پیچیدگی الگوریتمی حرکت های مالی

We survey the main applications of algorithmic (Kolmogorov) complexity to the problem of price dynamics in financial markets. We stress the differences between these works and put forward a general algorithmic framework in order to highlight its potential for financial data analysis. This framework is “general” in the sense that it is not constructed on the common assumption that price variations are predominantly stochastic in nature.

In driving the decisions made by investors, information fuels financial markets. But the market has proven to be very complex in its dynamics and therefore very hard to predict. Market price movements in themselves are unpredictable or barely predictable. Price movements can be seen as the outcome of interactions between investors following rules in their quest to reap a benefit. It has been suggested that the market alone is complex enough, even when isolated from external stimuli (see, e.g. Wolfram, 2002), yet external information can make it less or more predictable. These concepts are at the heart of one of the most famous hypotheses in finance: the Efficient Market Hypothesis (EMH), painstakingly reconstructed from the recently rediscovered works of Bachelier (1900) and subsequently refined. The concepts shaping this hypothesis, such as “information” or “randomness”, are undoubtedly of interest to computer scientists who have their own tradition of tackling these questions. Part of this tradition can be identified with the works of Shannon, and has provoked a burgeoning literature in finance. Another part of this tradition can clearly be linked with the works of Kolmogorov (1965) and Chaitin (1987). This paper attempts to assess the existing works in these fields, highlighting salient divergences and proposing a general algorithmic framework as an alternative to the mainstream probabilistic one used in financial analysis. This article is organized as follows: after a theoretical introduction to algorithmic complexity in Section 1, we take a quick look at the relation between financial theories and the randomness of price variations in Section 2. As this relation is studied by some existing works applying the notion of algorithmic complexity, we provide a survey of these works in Section 3 and show why they failed to propose a general algorithmic framework for financial pattern tracking, which was not available until the publication of our work in Ma (2010) and Zenil and Delahaye (2011). The main contributions of these two works are then sketched in 4 and 5, respectively.

The most obvious feature of financial markets is the apparent randomness with which prices tend to fluctuate and which most standard models try to capture. Nevertheless, the very idea of chance in financial markets clashes with our intuitive sense of the processes regulating the market. Traders do not just follow hunches, but act in accordance with specific rules, and even when they do appear to act on intuition, their decisions are not random but instead follow from the best of their knowledge of the internal and external state of the market. For example, traders copy other traders, or take the same decisions that have previously worked, sometimes reacting against information and sometimes acting in accordance with it. These deterministic processes could leave signatures (patterns) on financial data. To reveal their presence, algorithmic tools constitute a good alternative to stochastic models. In this paper, we have surveyed the principal applications of algorithmic (Kolmogorov) complexity to the problem of financial price motions and showed the relevance of the algorithmic framework to structure tracking in finance. Some empirical results are also provided to illustrate the power of the proposed estimators to take into account patterns in stock returns. Of course, the empirical tools reviewed above are only some of the uncountable possibilities opened up by the theory of algorithmic complexity. Just as one can always design new statistical tests for structure detection, the development of new algorithmic tools could enlarge the scope of patterns taken into account by the algorithmic framework and hence improve our comprehension of financial price dynamics.