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

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.

Appendices

Appendix

Further results and time series descriptions

See Fig. 7 and Tables 6, 7, 8, 9 and 10.

Table 6 Regressing future returns on exposure signals from a flexible exposure strategy

Table 7 Regressing future volatility on exposure signals from a flexible exposure strategy

Table 8 Regressing future returns on exposure signals from a flexible exposure strategy with leverage

Table 9 Regressing future volatility on exposure signals from a flexible exposure strategy with leverage

Table 10 K. R. French Dataset, 30 US Industry Portfolios

Methodology with mathematical notation

A simplicial complex S is a set of simplices $\{ \sigma \}$ of arbitrary dimensions, satisfying:

• 1. for any face $\sigma ' \subseteq \sigma$ of a simplex $\sigma \in S$, $\sigma ' \in S$ is true.

• 2. for all simplices $\sigma _1, \sigma _2 \in S$ the intersection $\sigma _1 \cap \sigma _2$ is either $\emptyset$ or a face of both $\sigma _1, \sigma _2$.

Regarding its properties, a topological space can be associated to a simplicial complex and therefore to a dataset, from which a simplicial complex is constructed. Given a dataset $X = \{x_1, \ldots ,x_n \}$ arranged as a point cloud in an Euclidian space $\mathbb {R}^d$ and a real number $\delta > 0$, as our threshold parameter, we construct a simplicial complex $VR(X, \delta )$, in this context called Vietoris–Rips complex, by defining that any subset of X is a simplex of $VR(X,\delta )$ if and only if

$$\begin{aligned} d(x_i, x_j) \le \delta \text { } \forall i,j\in \{1,\ldots ,n\} \end{aligned}$$

(7)

We note that $VR(X,\delta ) \subset VR(X,\delta ')$, whenever $\delta \le \delta '$. By satisfying this condition, the family of VR complexes $\{VR(X,\delta )\}_{\delta >0}$ are called a filtration. To this filtration, we can apply the i-dimensional homology, obtaining a family $\{H_i(VR(X,\delta ))\}_{\delta >0}$ of vector spaces, also having the filtration property $H_i(VR(X,\delta )) \subset H_i(VR(X,\delta '))$, whenever $\delta \le \delta '$.

The dimension of each such a vector space $H_i(VR(X,\delta ))$ corresponds to the i-th Betti number of the topological space constructed from the dataset X at a given $\delta$ and therefore to connected components, holes, cavities and so forth.

Corresponding to the inclusions of the filtration, canonical homomorphisms $H_i(VR(X,\delta )) \hookrightarrow H_i(VR(X,\delta '))$, whenever $\delta \le \delta '$, can be defined. Therefore, for each nonzero i-dimensional homology class h, there exists an interval [a, b] such that $\forall \delta \in [a,b]: h \in H_i(VR(X, \delta ))$ and $\forall \delta \notin [a,b]: h \notin H_i(VR(X, \delta ))$. Here, a and b correspond to the birth and death point of h.

This information can be illustrated by a persistence diagram $P_i$, which contains every nonzero i-dimensional homology class h represented by a point with coordinates (a, b) in $\mathbb {R}^2$ and all trivial homology generators being born and dying at the same $\delta$, represented as the positive diagonal of $\mathbb {R}^2$. A metric for measuring the distance of two persistence diagrams is the p-th Wasserstein distance:

$$\begin{aligned} W_{p}(P^1_i,P^2_i)= \underset{\phi }{\text {inf}} \left[ \sum _{x \in P^1_i} \Vert x- \phi (x) \Vert ^p_{\infty } \right] ^{(1/p)} \end{aligned}$$

(8)

where $\phi$ are all bijections between $P^1_i$ and $P^2_i$.