜. Further even steps will get our wanderers back to the bar with the probability of N raised to the power of 3/2, while the probability of their further walk will be counted as the root of N.
It means that a part of people will have a very long walk. E.g., if N=100, every tenth of the wanderers will still be in the street, if N=10,000, we will still be watching every 100th if them. In mathematics, such long returns are called Lévy flights. The interesting feature about them is that the increase in the experiment number causes the increase in the average time of the flight.
If we have 10 people, the most ‘adventurous’ of them will make around 100 steps, while the majority of the rest will not go further than 2 to 4 steps. The length of the walk can easily exceed the sum of other participants’ walks. As a result, the average return time will be 12-14 steps.
Same, is we have 100 participants, the most adventurous one will make 10,000, but the majority will still make 2 to 4 steps, thus making the average return time close to 100 steps.
The length of the longest streak will correspond with the square of the participants number, while the average return time will generally be close to the their number. The more wanderers we observe, the closer will the average return time be to the infinity (both pathways have their ends, though.) This is the strange phenomenon that can be, however proven mathematically — and even experimentally.
This experiment is important not only for the movement of pub crawlers, but also for the capital motion. The movement of asset prices, traders’ profits and losses can all be illustrated with the help of accidental processes often described mainly by the point wandering, including Lévy flights. These situations will be described in other parts of our series.