Asymmetries and Volatility Regimes in the European Equity Markets

This paper provides and empirical examination of four European equity indices between 1991 and 2005. We investigate the ability of fifteen different GARCH models to capture the characteristics of historical daily returns effectively and generate realistic implied volatility skews. Using many different model selection criteria we conclude that a normal mixture GARCH model with two volatility components, two sources of asymmetry and endogenous time-varying conditional higher moments provides the best fit overall. Since this model is relatively new in the literature we discuss the theoretical and empirical properties of such models. Examining the estimated parameters we show that they provide information on the likelihood of a crash and they specify the return and volatility behaviour, the leverage effect and the persistence of volatility during the two regimes (‘normal’ and ‘crash’). We also find that asymmetric normal mixture GARCH models, even without a volatility risk premium, afford a sufficiently rich structure to match the empirical characteristics of implied volatility skew surfaces, whereas single-state GARCH models give unrealistic shapes for the equity index skew.equity skew, market cras, GARCH process, normal mixture, skey peristence, leverage effect, volatility regimes

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.

On The Continuous Limit of GARCH

GARCH processes constitute the major area of time series variance analysis hence the limit of these processes is of considerable interest for continuous time volatility modelling. The limit of the GARCH(1,1) model is fundamental for limits of other GARCH processes yet it has been the subject of much debate. The seminal work of Nelson (1990) derived this limit as a stochastic volatility process that is uncorrelated with the price process but a subsequent paper of Corradi (2000) derived the limit as a deterministic volatility process and several other contradictory papers followed. In this paper we reconsider this continuous limit, arguing that because the strong GARCH model is not aggregating in time it is incorrect to examine its limit. Instead it is legitimate to use the weak definition of GARCH that is time aggregating. We prove that its continuous limit is a stochastic volatility model that reduces to Nelson’s GARCH diffusion only under certain assumptions. In general, the weak GARCH limit has correlated Brownian motions in which both the variance diffusion coefficient and the price-volatility correlation are related to the skewness and kurtosis of the physical returns density.GARCH, stochastic volatility, time agtregation, continuous limit

Time varying price discovery

We show how multivariate GARCH models can be used to generate a time-varying “information share” (Hasbrouck, 1995) to represent the changing patterns of price discovery in closely related securities. We find that time-varying information shares can improve credit spread predictions

Model risk of expected shortfall

In this paper we propose to measure the model risk of Expected Shortfall as the optimal correction needed to pass several ES backtests, and investigate the properties of our proposed measures of model risk from a regulatory perspective. Our results show that for the DJIA index, the smallest corrections are required for the ES estimates built using GARCH models. Furthermore, the 2.5% ES requires smaller corrections for model risk than the 1% VaR, which advocates the replacement of VaR with ES as recommended by the Basel Committee. Also, if the model risk of VaR is taken into account, then the corrections made to the ES estimates reduce by 50% on average

Rethinking Capital Structure Arbitrage

It is well known that the capital structure arbitrage strategy generated negative Sharpe ratios over the period 2005-2009. In this paper we introduce four new alternative strategies that, while still based on the discrepancy between the CDS market spread and its equity-implied spread, exploit the information provided by the time-varying price discovery of the equity and CDS markets. We implement the strategies for both US and European obligors and find that these outperform traditional arbitrage trading during the financial crisis. Moreover, the new strategies show higher Sharpe ratios when CDS and equity-implied spreads are cointegrated. The correlation of the new trading rules with hedge fund index returns is low or negative even during the crisis, which suggests that the new rules can be used for portfolio diversification at times when risk reduction is hard to achieve