Why are Asset paths unpredictable in the short run?
May 16, 2009
This question was asked by a history major in one of the boot camps of Delhi-Bombay bank in London and New York in 2005. Delhi-Bombay bank (DBB), has, in its last twenty year history perhaps, hired, only one history – and for that matter, any liberal arts – major in its associate and analyst programs. It’s always math / physics Ph.D.s or Ivy League MBAs. The trainer in the boot camp was a veteran who had memorized every single finance text book that exists on this planet and who had been in bed with AJ and the entire DBB HR team for as long as we can remember.
That history major got fired from DBB this summer (a proof that shows at least life is unpredictable) and the trainer’s company got acquired by a big league rating agency (who saw great value in the finance and banking training and education business). Anyway, back to the question.
Why are asset path unpredictable in the short run?
This is a fundamental question the answer to which underlies the entire asset price modelling in finance. Simply put, the asset prices are unpredictable in the short run because the stochastic (random) differential equation that describes the asset price path evolution over time is non-differentiable. Asset prices are driven by a deterministic term, which is the drift, and a stochastic, random term. The randomness enters the asset price equation through the volatility of the asset price (or the asset return).
If an equation is differentiable – in mathematical terms – then it cannot stochastic. All stochastic (random) equations are non-differentiable, because the presence of randomness prevents the system from having a bounded measure. In stochastic equations the rate of change of a function with respect to time does not converge. Therefore, randomness ensures the non-existence of mathematical derivative.
Let’s take the arithmetic Brownian Motion to describe an asset price path:
In a discrete, difference form the above equation can be written as:
In the above, is a Weiner process through which randomness is introduced in the asset path and, is a random normal number. is expressed as:
Therefore, the change in asset price, over a small interval of time, is given by the stochastic difference equation:
However, the stochastic difference equation will become a stochastic differential equation (SDE) in the limiting case of infinitesimal time intervals. In the limiting case, when we slice time into smaller and smaller intervals, i.e. when delta time tends to zero, , then we get:
Therefore, the above expression goes to plus or minus infinity depending on the random normal number, .
This shows that a typical sample asset path, defined as a realization of as a function of time – is non-differentiable. The expression does not converge. Therefore, in the short run the asset prices have no predictability.
The above explanation is based on Jamil Baz and George Chacko’s excellent work titled Financial Derivatives, Pricing, Applications and Mathematics. Perhaps the DBB trainer forgot to memorize this one before the DBB boot camp. Or perhaps he was clueless about stochastic calculus.
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