Path Dependence in Monte Carlo Simulation

From Javier...

Day three of the tournament and the path of the World Cup is slowly starting to reveal itself. The games played to date already have had a dramatic effect in reducing the potential number of combinations of the World Cup that we described in a previous blog entry on World Cup Statistics.

Monte Carlo simulation typically involves the modelling of dynamic processes. The models start at time To, and the interplay of the model structure and the outcomes of the assumed stochastic processes (governed by the random number generators of the model) leads to a calculation of the state of the model's world in T1, and the process is reiterated for T2, then T3 etc all the way to Tn. The final result is dependent on the path taken by the relevant variables and/or interim results. Therefore the term "path dependence".

Intuitively path dependence means something as common sensical as that "today is almost like yesterday" and a "lot like last week". However, over time small cumulative changes can lead to vastly different final outcomes. For example, take two 12 year old kids. One studies an extra hour a day than the other. At the end of 30 years, that extra effort may lead to a very significant gap in terms of educational degree completed and earnings.

Do not confuse path dependency with the "butterfly effect", which is the extreme sensitivity of the results of certain types of models to initial conditions and is studied in chaos theory.

In the Isthmus Partners World Cup Simulator there is path dependency. One is tempted to say that the first round of the playoff matches is path dependent on the results of the Group stages, but we need to be careful how we frame that statement. For example, the World Cup organisers have determined that the first team of Group A will play the second team of Group B in the first round of the playoffs. But that is not path dependency, that is a rigid structural feature of the Simulator. What is path dependent for that match-up are the results of the games in Group A and Group B which determine which team is number one in Group A and which team is number two in Group B. In the playoff stages path dependence is evident as the match-ups of each game in the second and successive rounds are directly determined by the two relevant games of the previous round.

The head to head distributions of results in the Isthmus Partners World Cup Simulator is one-dimensional. It is calculated once and applied throughout the tournament. But we could have, with lots of time, have made the model more nuanced and path dependent. For example, if a team wins its first two games in the group stages and is already qualified as first in the group, the model may take into account that that team would tend to field in the third game a lesser squad and play more relaxed. We could have also modelled the "propensity" to a tie in the third game between two teams which both pass guaranteed to the second round with the tie (reminiscences of 1982 scandal between W. Germany and Austria, where both teams agreed to a 1-0 result that meant both teams passed to the second stage in detriment of Algeria; This is why both third matches of each group are now played concurrently). And we could have also modelled that if a team has a string of good results that it would more likely continue to have a "hot hand" and outperform its original distribution.

Path dependence is very relevant to those of us who have been conditioned in MBAs and Finance to think that financial asset prices (such as stocks) follow a random walk. This approach was popularised in Burton Malkiel's classic "A Random Walk Down Wall Street" (first published 1973). A French mathematician, Bachelier, produced a thesis in 1900 in which he applied a stochastic process observed in nature, Brownian motion, to financial assets. Bachelier's work went unnoticed until re-discovered by the famous economist Paul Samuelson.

So once you know the initial price, the expected return and the expected volatility of a financial asset you can start simulating future paths of the financial asset. An intuitive way to understand volatility is using our previous description of today being "almost like yesterday". The higher the volatility, the wider the range of outcomes that today can take compared to yesterday. And the more likely that next month will be a lot different to today.

One note of caution here: a building is "path dependent" in that a sense that is looks like it was yesterday, the day before yesterday, etc. But a fire or earthquake can change that in a moment. So be careful with unexpected events, but not impossible (actually quite probable over an extended period of time) that have disproportionately large impacts. For this you can complement your Monte Carlo simulation with extreme scenario analysis and prepare contingency plans for those scenarios and buy insurance and/or hedge yourself to a feasible degree. But be humble and remember Nassim Taleb's warning in the "Black Swan" - there are events which will impact us which we cannot even contemplate or imagine. For example in the Simulator we have not modelled extreme events that may cancel the competition (God forbid). But that does not mean it cannot happen (I believe FIFA buys something akin to business continuity insurance for the World Cup - please comment if I am wrong).

The key point here is that when modelling Monte Carlo simulation be cognisant of path dependency and try to find out the insights that the modelling process reveals. Every time you produce a model you should strive to learn one or more unexpected lessons.

Let the path dependence of the World Cup continue!