Bent Jesper Christensen

The relation between implied and realized volatility

Abstract Previous research finds the volatility implied by S&P 100 index option prices to be a biased and inefficient forecast of future volatility and to contain little or no incremental information beyond that in past realized volatility. In contrast, we find that implied volatility outperforms past volatility in forecasting future volatility and even subsumes the information content of past volatility in some of our specifications. Our results differ from previous studies because we use longer time series and nonoverlapping data. A regime shift around the October 1987 crash explains why implied volatility is more biased in previous work.

Interest Rate Dynamics and Consistent Forward Rate Curves

We derive general necessary and sufficient conditions for the mutual consistency of a given parametrized family of forward rate curves and the dynamics of a given interest rate model. Consistency in this context means that the interest rate model will produce forward rate curves belonging to the parameterized family. The interest rate model may be driven by a multidimensional Wiener process as well as by a market point process. As an application, the Nelson-Siegel family of forward curves is shown to be inconsistent with the Ho-Lee interest rate model, and with the Hull-White extension of the Vasicec model, but it may be adjusted to achieve consistency with these models and with extensions that allow for jumps in interest rates. For a natural exponential-polynomial generalization of the Nelson-Siegel family, we give necessary and sufficient conditions for the existence of a consistent interest rate model with deterministic volatility.

On the Job Search and the Wage Distribution

The article structually estimates an on‐the‐job search model of job separations. Given each employer pays observably equivalent workers the same but wages are dispersed across employers, an employers separation flow is the sum of an exogenous outflow unrelated to the wage and a job‐to‐job flow that decreases with the employers wage. Using data from the Danish Integrated Database for Labour Market Research, the empirical results imply, as predicted by theory, that search effort declines with the wage. Furthermore, the estimates explain the employment effect, defined as the horizontal difference between the distribution of wages earned and the wage offer distribution.

The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets

We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets from variables in our information set, including implied volatility backed out from option prices. Realized volatility is separated into its continuous and jump components, and the heterogeneous autoregressive (HAR) model is applied with implied volatility as an additional forecasting variable. A vector HAR (VecHAR) model for the resulting simultaneous system is introduced, controlling for possible endogeneity issues. We find that implied volatility contains incremental information about future volatility in all three markets, relative to past continuous and jump components, and it is an unbiased forecast in the foreign exchange and stock markets. Out-of-sample forecasting experiments confirm that implied volatility is important in forecasting future realized volatility components in all three markets. Perhaps surprisingly, the jump component is, to some extent, predictable, and options appear calibrated to incorporate information about future jumps in all three markets.

New evidence on the implied-realized volatility relation

We consider the relation between the volatility implied in an options price and the subsequently realized volatility. Earlier studies on stock index options have found biases and inefficiencies in implied volatility as a forecast of future volatility. More recently, Christensen and Prabhala find that implied volatility in at-the-money one-month OEX call options on the S&P 100 index in fact is an unbiased and efficient forecast of ex-post realized index volatility after the 1987 stock market crash. In this paper, the robustness of the unbiasedness and efficiency result is extended to a more recent period covering April 1993 to February 1997. As a new contribution, implied volatility is constructed as a trade weighted average of implied volatilities from both in-the-money and out-of-the-money options and both puts and calls. We run a horse race between implied call, implied put, and historical return volatility. Several robustness checks, including a new simultaneous equation approach, underscore our conclusion, that implied volatility is an efficient forecast of realized return volatility.

Long Memory in Stock Market Volatility and the Volatility-in-Mean Effect: The FIEGARCH-M Model

We extend the fractionally integrated exponential GARCH (FIEGARCH) model for daily stock return data with long memory in return volatility of Bollerslev and Mikkelsen (1996) by introducing a possible volatility-in-mean effect. To avoid that the long memory property of volatility carries over to returns, we consider a filtered FIEGARCH-in-mean (FIEGARCH-M) effect in the return equation. The filtering of the volatility-in-mean component thus allows the co-existence of long memory in volatility and short memory in returns. We present an application to the daily CRSP value-weighted cum-dividend stock index return series from 1926 through 2006 which documents the empirical relevance of our model. The volatility-in-mean effect is significant, and the FIEGARCH-M model outperforms the original FIEGARCH model and alternative GARCH-type specifications according to standard criteria.

Panel data, local cuts and orthogeodesic models

Orthogeodesic models admit marginal local cuts and therefore separate inference on subparameters is asymptotically justified. Doubly-flat orthogeodesic models admit local cuts marginally and conditionally. Two important empirical models for panel data are used to illustrate this property and demonstrate its usefullness.

The Exact Likelihood Function for an Empirical Job Search Model

The exact likelihood function for a prototypal job search model is analyzed. The optimality condition implied by the dynamic programming framework is fully imposed. Using the optimality condition allows identification of an offer arrival probability separately from an offer acceptance probability. The estimation problem is nonstandard. The geometry of the likelihood function in finite samples is considered, along with asymptotic properties of the maximum likelihood estimator.

Inference in non-linear panel models with partially missing observations The case of the equilibrium search model

Abstract A thorough econometric analysis of the pure equilibrium search model is given. Minimal data requirements for estimation are unemployment durations, wages, and employment durations. An assessment of the information contribution of each data element is given. The results define the range of potential application of the equilibrium search framework and form the foundation for future econometric analysis of related models.

Measurement Error in the Prototypal Job-Search Model

Statistical analysis of the job-search model has isolated the role played by the minimum observed wage in identifying and estimating behavioral parameters. Estimators based on order statistics, however, are influenced by measurement error in ways that estimators based on averages are not, and there is ample evidence that wage data contain measurement errors. We propose a model in which worker behavior is captured by the prototypal search model but wages are observed with error.