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

Abstract

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. r 2005 Elsevier B.V. All rights reserved.JEL classification: C150

Non-parametric estimation of forecast distributions in non-Gaussian, non-linear state space models

“…Under the skewed Student-t DGP, the non-parametric estimates signi…cantly out-perform the misspeci…ed parametric estimates, for all four scoring measures. Table 1 Constants, ; c and !, used in the penalized likelihood function in (24), in the simulation experiments for the linear, SCD and RV models, as detailed in Sections (3.1.1), (3.1.2) and (3.1.3) respectively. Table 3 records (for the linear model) the test statistics associated with the three PIT tests described in Section 3.2, namely, the Pearson test for the uniformity of fu i T +1 ; i = 1; 2; :::; M g in (43), the LR test of the normality (and independence) of !…”

“…The …rst penalty component in (24) controls the smoothness of the estimated density function de…ned by the g j , with smaller values of 2 corresponding to smoother densities. The second penalty term in (24) penalizes values of g j associated with grid-points that are relatively far from the mean, with the value of c determining the size of the penalty. The constant !…”

Exact and asymptotic tests for possibly non-regular hypotheses on stochastic volatility models

“…Besides the quasi maximum likelihood approaches [Ruiz (1994)], or the generalized method of moments (GMM) procedures [Andersen and Sørensen (1996)], simulation-based estimation has become more attractive due to increasing computer power, and comprises: (1) indirect inference which has been used to estimate SV models by Monfardini (1998); (2) the efficient method of moments applied to SV models by Andersen, Chung and Sørensen (1999) and Chernov, Gallant, Ghysels and Tauchen (2003); (3) simulated maximum likelihood, which can be implemented in SV models using importance sampling; see Danielsson and Richard (1993), Danielsson (1994), Durham (2006Durham ( , 2007. Bayesian techniques can also be applied in this context through computer-intensive methods, such as Markov Chain Monte Carlo (MCMC) methods, and appear to yield relatively good results; see Jacquier, Polson and Rossi (1994), Chib, Nardari and Shephard (2002).…”

Risk-Neutral Modelling With Affine and Non-Affine Models

“…Jacquier, Polson, and Rossi (1994) demonstrate computationally efficient Bayesian techniques, which involve MCMC techniques for sampling over the latent state space. Jacquier, Polson, andRossi (2004), Eraker (2001), Eraker, Johannes, and Polson (2003), Shephard and Pitt (1997), Kim, Shephard, and Chib (1998), Gallant and Tauchen (1996), Durbin and Koopman (1997), Liesenfeld and Richard (2003), Bates (2006), and Durham (2006), among many others have added to this literature. Andersen, Benzoni, and Lund (2002) and Chernov, Gallant, Ghysels, and Tauchen (2003) provide comprehensive studies comparing a number of models using a simulated method of moments approach.…”