When building simulations of rational agents trading in dynamic markets over extended periods of time, tell your optimiser to solve everything down to 1 part in 100 million. If you don’t, all those tiny little errors will start to interact with each other… and 10 years down the track bad things will start to happen.
Mathematical finance can be really highly strung.
4 responses so far ↓
1 Richard Kennaway // Apr 5, 2008 at 11:15 am
If accumulated errors blow up that badly, doesn’t that suggest something wrong with the model? The real-world data you base it on are going to be far more uncertain than 1 part in 100 million.
2 mathemajician // Apr 5, 2008 at 11:41 am
True, if this was going to be fed read data it would blow up! The system would be chaotic.
In this case I’m using a simulation to study theoretical models of optimal portfolio choice. No data, just equations for the system’s dynamics. I’ve done the Merton model and now have a model by Wachter with mean reverting Sharpe ratios working. I started with these as they have analytic solutions. I’m now moving into behaviour finance models such as prospect theory for which analytic solutions can be very difficult.
Over short time periods matching the analytic models is no problem, but with a few hundred time steps tiny errors build up and things in the simulation deviate from the analytic solutions… which is bad news for us as we want to use this to understand the analytic models.
Nevertheless, I think it’s still valid to question whether analytic models which are so sensitive when simulated are good models of real behaviour over extended time periods.
In our particular case we are interested in prospect theory and in this theory we usually consider relatively passive investors who rebalance their portfolio about once a year. In this case simulating 30 years is no problem at all and the solutions at each point in time don’t need to be all that accurate. But for other models, simulating the behaviour can be problematic.
3 Falafulu Fisi // Apr 7, 2008 at 8:46 am
Yep. You are talking about the discipline of numerical analysis. Rounding-off errors and ill-conditions are important issues when a programmer develops numerical codes, because if they’re neglected, there will death as a result, and this is what happened in the 1991 Gulf war.
“Patriot Missile Failure”
http://ta.twi.tudelft.nl/nw/users/vuik/wi211/disasters.html
4 mathemajician // Apr 7, 2008 at 2:23 pm
Falafulu: Interesting page
Leave a Comment