Wavelet Analysis of Central European Stock Market Behaviour During the Crisis

In the paper we test for the different reactions of stock markets to the current financial crisis. We focus on Central European stock markets, namely the Czech, Polish and Hungarian ones, and compare them to the German and U.S. benchmark stock markets. Using wavelet analysis, we decompose a time series into frequency components called scales and measure their energy contribution. The energy of a scale is proportional to its wavelet variance. The decompositions of the tested stock markets show changes in the energies on the scales during the current financial crisis. The results indicate that each of the tested stock markets reacted differently to the current financial crisis. More important, Central European stock markets seem to have strongly different behaviour during the crisis.ewavelet analysis, multiresolution analysis, Central European stock markets, financial crisis

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

The multifractal model of asset returns captures the volatility persistence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregularities in the distribution of market returns over time and thereby of the kind of uncertainty or randomness in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or drawdowns. In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This paper focuses on these identifiable multifractal spectral patterns surrounding the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market preceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifractal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, therefore, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5Financial Markets, Persistence, Multi-Fractal Spectral Analysis, Wavelets

Interrelationships among international stock market indices: Europe, Asia and the Americas

In this paper, we investigate the price interdependence between seven international stock markets, namely Irish, UK, Portuguese, US, Brazilian, Japanese and Hong Kong, using a new testing method, based on the wavelet transform to reconstruct the data series, as suggested by Lee (2002). We find evidence of intra-European (Irish, UK and Portuguese) market co-movements with the US market also weakly influencing the Irish market. We also find co-movement between the US and Brazilian markets and similar intra-Asian co-movements (Japanese and Hong Kong). Finally, we conclude that the circle of impact is that of the European markets (Irish, UK and Portuguese) on both American markets (US and Brazilian), with these in turn impacting on the Asian markets (Japanese and Hong Kong) which in turn influence the European markets. In summary, we find evidence for intra-continental relationships and an increase in importance of international spillover effects since the mid 1990’s, while the importance of historical transmissions has decreased since the beginning of this century

Interdependence between emerging and major markets

In this paper, we investigate the price spillover effects among two developed markets, (the US and the UK ), and two developing markets, (Irish and Portuguese), using a new testing method suggested by Lee (2002). We find that there are interrelationships between any two of the Irish, the UK and Portuguese markets and that the co-movements between the emerging markets and the US are statistically significant but weak. We also found that the US market is slightly influenced by the UK but not vice versa

Long Memory Options: LM Evidence and Simulations

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence or long memory in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black-Scholes' geometric Brownian motion assumption. Moreover, Elliott and van der Hoek (2003) provide a theoretical framework for incorporating these findings into the Black- Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal Fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black-Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company's own shares for long expiration periods. It makes a significant difference in the valuation when an option is 'blue' or when it is 'red.' For a proper valuation of such stock options, the degrees of persistence of the companies' share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.Options, Long Memory, Persistence, Hurst Exponent, Identification, Simulation, Executive Remuneration

Measurement of Financial Risk Persistence

This paper discusses various ways of measuring the persistence or Long Memory (LM) of financial market risk in both its time and frequency domains. For the measurement of the risk, irregularity or 'randomness' of these series, we can compute a set of critical Lipschitz - Hölder exponents, in particular, the Hurst Exponent and the Lévy Stability Alpha, and relate them to the Mandelbrot-Hoskings' fractional difference operators, as occur in the Fractional Brownian Motion model (which is our benchmark). The main contribution of this paper is to provide a compaison table of the various critical exponents available in various scientific disciplines to measure the LM persistence of time seies. It also discusses why Markov- and (G)ARCH models cannot capture this LM, long term dependence or risk persistence, because these models have finite lag lengths, while the empirically observed long memory risk phenomenon is an infinite lag length phenomenon. Currently, there are three techniques of nonstationary time series analysis to measure time - varying financial risk: Range/Scale analysis, windowed Fourier analysis, and wavelet MRA. This paper relates these powerful analytic techniques to classical Box-Jenkins-type time series analysis and to Pearson's spectral frequency analysis, which both rely on the uncorroboated assumption of stationarity and ergodicity.Persistence, long memory, dependence, time series, frequency, critical exponents, fractional Brownian motion, (G)ARCH, risk measurement

Forecasting stock-return volatility in the time-frequency domain

Este estudo foca nos modelos autorregressivos de heterocedasticidade condicional, em especial nos modelos GARCH. A amostra principal usa dados do retorno do índice do S&P500 ajustados a divisão e dividendos de 1990 a 2008, usando uma janela fora da amostra de 2001 até ao final da amostra. O objetivo principal é analisar o desempenho das previsões do modelo num domínio tempo-frequência e, em seguida, compará-los com resultados em um cenário de domínio de tempo. Para fazer uma análise de domínio tempo-frequência, usamos técnicas de wavelets para decompor as séries temporais S&P500 originais em diferentes frequências, cada uma delas originalmente configurada no domínio do tempo. Em última análise, o objetivo é ver se a decomposição com wavelets traz um desempenho aprimorado na previsão/modelagem da volatilidade, observando a função de perdas de previsão de Quasi-Verossimilhança (QL), bem como os índices médios de perdas de previsão ao quadrado (MSFE). Embora a decomposição com wavelets ajude a capturar componentes periódicos ocultos das séries temporais originais, os resultados de domínio de frequência em termos de função de perda (QL e MSFE) não superam o resultado original do domínio do tempo para qualquer frequência dada. No entanto, a maioria das informações para a volatilidade futura é capturada em poucas frequências da série temporal do S&P500, especialmente, na parte de alta frequência dos espectros, representando horizontes de investimento muito curtos.This research focuses on generalized autoregressive conditional heteroskedasticity (GARCH) model. The main sample uses daily split-adjusted and dividend-adjusted log-return data of the S&P500 index ranging from 1990 to 2008, using an out-of-sample window from 2001 until the end of the sample. The main goal is to analyze the performance of the model forecasts in a time-frequency domain and then to compare them with results in a time-domain scenario. To make a time-frequency domain analysis, this research uses wavelets techniques to decompose the original S&P500 time series into different frequencies brands, each of them originally set in time-domain. Ultimately, the aim is to see if the wavelet decomposition brings an enhanced performance on forecasting/modelling volatility by looking at the Quasi-Likelihood forecasting losses (QL) as well as the mean squared forecasting losses ratios (MSFE). Although the wavelet decomposition helps to capture hidden periodic components of the original time-series, frequency-domain results in terms of loss function (QL e MSFE) don’t outperform the original time-domain result for any given frequency. Nevertheless, most of the information for future volatility is captured in few frequencies of the S&P500 time-series, specially in the high-frequency part of the spectra, representing very short investment horizons

Equity index returns predictability and fama-french factors : a frequency domain analysis

We extend the analysis of Faria and Verona (2020) tothe Fama French 5 factor model to predict the Equity Risk Premium in theS&P500 index. The Fama French 5 factor model is decomposed using themaximal overlap discrete wavelet transform so that we canstudy the forecasting performance of each factor’s different frequencies andtest them out-of-sample. The main findings of this study are that the factorsthemselves are not good predictors of the equity risk premiumout-of-sample, but the 16-128 month frequencyof the HML (high minus low) and, especially the RMW (robust minus weak),are good predictors of the Equity Risk Premium especially in post-2008 crisis. Ourresults support recent findings in the asset pricing literature that the business-cyclefrequency components of financial variables play a crucial role in forecastingthe equity premium. Thus, for both investors and academics, these findingsare of great relevance.Estendemos o trabalho de Faria e Verona (2020) para o modelo de 5 fatores Fama French para prever o prémio de risco de mercado do índice S&P500. O modelo de 5 factores Fama French é decomposto através do método maximal overlap discrete wavelet transform para que seja possível estudar o poder de previsão das várias frequências de cada fator. A principal conclusão deste trabalho é que os fatores por si não preveem o prémio de risco de mercado out-of-sample, mas a frequência de 16 a 128 meses dos fatores HML (high minus low) e RMW (robust minus weak) não só conseguem prever, como têm uma performance superior à média histórica no período a seguir à grande crise financeira de 2008. Os nossos resultados corroboram com os resultados da literatura mais recente, no sentido em que as frequências de médio prazo de variáveis financeiras são bons preditores do prémio de risco do mercado. Como tal, estes resultados são de elevada relevância para académicos e investidores

Long Memory Options: Valuation

This paper graphically demonstrates the significant impact of the observed financial market persistence, i.e., long term memory or dependence, on European option valuation. Many empirical researchers have observed non-Fickian degrees of persistence or long memory in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black-Scholes' geometric Brownian motion assumption. Moreover, Elliott and van der Hoek (2003) have now also provided a theoretical framework for incorporating these findings in the Black-Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. Risk-neutral valuation is equivalent to valuation by real world probabilities. By using a mono-fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the binomial and Black-Scholes pricing formulas. Long memory options are of considerable importance in Corporate remuneration packages, since warrants are written on a company's own shares for long expiration periods. Therefore, we recommend that for a proper valuation of such warrants, the degrees of persistence of the companies' share markets are measured and properly incorporated in the warrant valuation.Options, Long Memory, Persistence, Hurst Exponent, Executive Remuneration

A multiscale view on inverse statistics and gain/loss asymmetry in financial time series

Researchers have studied the first passage time of financial time series and observed that the smallest time interval needed for a stock index to move a given distance is typically shorter for negative than for positive price movements. The same is not observed for the index constituents, the individual stocks. We use the discrete wavelet transform to illustrate that this is a long rather than short time scale phenomenon -- if enough low frequency content of the price process is removed, the asymmetry disappears. We also propose a new model, which explain the asymmetry by prolonged, correlated down movements of individual stocks