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Financial Markets and Portfolio Management

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10-02-2021

The better turbulence index? Forecasting adverse financial markets regimes with persistent homology

Authors: Eduard Baitinger, Samuel Flegel

Published in: Financial Markets and Portfolio Management | Issue 3/2021

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Abstract

Persistent homology is the workhorse of modern topological data analysis, which in recent years becomes increasingly powerful due to methodological and computing power advances. In this paper, after equipping the reader with the relevant background on persistent homology, we show how this tool can be harnessed for investment purposes. Specifically, we propose a persistent homology-based turbulence index for the detection of adverse market regimes. With the help of an out-of-sample study, we demonstrate that investment strategies relying on a persistent homology-based turbulence detection outperform investment strategies based on other popular turbulence indices. Additionally, we conduct a stability analysis of our findings. This analysis confirms the results from the previous out-of-sample study, as the outperformance prevails for most configurations of the respective investment strategy and thereby mitigating possible data mining concerns.

next article Cross-validated covariance estimators for high-dimensional minimum-variance portfolios

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The trade-off between outperformance in turbulent times and underperformance in tranquil times can be controlled by the (indirect) weighting of negative/worst-case estimates of portfolio optimization inputs in the respective objective function.

See Scherer (2007).

Both methodologies can also be combined by estimating market regimes with the help of a risk indicator as shown in Kritzman et al. (2012).

An extensive discussion on TDA/PH can be found in Carlsson (2009, 2014).

Admittedly, even though it is a rule-based mathematical exercise, a PH-based analysis of very high-dimensional data still requires a lot of computing power.

As will be described more in detail below, we use a rolling window of daily data for the calculation of PH leading only to little changes in the underlying data.

See also Kritzman and Li (2010).

To be precise, the daily VIX data are constructed from three time series. The first time series contains the VIX calculated in real-time. It spans from the 22nd of September 2003 to the last trading day of December 2019. The second time series contains the reconstructed VIX (via back-calculation). It spans from the first trading day of January 1990 to the 19th of September 2003. Lastly, the third block contains the reconstruction of the S&P 100 Volatility Index (VXO). It spans from the first trading day of January 1986 to the last trading day of December 1989. Since the VXO and the VIX are very tightly correlated, this workaround should not affect the findings of our paper.

All methodologies and studies are implemented with the R programming language.

Furthermore, since our paper predominately targets finance practitioners, we do not always stick to a concise mathematical language throughout the paper.

Further applications are: breast cancer identification (Nicolau et al. 2011), semantic pattern recognition in political speeches (Gurciullo et al. 2015), critical transitions in genetic regulatory systems (Berwald and Gidea 2014) and spectroscopy (Offroy and Duponchel 2016). This list makes no claims to completeness.

Their work on the other hand was inspired by previous research of Edelsbrunner et al. (1983), Edelsbrunner and Mücke (1994) and Delfinado and Edelsbrunner (1995). Precursors to the full concept of PH were introduced by Frosini (1990) and Robins (1999).

Precisely, the dashed circle represents \(\delta /2\). We omit this fact in the main body of the text, as it leads to unnecessary complexity of the discussion.

To be precise, the hole emerges approximately at \(\delta =0.76537\). At \(\delta =0.76537\), all nearby points are connected by simple one-dimensional lines in such a way, that a “big hole” emerges.

To be precise, the hole disappears approximately at \(\delta =1.84776\). At \(\delta =1.84776\), all points in the cloud are connected by simple one-dimensional lines, leading to the disappearance of the “big hole”.

The selected period is characterized by the lowest return variance for a 60-trading-day window for the years 2006–2008. The variance was measured in monthly (month-end) intervals.

Analogous, this period exhibits the highest return variance for a 60-trading-day window for the years 2006–2008, whereby the variance was measured in monthly (month-end) intervals.

The interested reader is referred to the “Appendix” (Fig. 7), where we repeated the analysis of Sect. 3.3 for the period shortly before the bankruptcy of Lehman Brothers. This period can be considered as the transition phase towards turbulence.

Using more concise mathematical terms, W is a so-called earth mover’s distance. If we consider two distributions as two piles of earth, W quantifies the minimum cost of turning one pile into the other.

The interested reader is referred to the “Appendix” for a detailed and mathematical discussion of the Wasserstein-Distance. We calculated W up to the first dimension. Calculating W for higher dimension gets increasingly time consuming and, in our experience, does not meaningfully improve the performance of the PHTI.

We downloaded monthly Treasury Bills return data from the homepage of Amit Goyal (http://www.hec.unil.ch/agoyal/).

The original Mahalanobis distance is the square root of the distance indicator \(d_{t}\)

By using a trailing empirical window for quintile assignment, we follow the methodology of Kritzman and Li (2010). This approach deviates from Chow et al. (1999). They use theoretical quintile borders based on the assumption that \(d_{t}\) follows a chi-square distribution.

The complete results are available upon request.

The complete results are available upon request. As expected, the control variable is highly significant.

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Title: The better turbulence index? Forecasting adverse financial markets regimes with persistent homology
Authors: Eduard Baitinger
Samuel Flegel
Publication date: 10-02-2021
Publisher: Springer US
Published in: Financial Markets and Portfolio Management / Issue 3/2021
Print ISSN: 1934-4554
Electronic ISSN: 2373-8529
DOI: https://doi.org/10.1007/s11408-020-00377-x

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