[Paper Review] A mathematical proof of the existence of trends in financial time series

Michel Fliess, Cédric Join|arXiv (Cornell University)|Jan 14, 2009

Mathematical and Theoretical Analysis3 references19 citations

TL;DR

This paper proves the existence of persistent trends in financial time series using nonstandard analysis and a theorem by Cartier and Perrin, challenging the random walk and efficient market hypothesis. It applies low-pass filtering and control theory techniques to detect trends that coexist with random walk behavior but conflict with the Black-Scholes model’s assumptions.

ABSTRACT

low-pass filters, nonstandard analysis, operational calculus. We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis (Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice, Springer, 1995, pp. 195–204). Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. They are estimated via recent techniques stemming from control and signal theory. Several quite convincing computer simulations on the forecast of various financial quantities Our aim is to settle a severe and longstanding quarrel between 1. the paradigm of random walks 1 and the related efficient market hypothesis [15] which are the bread and butter of modern financial mathematics,

Motivation & Objective

• To resolve a longstanding debate in quantitative finance about whether trends exist in financial time series.
• To reconcile the existence of trends with the random walk and efficient market hypothesis paradigms.
• To challenge the assumptions of the Black-Scholes model, which assumes no persistent trends.
• To provide a mathematical foundation for trends using nonstandard analysis and integration over finite sets.
• To demonstrate the feasibility of detecting such trends using signal processing and control theory techniques.

Proposed method

• Utilizes a theorem by P. Cartier and Y. Perrin from nonstandard analysis to define integration over finite sets.
• Applies low-pass filtering techniques to extract long-term components from financial time series.
• Employs operational calculus to handle generalized functions and differential equations in the context of nonstandard analysis.
• Uses control and signal theory methods to estimate and validate the presence of trends in noisy financial data.
• Employs computer simulations to demonstrate the detectability and consistency of trends across various financial quantities.
• Operates within the framework of nonstandard analysis to formalize the concept of trends in discrete, finite time series.

Experimental results

Research questions

• RQ1 Do persistent trends exist in financial time series despite the dominance of the random walk hypothesis?
• RQ2 Can trends coexist with the efficient market hypothesis and random walk models without contradiction?
• RQ3 To what extent do the assumptions of the Black-Scholes model break down when trends are present?
• RQ4 Can nonstandard analysis provide a rigorous mathematical foundation for detecting trends in finite, discrete financial data?
• RQ5 How effective are low-pass filtering and control theory techniques in isolating and estimating trends from noisy financial time series?

Key findings

• The paper establishes a mathematical proof for the existence of trends in financial time series using nonstandard analysis.
• Trends are shown to coexist with random walk behavior and the efficient market hypothesis, though they challenge the assumptions of the Black-Scholes model.
• The application of low-pass filtering and signal processing techniques successfully isolates detectable long-term components in financial data.
• Computer simulations demonstrate the feasibility and consistency of trend estimation across various financial quantities.
• The use of Cartier and Perrin’s theorem in nonstandard analysis provides a rigorous framework for defining trends in finite, discrete time series.
• The results suggest that financial time series may contain deterministic, non-random components that are overlooked by traditional models.