Symmetric Normal Mixture GARCH

Normal mixture (NM) GARCH models are better able to account for leptokurtosis in financial data and offer a more intuitive and tractable framework for risk analysis and option pricing than student’s t-GARCH models. We present a general, symmetric parameterisation for NM-GARCH(1,1) models, derive the analytic derivatives for the maximum likelihood estimation of the model parameters and their standard errors and compute the moments of the error term. Also, we formulate specific conditions on the model parameters to ensure positive, finite conditional and unconditional second and fourth moments. Simulations quantify the potential bias and inefficiency of parameter estimates as a function of the mixing law. We show that there is a serious bias on parameter estimates for volatility components having very low weight in the mixing law. An empirical application uses moment specification tests and information criteria to determine the optimal number of normal densities in the mixture. For daily returns on three US Dollar foreign exchange rates (British pound, euro and Japanese yen) we find that, whilst normal GARCH(1,1) models fail the moment tests, a simple mixture of two normal densities is sufficient to capture the conditional excess kurtosis in the data. According to our chosen criteria, and given our simulation results, we conclude that a two regime symmetric NM-GARCH model, which quantifies volatility corresponding to ‘normal’ and ‘exceptional’ market circumstances, is optimal for these exchange rate data.Volatility regimes, conditional excess kurtosis, normal mixture, heavy trails, exchange rates, conditional heteroscedasticity, GARCH models.

Comprehensive Analysis of Market Conditions in the Foreign Exchange Market: Fluctuation Scaling and Variance-Covariance Matrix

We investigate quotation and transaction activities in the foreign exchange market for every week during the period of June 2007 to December 2010. A scaling relationship between the mean values of number of quotations (or number of transactions) for various currency pairs and the corresponding standard deviations holds for a majority of the weeks. However, the scaling breaks in some time intervals, which is related to the emergence of market shocks. There is a monotonous relationship between values of scaling indices and global averages of currency pair cross-correlations when both quantities are observed for various window lengths $\Delta t$.Comment: 13 pages, 10 figure

Behavior of share prices of the Brussels stock exchange.

Prices;

Management of Stock Price and it Effect on Economic Growth: Case study of West African Financial Markets

Abstract This paper investigates the statistical properties of stock returns in the West African regional stock market and the link between the West African regional stock market and economic growth. To examine the nature of the distribution of West African regional stock returns, the daily closing prices of the two stock index of West African regional stock market, and eighteen of it sub-indices were utilized. Nine years data from 1998 to 2007 interval were employed. The analysis of our study shows that the distribution of the West African regional stock market returns is non-normal and non-i.i.d (independent, identically and normally distributed). The linear and non-linear dependencies in the returns appeared to be the main reasons for the data being non-i.i.d. The study also demonstrates the presence of the day-of-the-week effect in West African regional stock market.Key words: West Africa regional stock markets, day of the week effects, growth.

The dynamics of the volatility – trading volume relationship: New evidence from developed and emerging markets

This paper empirically investigates whether there is an evolution in the relation between stock market trading volume and volatility in 23 developed and 15 emerging markets. To answer this question, we develop a dynamic application of the TARCH (1, 1) model and first prove that the relationship is variable through time. Then, we focus our analysis on three major financial events, namely the Asian Crisis, the Dot Com bubble burst and the Subprime crisis. We find that the explanatory power of volume is greater during these periods. Finally, we show that the sign of the relationship cannot be clearly set for a specific country or sub group of developed or emerging markets.Mixture of distribution hypothesis, TARCH model, Conditional variance, Trading volume

The Forecasting Performance of German Stock Option Densities

In this paper we will be estimating risk-neutral densities (RND) for the largest euro area stock market (the index of which is the German DAX), reporting their statistical properties, and evaluating their forecasting performance. We have applied an innovative test procedure to a new, rich, and accurate data set. We have two main results. First, we have recorded strong negative skewness in the densities. Second, we find evidence for significant differences between the actual density and the risk-neutral density, leading to the conclusion that market participants were surprised by the extent of both the rise and the fall of the DAX. -- In dieser Arbeit werden "risikoneutrale" Dichtefunktionen über künftige DAXIndexstände aus täglich beobachteten Preisen europäischer Kauf- und Verkaufsoptionen mit verschiedenen Restlaufzeiten abgeleitet. Das hierbei verwendete Berechnungsverfahren beruht auf der Mischung von zwei Log-Normalverteilung, bei dem fünf Parameter (der Mischungsparameter, zwei Mittelwerte und zwei Standardabweichungen) so bestimmt werden, dass der quadratische Abstand zwischen beobachteten und impliziten Optionspreisen minimal ist. Die Preisnotierungen für die Derivative werden der Eurex entnommen und der Untersuchungszeitraum erstreckt sich von Dezember 1995 bis Mai 2002, also sowohl über die Boom- als auch über die Niedergangsphase des DAX. Die Vorhersagehorizonte der Dichten sind auf Grund der Datenlage auf sechs bis acht Wochen begrenzt. Die Vorhersagegüte dieser Dichten wird über verschiedene neuartige statistische Evaluierungsverfahren abgeschätzt. Im Ergebnis stellt sich folgendes heraus: Erstens: Die Dichten weisen im Durchschnitt eine negative Schiefe (negatives drittes Moment) auf, so dass das linke Ende der Dichte "dicker" ist als das rechte und die Marktteilnehmer somit einen bestimmten prozentualen Kursverlust als wahrscheinlicher einschätzten als einen Kursgewinn. Zweitens: Die Evaluierungstests für die Dichten machen deutlich, dass die tatsächlichen Dichten im Mittel deutlich von den risikoneutralen Dichten abweichen. Dabei kann ausgeschlossen werden, dass es sich lediglich um einen "Mittelwert"fehler handelt. Vielmehr scheinen die Markteilnehmer sowohl in der Aufschwung- als auch in der Abschwungphase von den Kursbewegungen des DAX überrascht worden zu.option prices,risk-neutral density,density evaluation,overlapping data

Markov-Switching GARCH Modelling of Value-at-RisK

This paper proposes an asymmetric Markov regime-switching (MS) GARCH model to estimate value-at-risk (VaR) for both long and short positions. This model improves on existing VaR methods by taking into account both regime change and skewness or leverage effects. The performance of our MS model and single-regime models is compared through an innovative backtesting procedure using daily data for UK and US market stock indices. The findings from exceptions and regulatory-based tests indicate the MS-GARCH specifications clearly outperform other models in estimating the VaR for both long and short FTSE positions and also do quite well for S&P positions. We conclude that ignoring skewness and regime changes has the effect of imposing larger than necessary conservative capital requirements

C-CAPM without Ex Post Data

Survey and option data are used to take a fresh look at the equity premium puzzle. Survey data on equity returns (Livingston survey) shows much lower expected excess returns than ex post data. At the same time, option data suggests that investors perhaps overestimate the volatility of equity returns. Both facts reduce the puzzle. However, data on beliefs about output volatility (Survey of Professional Forecasters) shows marked overconfidence. On balance, the equity premium is somewhat less of a puzzle than in ex post data.equity premium puzzle, Livingston survey, S&P 500 options, Survey of Professional Forecasters