[Paper Review] What can we see from Investment Simulation based on Generalized (m,2)-Zipf law?

Hokky Situngkir, Yohanes Surya|CogPrints (University of Southampton)|Apr 28, 2005

Complex Systems and Time Series Analysis3 references4 citations

TL;DR

This paper proposes a novel investment simulation framework using the generalized (m,2)-Zipf law to analyze financial time series by converting price fluctuations into textual 'words' of length m. By ranking word frequencies and applying a strength parameter D(t), the method predicts market direction and simulates portfolio growth, revealing that longer dominant word lengths in larger time scales correspond to shorter ones in smaller scales, with wildness linked to persistence and memory in market trends.

ABSTRACT

The paper revisits the investment simulation based on strategies exhibited by Generalized (m,2)-Zipf law to present an interesting characterization of the wildness in financial time series. The investigations of dominant strategies on each specific time series shows that longer words dominant in larger time scale exhibit shorter dominant ones in smaller time scale and vice versa. Moreover, denoting the term wildness based on persistence over short term trend and memory represented by particular length of words, we can see how wild historical fluctuations over time series data coped with the Zipf strategies.

Motivation & Objective

• To develop a new analytical framework for characterizing financial market dynamics by transforming price fluctuations into text-based sequences.
• To investigate how the persistence of short-term trends and memory effects in financial time series relate to the structure of dominant word patterns under the generalized (m,2)-Zipf law.
• To simulate investment strategies based on word frequency rankings and assess performance across different time scales and financial instruments.
• To explore the relationship between the length of dominant words (m) and the perceived 'wildness' of financial time series data.
• To determine whether dominant strategies in larger time scales (e.g., daily) correlate with shorter or longer word lengths in smaller time scales (e.g., hourly or sessional).

Proposed method

• Transform financial time series data into binary 'words' of length m using 'u' for upward and 'd' for downward price movements at hourly intervals.
• Apply the generalized (m,2)-Zipf law to rank word frequencies, modeling the inverse relationship between rank R and frequency f as f ≈ R^(-a), with a = |2H - 1|.
• Adjust for bias in up/down probabilities using f' = p^m-n(u) * p^n(d), where p(u) and p(d) are empirical probabilities of upward and downward movements.
• Calculate prediction strength D(t) = |(p_up(t) - p_down(t)) / (p_up(t) + p_down(t))| to determine the fraction of investment to buy or sell at time t.
• Simulate total portfolio value using ψ_total = ψ_start + Σ(p_i - p_i-1) * D_i over time, with ψ(t) = ψ(t-1) + [p(t) - p(t-1)] * D(t).
• Compare performance across different m-values (e.g., Zipf(2,2), Zipf(7,2)) to identify dominant strategies and assess wildness based on persistence and memory.

Experimental results

Research questions

• RQ1 How does the length of dominant words (m) in the (m,2)-Zipf framework vary across different time scales (e.g., daily vs. hourly) for the same financial instrument?
• RQ2 What is the relationship between the persistence of short-term trends and the memory effect encoded in word sequences under the generalized (m,2)-Zipf law?
• RQ3 How does the wildness of financial time series, defined by fluctuation persistence and memory, correlate with the optimal m-value in investment simulations?
• RQ4 Do financial instruments with higher volatility (e.g., foreign exchange rates) exhibit shorter dominant word lengths compared to less volatile indices like DJIA or NASDAQ?
• RQ5 Is there a systematic inverse relationship between dominant word length in larger time scales and smaller time scales across multiple financial instruments?

Key findings

• The Jakarta Composite Index (IHSG) showed optimal performance under the Zipf(6,2) strategy, while DJIA and NASDAQ performed best with Zipf(4,2) and Zipf(5,2), respectively.
• In the Hang Seng Index, the dominant strategy shifted from Zipf(7,2) in daily data to Zipf(4,2) in sessional and Zipf(3,2) in hourly data, indicating shorter dominant words at finer time scales.
• For the NIKKEI225, the dominant strategy evolved from Zipf(3,2) in daily data to longer words (Zipf(6,2)) in sessional and hourly simulations, showing the inverse pattern observed in Hang Seng.
• GBP/USD exhibited a similar trend with dominant strategies of Zipf(7,2) on daily data and Zipf(8,2) on hourly data, suggesting increasing word length with time scale in this instrument.
• Foreign exchange rates like Yen/USD and Euro/USD showed better performance with shorter word lengths (Zipf(3,2) and Zipf(2,2)), indicating higher wildness due to shorter persistence and memory.
• The study confirms that longer dominant words in larger time scales correspond to shorter dominant words in smaller time scales, and vice versa, revealing a time-scale-dependent structure in market dynamics.