Distribution Of Log-returns Research Articles

Pendekatan Metode Monte Carlo untuk Simulasi Pergerakan Harga Saham

This research applies the Monte Carlo simulation method to predict the movement of Apple Inc.'s stock price over a long period of time. Using historical data of Apple's stock price from 12 December 1980 to 24 March 2022, this study aims to generate a probability distribution of the future stock price. The method involves several steps, including data collection, log return calculation, parameter estimation, and simulation of the stock price path through random iterations based on the log return distribution. The simulation results show that the closing price of Apple stock can be predicted by following the historical trend, although there are differences with the real data due to the stochastic nature of the Monte Carlo technique. This research also applies a variance reduction method to improve simulation efficiency. The findings provide a valuable perspective for investors and financial analysts in identifying investment risks and opportunities through an in-depth understanding of the dynamics of stock price movements using Monte Carlo simulation. Suggestions for future research include the use of VaR methods with historical variance and covariance approaches, as well as considering longer data periods and more stock indices for more comprehensive results.

Does COVID-19 Affect the Growth of the Fourth Industrial Revolution?

ABSTRACT The COVID-19 pandemic might accelerate Fourth Industrial Revolution such as FinTech, AI and Robotics. This study explores impact of the COVID-19 on Fourth Industrial Revolution by quantifying such an impact on stock market. Stock indices and ETFs are handled to find log-return distributions via kernel estimation. Before, during and after outbreak of the COVID-19, distributions and related measures are investigated. Results reveal that E-commerce and Fourth Industrial Revolution have grown 1.5-1.7 times in increase rates of daily net returns, compared to main stock indices. A statistical test supports that AI & Robotics sector has grown under influence of the COVID-19.

Occam’s razor, machine learning and stochastic modeling of complex systems: the case of the Italian energy market

In the spirit of Occam’s razor, we propose a parsimoniuos regime-switching model for describing the complex dynamics of electricity and natural gas prices observed in real markets. The model was built using a machine learning-based methodology, namely a cluster analysis to investigate the properties of the stable dynamics and a deep neural network appropriately trained on market data to drive transitions between different regimes. The main purposes of this study are twofold: (1) to build the simplest model capable of incorporating the main stylized facts of electricity and natural gas prices, including dynamic correlation; (2) to define an appropriate calibration procedure on market data. We applied this methodology to the Italian energy market. The results obtained show remarkable agreement with the empirical data, satisfactorily reproducing the first four moments of the empirical distributions of log-returns.

Fat and Heavy Tails in Asset Management

In this article, the authors explain non-normal probability distributions and the reasons it is important to properly model the tails of one or more distributions in applications to asset management. The authors illustrate the types of quantitative models needed in asset management and provide some basic concepts on random variables and stochastic processes useful to understand non-normal models. After having reviewed the stylized facts of log-returns, the authors describe, in nontechnical terms and with only a few formulas, univariate and multivariate non-normal models that are able to explain the fat (and heavy) tails empirically observed in the distribution of asset and portfolio log-returns.

A graph-based superframework for mixture model estimation using EM: an analysis of US wholesale electricity markets

A fully unsupervised graph-based superframework is proposed to handle the EM initialization problem for estimating mixture models on financial time series. Using a complex network approach that links time series and graphs, the graph-structured information derived from the observed data is exploited to produce a meaningful starting point for the EM algorithm. It is shown that structural information derived by complex graphs can definitely capture time series behavior and nonlinear relationships between different observations. The proposed methodology is employed to estimate Gaussian mixture models on US wholesale electricity market prices using two different configurations of the superframework. The obtained results show that the proposed methodology performs better than conventional initialization methods, such as K-means based techniques. The improvements are significant on the overall representation of the empirical distribution of log-returns and, in particular, on the first four moments. Moreover, this approach has a high degree of generalization and flexibility, exploiting graph manipulation and employing functional operating blocks, which can be adapted to very different empirical situations.

Market efficiency of cryptocurrency: evidence from the Bitcoin market

This study examines whether the Bitcoin market satisfies the (weak-form) efficient market hypothesis using a quantum harmonic oscillator, which provides the state-specific probability density functions that capture the superimposed Gaussian and non-Gaussian states of the log return distribution. Contrasting the mixed evidence from a variance ratio test, the high probability allocated to the ground state suggests a near-efficient Bitcoin market. Findings imply that as Bitcoin evolves into an efficient market, speculators might encounter difficulty in exploiting profitable trading strategies. Furthermore, when policymakers initiate tight regulations to control the market, they should closely monitor market efficiency as an index of price distortion.

Deep learning based regime-switching models of energy commodity prices

We discuss a deep learning based approach to model the complex dynamics of commodity prices observed in real markets. A regime-switching model is proposed to describe the time evolution of market prices. In this model, the base regime is described by a mean-reverting diffusion process and the second regime is driven by the predictions of a deep neural network trained on the market log-returns time series. A statistical technique, based on the method of simulated moments, is proposed to estimate the model on market data. We applied this methodology to energy commodity price time series with very different characteristics, namely the US wholesale electricity, natural gas and crude oil price daily time series. The obtained results show a good agreement with empirical data. In particular, the model seems to reproduce in a very interesting way the first four central moments of the empirical distributions of log-returns as well as the shape of the observed price time series.

Convolution Approach for Value at Risk Estimation

Formally, Value at risk (VaR) measures the worst expected loss over a given horizon under normal market conditions at a given confidence level. Very often, daily data are used to compute VaR and scale up to the required time horizon with the square root of time adjustment. This gives rise to an important question when we perform VaR estimation: whether the values of VaR (i.e., “loss”) should be interpreted as (1) exactly on [Formula: see text]th day and (2) within i days. This research attempted to answer the above question using actual data of SPX and HSI. It was found that, in determining the proportionality of the values, (i.e., slopes) of VaR versus the holding period, the slopes for within i days are generally greater than those for exactly on [Formula: see text]th day. This has great implications to risk managers as it would be inappropriate to simply scale up the one-day volatility by multiplying the square root of time (or the number of days) ahead to determine the risk over a longer horizon of [Formula: see text] days. The evolution of log return distribution over time using actual data has also been performed empirically. It provides a better understanding than a series of VaR values. However, the number of samples in actual data is limited. There may not be enough data to draw reliable observation after it has been evolved a few times. Numerical simulation can help solve the problem by generating, say, one million log returns. It has been used to provide many insights as to how the distribution evolves over time and reveals an interesting dynamic of minimum cumulative returns. Numerical simulation is time consuming for performing evolution of distribution. The convolution approach provides an efficient way to analyze the evolution of the whole data distribution that encompasses VaR and others. The convolution approach with modified/scaled input distribution has been developed and it matches the results of numerical simulation perfectly for independent data for both exactly on [Formula: see text]th day and within i days. As the autocorrelation of the single index is mostly close to zero, results show that the convolution approach is able to match empirical results to a large extent.

Technological Change and Market Conditions: Evidence from Bitcoin Fork

This article examines the impact of technological changes to cryptocurrency—known as “forking” that triggers blockchain splits—on market conditions. Despite the explicit distinction in log return distributions between the two splitting blockchains, adopting new technology does not result in a disparity in market conditions: no significant difference exists in market efficiency and long‐term market equilibrium between the two splitting blockchains. Technological changes accompanying market separation do not impede the underlying uniformity in market conditions. The findings suggest that mutual information flows linked to market liquidity explain the results between the new and old forks.

Revisiting the volatility of bitcoin with approximate entropy

Two distinct and non-redundant understandings of volatility, as deviation from consistency, exist for a time-series: (1) exhibiting high standard deviation and, closer to the dictionary definition of the term, (2) appearing highly irregular and unpredictable. We find that Bitcoin is a prime example of an asset for which the two concepts of volatility diverge. We show that, historically, Bitcoin combines high Standard Deviation and low Approximate Entropy, relative to Gold and S&P 500. Moreover, subsample analysis for different time-scales (daily, weekly, monthly) shows that lower sampling frequencies drastically reduce the Kurtosis of the distribution of log-returns of Bitcoin. The opposite effect is observed for Gold and S&P 500. These properties suggest that, contrary to the volatility of the two traditional assets, Bitcoin’s high volatility is essentially an intra-day phenomenon that is strongly attenuated for a weekly or monthly time-preference.

Combination of transition probability distribution and stable Lorentz distribution in stock markets

It is well known that the stock price is a stochastic series of time. The probability distribution of the stock price changes should be stable. However, the empirically observed probability distribution of log-returns is not Gaussian and has power-law tails. We present a combination of Lorentz stable distribution and transition probability distribution description for the stock index changes. The Lorentz stable distribution describes the stochastic changes of stock prices and the transition distribution presents features of stock price transiting from S(t) to S(t+Δt) for small time interval. By making use of the Fokker–Planck equation, we give the explicit expression of the transition probability distribution for the stock price changes. The two stage formalism is in agreement with empirical observations on high frequency time series of the S&P 500 Index.

A Statistical Classification of Cryptocurrencies

The aim of this paper is to derive the main factors that separate cryptocurrencies from the classical assets, by using various classification techniques applied to the daily time series of log-returns. In this sense, a daily time series of asset returns (either cryptocurrencies or classical assets) can be characterized by a multidimensional vector with statistical components like variance, skewness, kurtosis, tail probability, quantiles, conditional tail expectation or GARCH parameters. By using dimension reduction techniques (Factor Analysis) and classification models (Binary Logistic Regression, Discriminant Analysis, Support Vector Machines, K-means clustering, Variance Components Split methods) for a representative sample of cryptocurrencies, stocks, exchange rates and commodities, we are able to classify cryptocurrencies as a new asset class with unique features in the tails of the log-returns distribution. The main result of our paper is the complete separation of the cryptocurrencies from the other type of assets, by using the Maximum Variance Components Split method. In addition, we observe a synchronicity in the evolution of of the cryptocurrencies, compared to the classical assets, mainly due to the tails behaviour of the log-return distribution.

Evidence of inefficiency of the Brazilian stock market: The IBOVESPA future contracts

We present some indications of inefficiency of the Brazilian stock market based on the existence of strong long-time cross-correlations with foreign markets and indices. Our results show a strong dependence on foreign markets indices as the S&P 500 and CAC 40, but not to the Shanghai SSE 180, indicating an intricate interdependence. We also show that the distribution of log-returns of the Brazilian BOVESPA index has a discrete fat tail in the time scale of a day, which is also a deviation of what is expected of an efficient equilibrated market. As a final argument of the inefficiency of the Brazilian stock market, we use a neural network approach to forecast the direction of movement of the value of the IBOVESPA future contracts, with an accuracy allowing financial returns over passive strategies.

Modelling tail risk with tempered stable distributions: an overview

In this study, we investigate the performance of different parametric models with stable and tempered stable distributions for capturing the tail behaviour of log-returns (financial asset returns). First, we define and discuss the properties of stable and tempered stable random variables. We then show how to estimate their parameters and simulate them based on their characteristic functions. Finally, as an illustration, we conduct an empirical analysis to explore the performance of different models representing the distributions of log-returns for the S&P500 and DAX indexes.

Using High-Frequency Entropy to Forecast Bitcoin's Daily Value at Risk.

In this paper we investigate the ability of several econometrical models to forecast value at risk for a sample of daily time series of cryptocurrency returns. Using high frequency data for Bitcoin, we estimate the entropy of intraday distribution of logreturns through the symbolic time series analysis (STSA), producing low-resolution data from high-resolution data. Our results show that entropy has a strong explanatory power for the quantiles of the distribution of the daily returns. Based on Christoffersen’s tests for Value at Risk (VaR) backtesting, we can conclude that the VaR forecast build upon the entropy of intraday returns is the best, compared to the forecasts provided by the classical GARCH models.

The Impact of Adverse Selection on Stock Exchange Specialists’ Price Quotation Strategy

This paper focuses on the activity of the specialists – one of the key participants in stock exchange trading. We attempt to model the price quotations of specialists in a modelling framework where some of the parties involved in the transactions may be informed, while others are uninformed “liquidity traders”. It is in this adverse selection modelling framework that, relying on the technique of Monte Carlo simulation, we seek an answer to the following research questions: how does adverse selection impact the price quotation of specialists; to what extent are prices and logreturns influenced by uncertainty; to what degree of accuracy can specialists determine the proportion of informed traders and liquidity traders from trading volumes? Our model confirmed that as soon as uncertainty subsided in the simulated market, the number of transactions, wealth and the stock portfolio started to grow, while price fluctuations began to decline and the standard deviation and the distribution of logreturns edged closer and closer to a normal distribution, which points to improving market efficiency.

Identification of five time periods on the Indonesian stock exchange index historical data since 1997 to 2016

We are investigating the behavioural changes of the Indonesian financial systems in the last 20 years. Changes in the financial system behaviour were indicated by differences in the statistical properties of the daily log return distribution in two adjacent time windows. To measure how likely two distributions are differ, the Kolmogorov-Smirnov (KS) Test was applied. We have found that we can divide our time series data into five segments, where the KS probability values between two adjacent segments are maximum. This finding can be used to study the effect on the financial system imposed by, for example, the socio-economy and political policy by the government, by studying the changes in such factors in the identified time periods.

Modeling Tail Risk with Tempered Stable Distributions: An Overview

This paper investigates the performance of different parametric models, stable and tempered stable distributions, for capturing the tail behaviour of the log-returns. We first define and discuss the properties of the stable and tempered stable random variables. We then show how to estimate their parameters and simulate them from their characteristic functions. Finally, as an illustration, we perform an empirical experiment to explore the performance of different models representing the distribution of log-returns for the S&P500 and DAX indexes

Changes in The Statistical Properties of The Indonesian Stock Exchange Index Trough Five Different Time Periods Since 1998 Until 2016

A simple method based on Kolmogorov-Smirnov Test were applied to the Indonesian Stock Exchange Index (IHSG) daily log return distribution to identify it’s time segmentation . Five separated periods were found from the overall data since July 1997 until May 2016. Distribution of the daily log return price from each time periods are characterized by two shape parameters of the fitted Levy-stable function, where the parameters are directly related to the variance and skewness of the distribution. The resulting time segmentations does indeed show systematic differences on the evolution of the daily index price, and the varying two fit parameters of the daily return distribution can be used to explain such diverse behavior.

Volatility Analysis of Bitcoin Price Time Series

Bitcoin has the largest share in the total capitalization of cryptocurrency markets currently reaching above 70 billion USD. In this work we focus on the price of Bitcoin in terms of standard currencies and their volatility over the last five years. The average day-to-day return throughout this period is 0.328%, amounting in exponential growth from 6 USD to over 4,000 USD per 1 BTC at present. Multi-scale analysis is performed from the level of the tick data, through the 5 min, 1 hour and 1 day scales. Distribution of trading volumes (1 sec, 1 min, 1 hour and 1 day) aggregated from the Kraken BTCEUR tick data is provided that shows the artifacts of algorithmic trading (selling transactions with volume peaks distributed at integer multiples of BTC unit). Arbitrage opportunities are studied using the EUR, USD and CNY currencies. Whereas the arbitrage spread for EUR-USD currency pair is found narrow at the order of a percent, at the 1 hour sampling period the arbitrage spread for USD-CNY (and similarly EUR-CNY) is found to be more substantial, reaching as high as above 5 percent on rare occasions. The volatility of BTC exchange rates is modeled using the day-to-day distribution of logarithmic return, and the Realized Volatility, sum of the squared logarithmic returns on 5-minute basis. In this work we demonstrate that the Heterogeneous Autoregressive model for Realized Volatility Andersen et al. (2007) applies reasonably well to the BTCUSD dataset. Finally, a feed-forward neural network with 2 hidden layers using 10-day moving window sampling daily return predictors is applied to estimate the next-day logarithmic return. The results show that such an artificial neural network prediction is capable of approximate capture of the actual log return distribution; more sophisticated methods, such as recurrent neural networks and LSTM (Long Short Term Memory) techniques from deep learning may be necessary for higher prediction accuracy.