Mustafa U. Torun

Portfolio Risk in Multiple Frequencies

Portfolio risk, introduced by Markowitz in 1952 and defined as the standard deviation of the portfolio return, is an important metric in the modern portfolio theory (MPT). A popular method for portfolio selection is to manage the risk and return of a portfolio according to the cross-correlations of returns for various financial assets. In a real-world scenario, estimated empirical financial correlation matrix contains significant level of intrinsic noise that needs to be filtered prior to risk calculations.

Toeplitz Approximation to Empirical Correlation Matrix of Asset Returns: A Signal Processing Perspective

Empirical correlation matrix of asset returns has its intrinsic noise component. Eigen decomposition, also called Karhunen-Loeve Transform (KLT), is employed for noise filtering where an identified subset of eigenvalues replaced by zero. The filtered correlation matrix is utilized for calculation of portfolio risk and rebalancing. We introduce Toeplitz approximation to symmetric empirical correlation matrix by using auto-regressive order one, AR(1), signal model. It leads us to an analytical framework where the corresponding eigenvalues and eigenvectors are defined in closed forms. Moreover, we show that discrete cosine transform (DCT) with implementation advantages provides comparable performance as a good approximation to KLT for processing the empirical correlation matrix of a portfolio with highly correlated assets. The energy packing of both transforms degrade for lower values of correlation coefficient. The theoretical reasoning for such a performance is presented. It is concluded that the proposed framework has a potential use for quantitative finance applications.

An Efficient Method to Derive Explicit KLT Kernel for First-Order Autoregressive Discrete Process

Signal dependent Karhunen-Loève transform (KLT), also called factor analysis or principal component analysis (PCA), has been of great interest in applied mathematics and various engineering disciplines due to optimal performance. However, implementation of KLT has always been the main concern. Therefore, fixed transforms like discrete Fourier (DFT) and discrete cosine (DCT) with efficient algorithms have been successfully used as good approximations to KLT for popular applications spanning from source coding to digital communications. In this paper, we propose a simple method to derive explicit KLT kernel, or to perform PCA, in closed-form for first-order autoregressive, AR (1), discrete process. It is a widely used approximation to many real world signals. The merit of the proposed technique is shown. The novel method introduced in this paper is expected to make real-time and data-intensive applications of KLT, and PCA, more feasible.

Novel GPU implementation of Jacobi algorithm for Karhunen-Loève transform of dense matrices

Jacobi algorithm for Karhunen-Loève transform of a symmetric real matrix, and its parallel implementation using chess tournament algorithm are revisited in this paper. Impact of memory access patterns and significance of memory coalescing on the performance of the GPU implementation for the parallel Jacobi algorithm are emphasized. Two novel memory access methods for the Jacobi algorithm are proposed. It is shown with simulation results that one of the proposed methods achieves 77.3% computational performance improvement over the traditional GPU methods, and it runs 73.5 times faster than CPU for a dense symmetric square matrix of size 1,024.

On basic price model and volatility in multiple frequencies

This paper revisits volatility and emphasizes interrelationships of risk metrics at various time horizons expressed in multiple frequencies. The basic price model defined by Black-Scholes equation and its extensions for varying variance scenarios are presented, i.e. Heston and GARCH models. Moreover, we highlight the significance of abrupt changes in the price of an asset on price modeling and volatility estimation. We extend basic price model where price jumps are taken into account as well. The proposed approach is validated by simulations, and shown that it improves volatility estimation.

FPGA, GPU, and CPU implementations of Jacobi algorithm for eigenanalysis

Parallel implementations of Jacobi algorithm for eigenanalysis of a matrix on most commonly used high performance computing (HPC) devices such as central processing unit (CPU), graphics processing unit (GPU), and field-programmable gate array (FPGA) are discussed in this paper. Their performances are investigated and compared. It is shown that CPU, even with multi-threaded implementation, is not a feasible option for large dense matrices. For the GPU implementation, performance impact of the global memory access patterns on the GPU board and the memory coalescing are emphasized. Three memory access methods are proposed. It is shown that one of them achieves 81.6% computational performance improvement over the traditional GPU methods, and it runs 68.5 times faster than a single-threaded CPU for a dense symmetric square matrix of size 1,024. Furthermore, FPGA implementation is presented and its performance running on chips from two major manufacturers are reported. A comparison of GPU and FPGA implementations is quantified and ranked. It is reported that FPGA design delivers the best performance for such a task while GPU is a strong competitor requiring less development effort with superior scalability. We predict that emerging big data applications will benefit from real-time and high performance computing implementations of eigenanalysis for information inference and signal analytics in the future.

A novel GPU implementation of eigenanalysis for risk management

Portfolio risk is commonly defined as the standard deviation of its return. The empirical correlation matrix of asset returns in a portfolio has its intrinsic noise component. This noise is filtered for more robust performance. Eigendecomposition is a widely used method for noise filtering. Jacobi algorithm has been a popular eigensolver technique due to its stability. We present an efficient GPU implementation of parallel Jacobi eigensolver for noise filtering of empirical correlation matrix of asset returns for portfolio risk management. The computational efficiency of the proposed implementation is about 34% better than our most recent study for an investment portfolio of 1024 assets.

Full length article: Optimal design of phase function in Generalized DFT

Recently, a theoretical framework for Generalized DFT (GDFT) with nonlinear phase functions exploiting the phase space to design various constant modulus orthogonal transforms was introduced. This paper extends prior work to design longer bases widely used in real world communications systems. GDFT provides flexibilities in phase space yielding performance improvements over other code families including DFT in correlation metrics. GDFT phase functions are optimally designed in order to reduce inter carrier interference (ICI) and inter-symbol interference (ISI) that dictate the performance of a multiuser channel. It is presented in the paper that superior correlation of GDFT over DFT manifests itself in improved BER performance in direct sequence CDMA based multi-user communications systems

FPGA based eigenfiltering for real-time portfolio risk analysis

The empirical correlation matrix of asset returns in an investment portfolio has its built-in noise due to market microstructure. This noise is usually eigenfiltered for robust risk analysis and management. Jacobi algorithm (JA) has been a popular eigensolver method due to its stability and efficient implementations. We present a fast FPGA implementation of parallel JA for noise filtering of empirical correlation matrix. Proposed FPGA implementation is compared with CPU and GPU implementations. It is shown that FPGA implementation of eigenfiltering operator in real-time significantly outperforms the others. We expect to see such emerging high performance DSP technologies to be widely used by the financial sector for real-time risk management and other tasks in the coming years.

On toeplitz approximation to empirical correlation matrix of financial asset returns

We present a Toeplitz approximation to symmetric empirical correlation matrix of asset returns by auto-regressive order one, AR(1), signal source modeling. AR(1) approximation provides an analytical framework where the corresponding eigenvalues and eigenvectors are defined in closed forms. Furthermore, we show discrete cosine transform (DCT) offers comparable performance to Karhunen-Loeve transform (KLT) for decomposition of empirical correlation matrix of a given portfolio where the first is significantly more efficient to implement. It is concluded that the proposed framework has a potential use for noise filtering and risk management in quantitative finance.