Estimating GARCH models using support vector machines*

Abstract

Support vector machines (SVMs) are a new nonparametric tool for regression estimation. We will use this tool to estimate the parameters of a GARCH model for predicting the conditional volatility of stock market returns. GARCH models are usually estimated using maximum likelihood (ML) procedures, assuming that the data are normally distributed. In this paper, we will show that GARCH models can be estimated using SVMs and that such estimates have a higher predicting ability than those obtained via common ML methods.     

Normally distributed high-frequency returns: a subordination approach

In this paper, market microstructure models are considered to assess the influence of the key components that drive asset prices. Synchronization techniques and sampling methods are reviewed. Alternatives to normally distributed asset returns, specifically subordinated Brownian motion, are considered. The influential components of the price processes are then combined under tick time to recover normality of asset returns via subordination, a process I denote as ‘natural time’. Normally distributed returns are obtained with the natural time approach which is also found to dominate GARCH results.     

Announcement Effects and Asymmetric Volatility in Industry Stock Returns: Evidence from Taiwan

This study employs financial econometric models to examine the asymmetric volatility of equity returns in response to monetary policy announcements in the Taiwanese stock market. The meetings of the board of directors at the Central Bank of the Republic of China (Taiwan) are considered for testing the announcement effects. The asymmetric generalized autoregressive conditional heteroskedasticity (GARCH) model and the smooth transition autoregression with GARCH model are used to measure equity returns' asymmetric volatility. We conclude that the asymmetric volatility of countercyclical equity returns can be identified. Our findings support the leverage effect of stock price changes for most industry equity returns in Taiwan.     

Managing extreme risks in tranquil and volatile markets using conditional extreme value theory

Financial risk management typically deals with low-probability events in the tails of asset price distributions. To capture the behavior of these tails, one should therefore rely on models that explicitly focus on the tails. Extreme value theory (EVT)-based models do exactly that, and in this paper, we apply both unconditional and conditional EVT models to the management of extreme market risks in stock markets. We find conditional EVT models to give particularly accurate Value-at-Risk (VaR) measures, and a comparison with traditional (Generalized ARCH (GARCH)) approaches to calculate VaR demonstrates EVT as being the superior approach both for standard and more extreme VaR quantiles.     

Extreme observations and risk assessment in the equity markets of MENA region: Tail measures and Value-at-Risk

The standard “delta-normal” Value-at-Risk methodology requires that the underlying returns generating distribution for the security in question is normally distributed, with moments which can be estimated using historical data and are time-invariant. However, the stylized fact that returns are fat-tailed is likely to lead to under-prediction of both the size of extreme market movements and the frequency with which they occur. In this paper, we use the extreme value theory to analyze four emerging markets belonging to the MENA region (Egypt, Jordan, Morocco, and Turkey). We focus on the tails of the unconditional distribution of returns in each market and provide estimates of their tail index behavior. In the process, we find that the returns have significantly fatter tails than the normal distribution and therefore introduce the extreme value theory. We then estimate the maximum daily loss by computing the Value-at-Risk (VaR) in each market. Consistent with the results from other developing countries [see Gencay, R. and Selcuk, F., (2004). Extreme value theory and Value-at-Risk: relative performance in emerging markets. International Journal of Forecasting, 20, 287–303; Mendes, B., (2000). Computing robust risk measures in emerging equity markets using extreme value theory. Emerging Markets Quarterly, 4, 25–41; Silva, A. and Mendes, B., (2003). Value-at-Risk and extreme returns in Asian stock markets. International Journal of Business, 8, 17–40], generally, we find that the VaR estimates based on the tail index are higher than those based on a normal distribution for all markets, and therefore a proper risk assessment should not neglect the tail behavior in these markets, since that may lead to an improper evaluation of market risk. Our results should be useful to investors, bankers, and fund managers, whose success depends on the ability to forecast stock price movements in these markets and therefore build their portfolios based on these forecasts.     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

Model risk of the implied GARCH-normal model

Model risk causes significant losses in financial derivative pricing and hedging. Investors may undertake relatively risky investments due to insufficient hedging or overpaying implied by flawed models. The GARCH model with normal innovations (GARCH-normal) has been adopted to depict the dynamics of the returns in many applications. The implied GARCH-normal model is the one minimizing the mean square error between the market option values and the GARCH-normal option prices. In this study, we investigate the model risk of the implied GARCH-normal model fitted to conditional leptokurtic returns, an important feature of financial data. The risk-neutral GARCH model with conditional leptokurtic innovations is derived by the extended Girsanov principle. The option prices and hedging positions of the conditional leptokurtic GARCH models are obtained by extending the dynamic semiparametric approach of Huang and Guo [Statist. Sin., 2009, 19, 1037–1054]. In the simulation study we find significant model risk of the implied GARCH-normal model in pricing and hedging barrier and lookback options when the underlying dynamics follow a GARCH-t model.     

El efecto de la volatilidad del peso mexicano en los rendimientos y riesgo de la Bolsa Mexicana de Valores

The effect of heavy tails due to rare events and different levels of asymmetry associated with high volatility clustering in the emerging financial markets requires sophisticated models for statistical modelling of such stylized facts. This article applies extreme value theory (EVT) to quantify tail risk on the daily returns of Mexican stock market under aggregation of foreign exchange rate risk from January 1971 to December 2010. This study focuses on the maximum-block method and generalized extreme value distribution (GEVD) to model the asymptotic behavior of extreme returns in US dollars. The empirical results show that EVT-Based VaR measured at high confidence levels performs better than simulation historical and delta-normal VaR models on capturing fat-tails in the returns of highly volatile stock markets. Additionally, international investors holding long positions in Mexican stock market are more prone to experience larger potential losses than investors with short positions during local currency depreciation and financial crisis periods.     

An analysis of option pricing under systematic consumption risk using GARCH

An issue in the pricing of contingent claims is whether to account for consumption risk. This is relevant for contingent claims on stock indices, such as the FTSE 100 share price index, as investor’s desire for smooth consumption is often used to explain risk premiums on stock market portfolios, but is not used to explain risk premiums on contingent claims themselves. This paper addresses this fundamental question by allowing for consumption in an economy to be correlated with returns. Daily data on the FTSE 100 share price index are used to compare three option pricing models: the Black–Scholes option pricing model, a GARCH (1, 1) model priced under a risk-neutral framework, and a GARCH (1, 1) model priced under systematic consumption risk. The findings are that accounting for systematic consumption risk only provides improved accuracy for in-the-money call options. When the correlation between consumption and returns increases, the model that accounts for consumption risk will produce lower call option prices than observed prices for in-the-money call options. These results combined imply that the potential consumption-related premium in the market for contingent claims is constant in the case of FTSE 100 index options.     

VARIANCE,RETURN, AND HIGH-LOW PRICE SPREADS

We report three new findings that rely upon the high-low price range as an estimate of stock return variance. The predictability of variance is associated with persistence in high prices and with correlated shocks to high and low prices. Excess stock returns are positively related to anticipated variance and inversely related to unanticipated variance. Lagged squared residuals in GARCH(1,1) models have no incremental explanatory power in the presence of forecasts of conditional volatility generated from high-low price spread models.     

Financial market models with Lévy processes and time-varying volatility

Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the “KR distribution”. We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr–Geman–Madan–Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the “change of measure” for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.     

Volatility transmission between oil prices and equity sector returns

This paper employs bivariate GARCH models to simultaneously estimate the mean and conditional variance between five different US sector indexes and oil prices. Since many different financial assets are traded based on these market sector returns, it is important for financial market participants to understand the volatility transmission mechanism over time and across these series in order to make optimal portfolio allocation decisions. We examine weekly returns from January 1, 1992 to April 30, 2008 and find evidence of significant transmission of shocks and volatility between oil prices and some of the examined market sectors. The findings support the idea of cross-market hedging and sharing of common information by investors.     

Robust outlier detection for Asia–Pacific stock index returns

Outliers can lead to model misspecifications, poor forecasts and invalid inferences. Their identification and correction is therefore an important objective of financial modeling.This paper introduces a simple method to detect outliers in a financial series. It uses an AR(1)–GARCH(1,1) model to calculate interval forecasts for one-step ahead returns that are then compared to realized returns to determine whether or not we are in the presence of an aberrant observation. The GARCH model, however, is only used as a filter and the identification algorithm remains robust to model misspecifications.The efficiency of this outlier-correction technique is first tested with a simulation study, before being applied to five Asian stock market returns to identify the outlying observations. After an analysis of these extreme fluctuations, the out-of-sample forecasting performance of our outlier-corrected model is then compared to the classical forecasts of a GARCH model in which no account is taken of outliers.     

Volatility spillovers between oil prices and the stock market under structural breaks

This paper employs univariate and bivariate GARCH models to examine the volatility of oil prices and US stock market prices incorporating structural breaks using daily data from July 1, 1996 to June 30, 2013. We endogenously detect structural breaks using an iterated algorithm and incorporate this information in GARCH models to correctly estimate the volatility dynamics. We find no volatility spillover between oil prices and US stock market when structural breaks in variance are ignored in the model. However, after accounting for structural breaks in the model, we find strong volatility spillover between the two markets. We compute optimal portfolio weights and dynamic risk minimizing hedge ratios to highlight the significance of our empirical results which underscores the serious consequences of ignoring these structural breaks. Our findings are consistent with the notion of cross-market hedging and sharing of common information by financial market participants in these markets.     

The choice of the distribution of asset returns: How extreme value theory can help?

One of the issues of risk management is the choice of the distribution of asset returns. Academics and practitioners have assumed for a long time (for more than three decades) that the distribution of asset returns is a Gaussian distribution. Such an assumption has been used in many fields of finance: building optimal portfolio, pricing and hedging derivatives and managing risks. However, real financial data tend to exhibit extreme price changes such as stock market crashes that seem incompatible with the assumption of normality. This article shows how extreme value theory can be useful to know more precisely the characteristics of the distribution of asset returns and finally help to chose a better model by focusing on the tails of the distribution. An empirical analysis using equity data of the US market is provided to illustrate this point.     

A note on cointegration of international stock market indices

Cointegration is frequently used to assess the degree of interdependence of financial markets. We show that if a stock's price follows a stock specific random walk, market indices cannot be cointegrated. Indices are a mere combination of n different random walks which itself is non-stationary by construction. We substantiate the theoretical propositions using a sample of 28 stock indices as well as a simulation study. In the latter we simulate stock prices, construct indices and test whether these indices are cointegrated. We show that while heteroscedasticity misleads cointegration tests, it is not sufficient to explain the high correlation between stock market index returns. A common random walk component and correlated price innovations are necessary to reproduce this feature.     

Tempered stable and tempered infinitely divisible GARCH models

In this paper, we introduce a new GARCH model with an infinitely divisible distributed innovation. This model, which we refer to as the rapidly decreasing tempered stable (RDTS) GARCH model, takes into account empirical facts that have been observed for stock and index returns, such as volatility clustering, non-zero skewness, and excess kurtosis for the residual distribution. We review the classical tempered stable (CTS) GARCH model, which has similar statistical properties. By considering a proper density transformation between infinitely divisible random variables, we can find the risk-neutral price process, thereby allowing application to option-pricing. We propose algorithms to generate scenarios based on GARCH models with CTS and RDTS innovations. To investigate the performance of these GARCH models, we report parameter estimates for the Dow Jones Industrial Average index and stocks included in this index. To demonstrate the advantages of the proposed model, we calculate option prices based on the index.     

Asymmetric and leptokurtic distribution for heteroscedastic asset returns: The SU-normal distribution

This paper proposes the S_U-normal distribution to describe non-normality features embedded in financial time series, such as: asymmetry and fat tails. Applying the S_U-normal distribution to the estimation of univariate and multivariate GARCH models, we test its validity in capturing asymmetry and excess kurtosis of heteroscedastic asset returns. We find that the S_U-normal distribution outperforms the normal and Student-t distributions for describing both the entire shape of the conditional distribution and the extreme tail shape of daily exchange rates and stock returns. The goodness-of-fit (GoF) results indicate that the skewness and excess kurtosis are better captured by the S_U-normal distribution. The exceeding ratio (ER) test results indicate that the S_U-normal is superior to the normal and Student-t distributions, which consistently underestimate both the lower and upper extreme tails, and tend to overestimate the lower tail in general.     

Additional evidence on transparency and bank financial performance

This paper examines the relationship between volatility and the probability of occurrence of expected extreme returns in the Canadian market. Four measures of volatility are examined: implied volatility from firm option prices, conditional volatility calculated using an EGARCH model, idiosyncratic volatility, and expected shortfall. A significantly positive relationship is observed between a firm's idiosyncratic volatility and the probability of occurrence of an extreme return in the subsequent month for firms. A 10% increase in idiosyncratic volatility in a given month is associated with the probability of an extreme shock in the subsequent month (top or bottom 1.5% of the returns distribution) of 26.4%. Other firm characteristics, including firm age, price, volume and book‐to‐market ratio, are also shown to be significantly related to subsequent firm extreme returns. The effects of conditional and implied volatility are mixed. The E‐GARCH and expected shortfall measures of conditional volatility are consistent with mean reversion: high short term realizations of conditional volatility foreshadow a lower probability of extreme returns.