28 December 2009

Postscript to the preceding discussion

To those interested in the financial markets we may point out a partially related meditation resulting from our preceding discussion. Many (not all) financial models, having been developed by mathematically inclined persons used to thinking of continuous quantities, model asset prices as function taking place in continuous time. Hopefully we are all beginning to question now, given that we have shown that space is discrete, whether time is discrete as well (or whether these mathematical categories are even meaningful in relation to the essence of time and space). However, ignoring that point for the moment, we must point out as fact that time in the financial markets is not continuous, since the markets open and close at definite points, placing discrete cuts in the apparently continuous whole of market activity. This would be irrelevant if asset prices could not change after market close. However, we can observe, albeit rarely, enormous changes in value that occur (practically speaking) instantaneously at the market’s opening that result from accumulated orders placed overnight, usually as a result of news that came after the previous day’s market close.

This is especially important if we consider portfolios that attempt to remain neutral to market fluctuations through the use of derivatives. If asset prices are modeled using Brownian motion (the usual method), which was originally designed to model the behavior of particles of pollen suspended in water, then the model will not be able to appropriately price the risk of a discontinuous overnight movement, since the objects Brownian motion models cannot move discontinuously. This poses no problem to the original intended uses of Brownian motion, but it does pose a significant problem to attempts to hedge portfolio value, since it will result in the failure of such methods at precisely the point when they are most necessary. The financial engineers who designed such portfolios, instead of questioning the validity of their underlying theoretical assumptions, will all-too-often simply accept that the infinitely small probability of hedging failure predicted by their financial models had in fact transpired, effectively blaming fate instead of their own mathematically brilliant models, models that have the one drawback of not entirely corresponding to that which they seek to describe.

There are other issues with the usual formulas used to construct hedged positions, but these are beyond the scope of our present inquiry into the essence of space.

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