Noncomputability, Unpredictability, and Financial Markets

摘要

One of the most significant achievements from theoretical computer science was to show that there are noncomputable problems, which cannot be solved through algorithms. Although the formulation of such problems is mathematical, they often can be interpreted as problems derived fromother fields, like physics or computer science. However, no non-computable problem with economical or financial inspiration has been presented before. Here, we study the problem of valuation: given some adequate data, find the value of an asset. Valuation is modeled mathematically by the discounted cash flow operator. We show, using surprisingly simple arguments, that this operator is not computable. As theoretically, financial markets should trade assets based on their fair value, our result suggests that unpredictability of such markets may partially stem from inherent noncomputable behavior. A discussion of this result is also included.