He ain’t ergodic, he’s my brother November 17, 2013 at 1:57 pm
I have been meaning to blog for a while about ergodicity. I know, exciting stuff. Here’s the skinny:
Roughly speaking, a system is ergodic if the time average is the state space average. Suppose we have a financial asset with genuinely IID returns: then if we look at the average return over time, that will be same as the average return over all possible paths.
The key point here is that computing the phase space average requires that we can reasonably take multiple copies of the system and observe different paths. A coin, say, can be tossed multiple times, allowing us to see the whole phase space.
For most financial systems, this copyability is not present. It might be reasonable to attribute a probability of default to a company a priori, for instance, but a posteriori it either defaults or it doesn’t, and we cannot take multiple copies of it and see how many times it does in repeated experiments. All we can do is look at it through time.
Given that we can’t often measure a phase space average, it would be handy if many financial systems were ergodic. Unfortunately, as this Towers Watson post points out, they often aren’t.
Risk managers therefore need to be very careful to distinguish two situations:
- I have a lot of genuinely independent bets going at once; and
- I have one bet that I repeat multiple times.
The former might, for instance, be hedging lots of different single stock options (on uncorrelated stocks, not that there are such things); the latter would be hedging a rolling position on one stock. In the first case you can reasonably take the phase space average – so if I sell the options for more than the historic volatility and I have enough of them, I will on average make money. In the second, you can’t. Here running out of money/hitting your limits and being forced to close out are much bigger issues.
Do read the PDF linked to in the Towers Watson post for more: it’s insightful.