Monte Carlo methods for estimating, smoothing, and filtering one- and two-factor stochastic volatility models

One- and two-factor stochastic volatility models are assessed over three sets of stock returns data: S&P 500, DJIA, and Nasdaq. Estimation is done by simulated maximum likelihood using techniques that are computationally efficient, robust, straightforward to implement, and easy to adapt to different models. The models are evaluated using standard, easily interpretable time-series tools. The results are broadly similar across the three data sets. The tests provide no evidence that even the simple single-factor models are unable to capture the dynamics of volatility adequately; the problem is to get the shape of the conditional returns distribution right. None of the models come close to matching the tails of this distribution. Including a second factor provides only a relatively small improvement over the single-factor models. Fitting this aspect of the data is important for option pricing and risk management

SV mixture models with application to S&P 500 index returns

Understanding both the dynamics of volatility and the shape of the distribution of returns conditional on the volatility state is important for many financial applications. A simple single-factor stochastic volatility model appears to be sufficient to capture most of the dynamics. It is the shape of the conditional distribution that is the problem. This paper examines the idea of modeling this distribution as a discrete mixture of normals. The flexibility of this class of distributions provides a transparent look into the tails of the returns distribution. Model diagnostics suggest that the model, SV-mix, does a good job of capturing the salient features of the data. In a direct comparison against several affine-jump models, SV-mix is strongly preferred by Akaike and Schwarz information criteria

Likelihood-Based Specification Analysis of Continuous-Time Models of the Short-Term Interest Rate

An extensive collection of continuous-time models of the short-term interest rate are evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (1999), provides new insights regarding several previously unresolved questions. For single factor models, I find that the volatility rather than the drift is the critical component in model specification. Allowing for additional flexibility beyond a constant term in the drift provides negligible benefit. While constant drift would appear to imply that the short rate is nonstationary, in fact stationarity is volatility-induced. The simple constant elasticity of volatility model fits weekly observations of the three-month Treasury bill rate remarkably well but is easily rejected when compared to more flexible volatility specifications over daily data. The methodology of Durham and Gallant can also be used to estimate stochastic volatility models. While adding the latent volatility component provides a large improvement in the likelihood for the physical process, it does little to improve bond-pricing performance

Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes

Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, since the transition density is generally unknown, one is forced to approximate it. The simulation-based approach suggested by Pedersen (1995) has great theoretical appeal, but previously available implementations have been computationally costly. We examine a variety of numerical techniques designed to improve the performance of this approach. Synthetic data generated by a CIR model with parameters calibrated to match monthly observations of the U.S. short-term interest rate are used as a test case. Since the likelihood function of this process is known, the quality of the approximations can be easily evaluated. On data sets with 1000 observations, we are able to approximate the maximum likelihood estimator with negligible error in well under one minute. This represents something on the order of a 10,000-fold reduction in computational effort as compared to implementations without these enhancements. With other parameter settings designed to stress the methodology, performance remains strong. These ideas are easily generalized to multivariate settings and (with some additional work) to latent variable models. To illustrate, we estimate a simple stochastic volatility model of the U.S. short-term interest rate