The eigenvalues of data-driven exponentially weighted moving average (DDEWMA) covariance matrix of stock returns plays an important role in portfolio optimization. Moreover, the eigenvalues of the Laplacian matrix play a fundamental role in financial network analysis. The novelty of this paper is to apply Random Matrix Theory (RMT) to denoise three covariance matrices ( empirical covariance, neuro covariance and DDEWMA covariance). Simulation study shows that portfolio weights generated using RMT filtered empirical covariance matrices based on both normal and t-distributed returns are more resilient. The networks built from t-distributed returns are more resilient (fewer but stronger connections with smaller average degree of nodes) than the networks built from normally distributed returns. Unlike the existing work, the novelty of this paper is to first show that the superiority of the RMT-filtered DDEWMA covariance model in portfolio optimization and, then show that the networks built from t-distributed returns are more resilient.

Forecasting stock prices and volatility plays a crucial role in making better investment decisions. Recently, there has been a growing interest in studying the Forecast Linear Augmented Projection (FLAP) method for multivariate normally distributed time series to reduce the forecast error variance and mean squared error (MSE) of long-term forecasts. FLAP method consists of three steps: formation of principal components, forecasting original and component series (base forecasts), and then projecting those base forecasts. First, this paper shows that the non-linear prediction (conditional mean) has a smaller MSE than the linear prediction. Then, it demonstrates that for multivariate time series, forecasts using the neural network autoregressive (NNAR) model with FLAP outperform those obtained by the commonly used autoregressive integrated moving average (ARIMA) model forecasts with FLAP (resulting in lower out-of-sample root mean squared error (RMSE) in general). For simulated multivariate data with t-distributed errors, the combined forecast model with NNAR and FLAP outperforms the NNAR forecast model. In the empirical study, we forecast stock price and volatility for 55 most traded stocks and S&P500 index using the ARIMA model with FLAP and the NNAR model with FLAP. The results show that FLAP is more effective in reducing forecast error variance and RMSE for non-stationary multivariate price and volatility forecasts.

Pairs trading and multiple trading strategies are types of market-neutral strategies to use a pair or a combination of stocks and other financial instruments with co-integration or co-movements to generate potential profits, which may not be affected by the direction of the overall market. Commonly used pairs and multiple trading strategies are constructed using the Kalman Filter (KF) to utilize mean reversion in nonstationary but co-integrated asset prices. In this paper, we propose novel resilient pairs trading and multiple trading strategies using the combination of the KF algorithm and the KF innovation volatility interval forecasts using neural networks. The proposed trading strategies are implemented and investigated using the hourly prices of Bitcoin, Ethereum and Bitcoin Cash in the bear market. Those crypto assets are selected because they move in the same direction in the long term and have high trading volumes. The experimental results reveal performance for the proposed trading strategies with the upper and lower trading intervals using KF innovation volatility interval forecasts superior to that of the trading strategies with upper and lower trading bands using KF innovation volatility point forecasts. The performance and robustness of the proposed trading strategies using a proper assumption of transaction costs have also been examined. The strategies using innovation volatility interval forecasts consistently generate higher profits and a more robust number of transactions with or without transaction costs than those using innovation volatility point forecasts.

One of the recent developments in computational finance has been the rise in using graph-based approaches to analyse stock market dynamics systematically. In algo trading literature, stocks for pairs trading and multiple trading are commonly selected by using Engle Granger and Johansen co-integration tests. Identifying all pairs eligible for trading in large datasets, entails a significant computational burden. In this paper, price correlation-based dynamic networks and differenced price series-based correlation networks and their importance ranks are used to pre-select pairs for trading. Algorithmic trading profits using commonly used co-integration methods are compared with the profits made by proposed correlation-based financial networks. Unlike the existing work, the novelty of the paper is that it uses data-driven correlation-based financial network approach to propose a stock selection method for pairs trading. In this paper, profit per transaction is used to compare the trading strategies. The superiority of the financial network stock selection method is discussed as well in some detail.

Recently, there has been a growing interest in utilizing empirical correlations of log returns to examine financial network models for stock prices by representing stocks as nodes and their relationships as edges. In this paper, empirical cor-relation, data-driven correlation, and cosine similarity matrices are used to obtain adjacency matrices for connectedness. For a given number of nodes, the simplest Erdos and Rényi (ER) model can be obtained by fixing the number of stocks and estimating the probability for any two vertices to be connected by an edge. Unlike existing work, the driving idea in this paper is to estimate the probability with the median cross-correlation to construct the ER network for observed volatility. Moreover, to address the uncertainty associated with these estimates of connectedness, this study introduces data-driven non-linear adaptive symmetric fuzzy adjacency matrices. In the literature, a certain thresh-old is identified for a network considering its connectedness. In this study, threshold and fuzzy parameters for the corre-sponding minimally connected fuzzy networks are determined, revealing unique network structures and the most significant stocks/cryptocurrencies (nodes with the highest number of links). Furthermore, as an application of the suggested fuzzy networks, three clustering approaches, fuzzy network clustering, k-means clustering, and Sharpe ratio clustering, are applied followed by the PageRank algorithm to construct optimal portfolios. A comparison of the constructed portfolios suggests that the fuzzy networks and clustering techniques are capable of yielding high cumulative returns during the study period.

A profitable data-driven algorithmic trading algorithm will benefit from a dynamic system that can produce accurate hedge ratio estimates and short-term innovation volatility forecasts. Commonly used pairs and multiple trading strategies are constructed using the Kalman Filter (KF) and exploiting mean reversion in co-integrated nonstationary stock prices. However, KFs are sensitive to model errors. Misspecified modelling produces unstable solutions for dynamic systems. Fading-Memory Filter (FMF) uses a discounting weight to past observations. Compared to a standard KF, FMF addresses more recent observations and is more resilient (less sensitive) to modelling errors. However, the FMF algorithm does not provide slope parameter covariance matrix updates and innovation volatility forecasts. This paper proposes a novel resilient FMF algorithm for pairs trading and multiple trading by defining an appropriate data-driven innovation volatility forecasting model. The FMF-based strategies are implemented through some experiments on the hourly prices (high-frequency data) of Bitcoin, Ethereum and Litecoin. It is shown that the proposed FMF trading strategies outperform the existing KF trading strategies and they are more profitable in the bear market over time, especially for continuous falling of prices and the short-lived and sharp rally recovery where prices are not stationary.

The market focuses on the Cboe Volatility Index (VIX) or Fear Index, an option-implied forecast of 30 calendar-day realized volatility of S&P 500 returns derived from a cross-section of vanilla options. The VIX is determined using a formula that derives the market’s expectation of realized one-month standard deviation of returns backed out from the near-term call and put options on the S&P 500 index. Market participants such as traders, asset managers, and risk managers, keenly watch the VIX index, and are interested in achieving accurate intelligent probabilistic forecasts of the VIX, and also of the realized volatility of individual stocks. These volatility forecasts are useful to options traders placing bets on the future volatility of individual stocks. This paper examines models that only utilize past values of the VIX and document improvements in forecasting the VIX (and its volatility) over different horizons. The approaches include long short-term memory (LSTM) models, simple moving average methods, data-driven neuro volatility techniques, and industry models like Prophet. Uniquely, we propose a novel VIX price interval forecasting model. The driving idea, unlike the existing VIX price forecasting models, is that the proposed novel LSTM interval forecasting method trains two LSTMs to obtain price forecasts and the forecast error volatility forecasts. All the proposed forecasting methods also avoid model identification and estimation issues, especially for a series like the VIX which is non-stationary. We compare models and document which ones perform best for varied horizons.

Long-term electricity load forecasts are the main in-puts of production planning and scheduling at different horizons and load forecasting plays an important role in balancing the electricity grid. Forecast (prediction) intervals provide the mea-sure of uncertainty of the point forecasts (predictions). However, a data-driven innovation distribution approach is not available to calculate long-term prediction intervals when using deep learning neural networks dynamic regression (NNDR) models. In this paper, a feedforward NNDR model for long-term electricity demand forecasting is introduced and the corresponding data-driven prediction intervals (PIs) are obtained. The novelty of this paper is to use long-term point forecasts and model innovation residuals to obtain three classes of PIs by using data-driven innovation distribution, nonlinear adaptive trapezoidal fuzzy numbers and bootstrapping. It is shown that NNDR models and dynamic regression models with Seasonal Autoregressive Integrated Moving Average (SARIMA) errors (DRSARIMA) are capable of modelling seasonality as well as nonlinearity for demand and other features such as temperature and a day type indicator. NNDR models and DRSARIMA models are evaluated through numerical experiments and the results show that the proposed NNDR model outperforms the DRSARIMA model to forecast demand during the volatile period. It is also shown that innovation residuals from DRSARIMA and NNDR follow heavy-tailed$t$distributions. The data-driven probabilistic Student$t$PIs have higher coverage probabilities than either trapezoidal fuzzy PIs or the bootstrap PIs for both DRSARIMA and NNDR models.

Pairs trading strategies are constructed based on exploiting mean reversion in security prices, which have been demonstrated to perform well for stocks. However, their performance is not widely studied for cryptocurrencies, which are usually discerned as inefficient and unpredictable. One significant advantage of pairs trading is that potential profits can be generated regardless of the overall market movement. The pairs trading has the potential to be profitable for cryptocurrencies in bear markets and with intraday data. Kalman filter (KF) algorithms are popular for pairs trading to update the hedge ratio dynamically. They reduce the arbitrariness in parameter optimization by putting constraints on the parameter space. However, a major drawback is that the innovation volatility estimate calculated by using a KF algorithm is always affected by the initial values and outliers. An effective resilient filtering approach to estimate the innovation volatility is presented in this paper for cryptocurrencies. This paper presents rolling regression pairs trading strategies, traditional KF pairs trading strategies and resilient filter pairs trading strategies. The proposed trading strategies have been evaluated through some experiments on hourly Bitcoin USD and Ethereum USD prices and it is shown that the proposed resilient filter trading strategy is much more stable to initial values than the traditional KF trading strategy.

Recently there has been a growing interest in applying neural network modelling from natural language processing to financial time series prediction problems in computational finance. Cryptocurrency price prediction is a challenging problem with non-stationary market price and volatility clustering. Cryp-tocurrency data tends to be non-stationary, which means that predictive information extracted using deep learning techniques on observed data can not be used with future data. Moreover, there is a very little signal in cryptocurrency data to indicate the future direction of the market. This paper proposes a sensible way to frame the prediction problem as a dynamic regression problem by defining the features in the feedforward neural networks and the target as an appropriate average of the historical data. The novelty of this paper is to use deep learning algorithms and statistical bootstrapping to obtain cryptocurrency price prediction and the corresponding prediction intervals. It is shown that neural networks are capable of modelling nonlinearity directly for nonlinear time series models. The proposed hybrid approach is evaluated using simulated and cryptocurrency data through numerical experiments. Moreover, Gaussian and boot-strap prediction intervals for the price and the volatility of the prediction errors, are also discussed in some detail.

The development of algorithmic trading has been one of the most prominent trends in finance and its applications. Hidden Markov Models (HMMs) help enhance the predictive power of statistical models and improve trading strategies for data scientists and algorithmic traders. In recent years there has been growing interest in investigating the pairs trading and multiple trading based on robust Kalman filtering (KF) using data-driven innovation volatility forecasts (DDIVF). KF algorithms were successfully applied in pairs trading with two cointegrated assets using DDIVF as a method for forecasting non-normal innovation volatility. In this paper a novel combined pairwise trading strategy is proposed by combining HMM and DDIVF to further optimize trading signals in different market regimes. The results of the numerical experiments on two cointegrated stocks show that the proposed profitable trading strategy using DDIVF-HMM outperforms the recently studied robust trading strategy using DDIVF alone.

There has been a growing interest in studying risk forecasting using data-driven exponential weighted moving average (DD-EWMA) volatility models as well as nonlinear neuro volatility models based on Neural network (NN). However, the recently proposed volatility forecasting models have not been used to study portfolio optimization. Large kurtosis of the portfolio return sequence shows that it follows a heavy-tailed t distribution. Significant sample autocorrelations of the absolute portfolio returns and squared portfolio returns suggest that time-varying volatility models are more appropriate to model the volatility. In this paper, a DD-EWMA portfolio volatility forecasting model is used to study the generalized portfolio optimization using the intelligent probabilistic risk forecasts based on data-driven t distribution of the portfolio returns. Optimal portfolio weights are obtained by minimizing the corresponding risk forecasts of portfolio volatility, mean absolute deviation (MAD), Value-at-Risk (VaR), and conditional Value-at-Risk (CVaR) for minimum risk forecast portfolios. Moreover the portfolio weights of the generalized tangency portfolios with different risk measures are obtained by maximizing the corresponding portfolio Sharpe ratio (PSR) forecasts. Experiments are conducted to show that the DD-EWMA volatility forecasting model is most computationally efficient (less computing time), whereas neuro volatility model takes longer time to obtain one-step ahead portfolio volatility forecasts. Moreover, the superiority of the portfolio selection based on data-driven volatility forecasts over the portfolio selection based on volatility estimates is demonstrated through numerical experiments using ten frequently traded stocks.

Portfolio risk management is an important success factor in an organization’s ability to deliver more business value. Moreover, constructing a true resilient portfolio is impossible without the inclusion of higher-order moments such as skewness and kurtosis. Recently there has been a growing interest in using machine learning methods with empirical variance covariance matrix of returns to study Markowitz portfolio optimization. A major drawback is that the tangency portfolios constructed by using the existing portfolio risk measures such as portfolio standard deviation, value-at-risk (VaR), conditional value-at-risk (CVaR), maximum absolute deviation (MAD) are always affected by the larger skewness and kurtosis of the portfolio return. This paper develops a set of metrics that extend the traditional portfolio Sharpe ratio (PSR) to measures that include skewness and kurtosis. Using a random portfolio approach, the paper demonstrates how to use these new metrics and optimize portfolios. Inclusion of higher moments such as skewness and kurtosis in portfolio risk management acknowledges the risk of asymmetric and heavy-tailed returns and can help in constructing resilient portfolios. For portfolio optimization, simple yet effective novel data-driven resilient portfolio risk measures incorporating skewness and kurtosis are presented in this paper. The results show that the performance of maximum mean-risk portfolios using the proposed portfolio risk measures based on volatility correlation and sign correlation outperform the commonly used tangency portfolio using portfolio standard deviation.

Recently, there has been a growing interest in portfolio optimization using graphical LASSO (GL) machine learning method, by assuming normality for asset returns. However, a major drawback is that most of the asset returns follow non-normal distributions and sample percentiles are used to study the portfolio optimization with Value-at-Risk (VaR) as a risk measure. In this paper, a data-driven random weights algorithm (RWA) and a sign correlation based portfolio return distribution are used to study the fuzzy portfolio optimization. The superiority of RWA over the commonly used genetic algorithm (GA) in computing the optimal portfolio weights is demonstrated by comparing the computing time. When comparing the estimate of the risk tolerance coefficient and the theoretical value for tangency portfolios with volatility as a risk measure, RWA outperforms (smaller absolute error) the GA. The novelty of this paper is the use of RWA and GA to calculate the fuzzy estimates (interval estimates) of the risk tolerance coefficient/optimal weights and using the sign correlation to obtain the data-driven distribution of the portfolio returns. More specifically the novelty is to obtain the fuzzy estimates of the risk tolerance coefficient and portfolio weights by modelling the portfolio volatility as an asymmetric triangular fuzzy number from the data-driven observed portfolio volatilities. In particular, the proposed RWA as well as GA lead to machine learning solutions for the portfolio optimization problems without a closed form solution and provide fuzzy estimates of the risk tolerance coefficient and the optimal portfolio weights.

Volatility forecasts and stock price forecasts play major roles in algorithmic trading. In this paper, joint forecasts of volatility and stock price are first obtained and then applied to algorithmic trading. Interval forecasts of stock prices are constructed using generalized double exponential smoothing (GDES) for stock price forecasts and data-driven exponentially weighted moving average (DD-EWMA) for volatility forecasts. Multi-stepahead interval forecasts for nonstationary stock price series are obtained. As an application, one-step-ahead interval forecasts are used to propose a novel dynamic data-driven algorithmic trading strategy. Commonly used simple moving average (SMA) crossover trading strategy and Bollinger bands trading strategy depend on unknown parameters (moving average window sizes) and the window sizes are usually chosen in an ad hoc fashion. However the proposed trading strategy does not depend on the window size, and is data-driven in the sense that the optimal smoothing constants of GDES and DD-EWMA are chosen from the data. In the proposed trading strategy, a training sample is used to tune the parameters: smoothing constant for GDES price forecasts, smoothing constant for DD-EWMA volatility forecasts, and the tuning parameter which maximizes Sharpe ratio (SR). A test sample is then used to compute cumulative profits to measure the out-of-sample trading performance using optimal tuning parameters. An empirical application on a set of widely traded stock indices shows that the proposed GDES interval forecast trading strategy is able to significantly outperform SMA and the buy and hold strategies for the majority of stock indices.

Many of the challenges and opportunities of data science in finance involve recursive smoothing, forecasting, filtering and pattern mining. Recently there has been a growing interest in using filtered estimates for dynamic hedge ratios for pairs trading. Moreover, rolling estimates and forecasts are used for pattern mining in technical analysis. Kalman filtering algorithms were successfully applied in pairs trading with only two co-integrated assets. In this paper, pairs trading strategy is extended to multiple trading (with more than two assets) strategy. Recently proposed non-Gaussian maximum informative filtering algorithms for dynamic state space models are used to obtain the filtered estimates of hedge ratios and applied in multiple trading. It is shown that the proposed multiple trading strategy outperforms (with higher profits) the commonly used pairs trading strategy using real data. A data-driven approach for selecting a parameter which maximizes the Sharpe ratio (SR) to generate optimal trading signals is also discussed in some detail.

Regularization methods allow data scientists and risk managers to enhance the predictive power of a statistical model and improve the quality of risk forecasts. Financial risk forecasting is about forecasting volatility, Value at Risk (VaR), expected shortfall (ES) and model risk ratio. While regularized estimates have been shown to perform well in model selection and parameter estimation, their applications in financial risk forecasting has not yet been studied. In this paper, regularized adaptive forecasts and computationally efficient forecasting algorithms for volatility, VaR, ES and model risk are studied using various regularization methods such as ridge, lasso and elastic net. Sample sign correlation of standardized log returns (standardized by volatility forecasts) is used to identify the conditional distribution of the log returns series and provide regularized interval forecasts as well as regularized probability forecasts. Superiority of the regularized risk forecasts is demonstrated using different volatility models including a recently proposed generalized data-driven volatility model in [8]. Validation of the regularized risk forecasts using real financial data is given. Regularized probabilistic forecasts for stationary time series models are also discussed in some detail.

The explosion of algorithmic trading has been one of the most prominent trends in the finance industry. In this paper, two strategies for algorithmic trading such as Bollinger bands and the simple moving average (SMA) crossover strategy are studied in the fuzzy settings. The commonly used Bollinger bands trading strategy assumes that the difference between an asset's price and its SMA is normally distributed. However, it is shown that a data-driven t distribution is more appropriate to model the difference between an asset's price and its SMA. A novel data-driven fuzzy Bollinger bands strategy is proposed for algo trading. A good strategy should have a good algo return on investment with low algo volatility. Therefore, forecasting algo volatility and identifying an appropriate distribution of algo returns play a crucial role in algo trading. Sharpe Ratio (SR) is a measure of average algo return earned in excess of the risk-free rate per unit of algo volatility. For a class of SMA crossover strategies with varying window sizes, fuzzy estimates of SR are computed based on various risk measures including the data-driven volatility estimate (DDVE). SR fuzzy forecasts are computed using two recently proposed volatility forecasting models such as data-driven exponentially weighted moving average (DD-EWMA) and data-driven neuro volatility models. The main reason of using the fuzzy approach is to provide α-cuts (interval forecasts) of the SR. An empirical application on a set of widely traded technology stocks shows that the proposed models deliver forecasts of SR with small errors.

It has recently been suggested that movement variability directly increases the speed of motor learning. Here we use computational modeling of motor adaptation to show that variability can have a broad range of effects on learning, both negative and positive. Experimentally, we also find contributing and decelerating effects. Lastly, through a meta-analysis of published papers, we verify that across a wide range of experiments, movement variability has no statistical relation with learning rate. While motor learning is a complex process that can be modeled, further research is needed to understand the relative importance of the involved factors.