Risk and Volatility: Econometric Models and Financial Practice

The advantage of knowing about risks is that we can change our behavior to avoid them. Of course, it is easily observed that to avoid all risks would be impossible; it might entail no flying, no driving, no walking, eating and drinking only healthy foods and never being touched by sunshine. Even a bath could be dangerous. I could not receive this prize if I sought to avoid all risks. There are some risks we choose to take because the benefits from taking them exceed the possible costs. Optimal behavior takes risks that are worthwhile. This is the central paradigm of finance; we must take risks to achieve rewards but not all risks are equally rewarded. Both the risks and the rewards are in the future, so it is the expectation of loss that is balanced against the expectation of reward. Thus we optimize our behavior, and in particular our portfolio, to maximize rewards and minimize risks.time series;

Hedging Options in a GARCH Environment: Testing the Term Structure of Stochastic Volatility Models

This paper develops a methodology for testing the term structure of volatility forecasts derived from stochastic volatility models, and implements it to analyze models of S&P 500 index volatility. Volatility models are compared by their ability to hedge options positions sensitive to the term structure of volatility. Overall, the most effective hedge is a Black-Scholes (BS) delta-gamma hedge, while the BS delta-vega hedge is the least effective. The most successful volatility hedge is GARCH components delta-gamma, suggesting that the GARCH components estimate of the term structure of volatility is most accurate. The success of the BS delta-gamma hedge may be due to mispricing in the options market over the sample period.

Value at risk models in finance

The main objective of this paper is to survey and evaluate the performance of the most popular univariate VaR methodologies, paying particular attention to their underlying assumptions and to their logical flaws. In the process, we show that the Historical Simulation method and its variants can be considered as special cases of the CAViaR framework developed by Engle and Manganelli (1999). We also provide two original methodological contributions. The first one introduces the extreme value theory into the CAViaR model. The second one concerns the estimation of the expected shortfall (the expected loss, given that the return exceeded the VaR) using a regression technique. The performance of the models surveyed in the paper is evaluated using a Monte Carlo simulation. We generate data using GARCH processes with different distributions and compare the estimated quantiles to the true ones. The results show that CAViaR models perform best with heavy-tailed DGP. JEL Classification: C22, G22CAViaR, extreme value theory, Value at Risk

Theoretical and Empirical properties of Dynamic Conditional Correlation Multivariate GARCH

In this paper, we develop the theoretical and empirical properties of a new class of multi-variate GARCH models capable of estimating large time-varying covariance matrices, Dynamic Conditional Correlation Multivariate GARCH. We show that the problem of multivariate conditional variance estimation can be simplified by estimating univariate GARCH models for each asset, and then, using transformed residuals resulting from the first stage, estimating a conditional correlation estimator. The standard errors for the first stage parameters remain consistent, and only the standard errors for the correlation parameters need be modified. We use the model to estimate the conditional covariance of up to 100 assets using S&P 500 Sector Indices and Dow Jones Industrial Average stocks, and conduct specification tests of the estimator using an industry standard benchmark for volatility models. This new estimator demonstrates very strong performance especially considering ease of implementation of the estimator.

CAViaR: Conditional Value at Risk by Quantile Regression

Value at Risk has become the standard measure of market risk employed by financial institutions for both internal and regulatory purposes. Despite its conceptual simplicity, its measurement is a very challenging statistical problem and none of the methodologies developed so far give satisfactory solutions. Interpreting Value at Risk as a quantile of future portfolio values conditional on current information, we propose a new approach to quantile estimation which does not require any of the extreme assumptions invoked by existing methodologies (such as normality or i.i.d. returns). The Conditional Value at Risk or CAViaR model moves the focus of attention from the distribution of returns directly to the behavior of the quantile. We postulate a variety of dynamic processes for updating the quantile and use regression quantile estimation to determine the parameters of the updating process. Tests of model adequacy utilize the criterion that each period the probability of exceeding the VaR must be independent of all the past information. We use a differential evolutionary genetic algorithm to optimize an objective function which is non-differentiable and hence cannot be optimized using traditional algorithms. Applications to simulated and real data provide empirical support to our methodology and illustrate the ability of these algorithms to adapt to new risk environments.

The ACD Model: Predictability of the Time Between Concecutive Trades

Forecasting ability of several parameterizations of ACD models are compared to benchmark linear autoregressions for inter-trade durations. The estimation of parametric ACD models requires both the choice of a conditional density for durations and the specification of a functional form for the conditional mean duration. Our results provide guidance for choosing among different parameterizations and for developing better forecasting models to predict one-step-ahead, multi-step-ahead, and the whole density of time durations. For evaluating density forecasts, we propose a new constructive test, which is based on the series of probability integral transforms. The choice of the conditional distribution for inter-trade durations does not seem to affect the out-of sample performances of the ACD at short, as well as longer, horizons. Yet, this choice becomes critical when forecasting the density.