Markov chains say that any information that can be used to predict tomorrow’s price is contained in today’s price. Let’s do a little talking about that.
If we assume the previous statement, the probability of an event occurring depends only on the previous event.
An event can be: let the series go down (E1) or go up (E2).
– So, if I watch the movement today, can I know if I will win or lose tomorrow?
– Yes, by constructing a diagram that models the transition probability between the defined events.
…where P is the probability of passing from one state (event) to the other. As an example:
- P12 shows the probability of having a positive performance tomorrow, when today has been negative.
- P21 shows the probability of having a negative performance tomorrow, when today has been positive.
- P11 shows the probability of having a negative performance tomorrow, when today has been negative.
- P22 shows the probability of having a positive performance tomorrow, when today has been positive.
– And how do you know these probabilities?
– You can observe the probabilities in the past. In the last 20 years of an equity fund, this happened:
– Okay, so if today’s performance has been negative…that means tomorrow’s performance will be negative too?
– Exactly! If today (T0) is negative, we express it as:
We can calculate what will happen tomorrow (T1) as:
So the probability that tomorrow will be negative is 0.6, and that tomorrow is positive is 0.4.
– And in n days?
– So D will always have the first eigenvectors = 1 and its associated eigenvectors will be the probability ratio between E1 and E2 when n is at the limit. In this example, at infinity, E2 is twice as likely as E1.
Notice that the system seems to be forgetting the initial state where we left off.
– Okay, but why does P always have the first eigenvectors equal to 1?Because it’s a property of this type of transaction probability matrices, in which the columns always add up to one.
– And how does this help me?
– It doesn’t, very much. But you can enlarge the problem by defining more states that include the relevant information you use to model your time series. I have chosen the S&P 500. I’ve defined 12 states that characterise the magnitude of yields measured in standard deviations and have learned the P transition probability matrix across 40 years of history.
With it I can predict what will happen tomorrow, and that the performance will be between zero and two standard deviations.
-So the S&P500 model has been built.
– Yes, and I can generate as many S&P 500’s as I want with the same pattern of dependencies.
In the graph, each series is a different realisation and – perhaps in some – the S&P of the future.
Dare to read the Spanish version!