@bucciGeometricDeepLearning2024

Geometric deep learning for realized covariance matrix forecasting (2024)

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1 Abstract

Traditional methods employed in matrix volatility forecasting often overlook the inherent Riemannian manifold structure of symmetric positive definite matrices, treating them as elements of Euclidean space, which can lead to suboptimal predictive performance. Moreover, they often struggle to handle high-dimensional matrices. In this paper, we propose a novel approach for forecasting realized covariance matrices of asset returns using a Riemannian-geometry-aware deep learning framework. In this way, we account for the geometric properties of the covariance matrices, including possible non-linear dynamics and efficient handling of high-dimensionality. Moreover, building upon a Fr’echet sample mean of realized covariance matrices, we are able to extend the HAR model to the matrix-variate. We demonstrate the efficacy of our approach using daily realized covariance matrices for the 50 most capitalized companies in the S&P 500 index, showing that our method outperforms traditional approaches in terms of predictive accuracy.

2 NOTES

In this paper the authors propose two DRNNs: ReSPDNet and GeoHAR (Geometric HAR).

2.1 ReSPDNet

The ReSPDNet is a SPDNet modified to work as a regression model, whose output are covariance matrices. Its inputs is as follows:
Let $n$ be the number of stocks, the dataset is a time series of RCOV matrices $R{C}_1,\dots ,R{C}_T$, each of size $n\times n$. Let $k$ be the number of input lags to be used, then the resulting dataset used is as follows: $\{(D_t,Y_t)\mid D_t\in \mathcal{I},Y_t\in \mathcal{O}\}$ where $\mathcal{O}=\{R{C}_k,\dots ,R{C}_T\}$ and $\mathcal{I}=\{D_k,\dots ,D_T\}$ and each $D_t$ is obtained as a block-diagonal matrix of $k$ input-lagged matrices preceding $Y_t$:

$$D_t=\left[\begin{array}{ccccc}{Y}_{t-1}& 0& 0& \cdots & 0\\ 0& Y_{t-2}& 0& \cdots & 0\\ 0& 0& Y_{t-3}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & Y_{t-k}\end{array}\right]\text{\hspace{1em}(4)}$$

At each training step, $D_t$ will be the input for the first layer of our network.

2.2 GeoHAR

This model extends the idea of Corsi, incorporating in the input the realized volatilities over different time horizons — daily, weekly, and monthly.
For instance, by computing the Fréchet mean over the previous 5 and 22 observations — mirroring the weekly and monthly horizons in the original HAR model — they can obtain two $n\times n$ sample means of covariance matrices:

$$R{C}_{t-1}^w=\arg \underset{RC\in {\mathcal{S}}_{++}^n}{\min }\frac{1}{5}\sum _{i=1}^{5}d(R{C}_{t-i},RC{)}^2,$$ $$R{C}_{t-1}^m=\arg \underset{RC\in {\mathcal{S}}_{++}^n}{\min }\frac{1}{22}\sum _{i=1}^{22}d(R{C}_{t-i},RC{)}^2,$$

which can be used to construct the input diagonal matrix in a HAR-like fashion as follows:

$$D_t=\left[\begin{array}{ccc}R{C}_{t-1}^d& 0& 0\\ 0& R{C}_{t-1}^w& 0\\ 0& 0& R{C}_{t-1}^m\end{array}\right],\text{\hspace{1em}(11)}$$

where $R{C}_{t-1}^d$ is the previous lag, and the suffix “d” only denotes the daily nature of the memory component.
In this way, we extend the HAR model to the matrix-variate framework and capture the dynamics of the entire covariance structure.