Simulated maximum likelihood for general stochastic volatility models: a change of variable approach

Maximum likelihood has proved to be a valuable tool for fitting the log-normal stochastic volatility model to financial returns time series. Using a sequential change of variable framework, we are able to cast more general stochastic volatility models into a form appropriate for importance samplers based on the Laplace approximation. We apply the methodology to two example models, showing that efficient importance samplers can be constructed even for highly non-Gaussian latent processes such as square-root diffusions.Change of Variable; Heston Model; Laplace Importance Sampler; Simulated Maximum Likelihood; Stochastic Volatility

Simulated Maximum Likelihood Estimation for Latent Diffusion Models

In this paper a method is developed and implemented to provide the simulated maximum likelihood estimation of latent diffusions based on discrete data. The method is applicable to diffusions that either have latent elements in the state vector or are only observed at discrete time with a noise. Latent diffusions are very important in practical applications in nancial economics. The proposed approach synthesizes the closed form method of Aït-Sahalia (2008) and the ecient importance sampler of Richard and Zhang (2007). It does not require any inll observations to be introduced and hence is computationally tractable. The Monte Carlo study shows that the method works well in finite sample. The empirical applications illustrate usefulness of the method and find no evidence of infinite variance in the importance sampler.Closed-form approximation; Diusion Model; Ecient importance sampler

Estimating the GARCH Diffusion: Simulated Maximum Likelihood in Continuous Time

A new algorithm is developed to provide a simulated maximum likelihood estimation of the GARCH diffusion model of Nelson (1990) based on return data only. The method combines two accurate approximation procedures, namely, the polynomial expansion of Aït-Sahalia (2008) to approximate the transition probability density of return and volatility, and the Efficient Importance Sampler (EIS) of Richard and Zhang (2007) to integrate out the volatility. The first and second order terms in the polynomial expansion are used to generate a base-line importance density for an EIS algorithm. The higher order terms are included when evaluating the importance weights. Monte Carlo experiments show that the new method works well and the discretization error is well controlled by the polynomial expansion. In the empirical application, we fit the GARCH diffusion to equity data, perform diagnostics on the model fit, and test the finiteness of the importance weights.Ecient importance sampling; GARCH diusion model; Simulated Maximum likelihood; Stochastic volatility

Stimulated Maximum Likelihood Estimation of Continuous Time Stochastic Volatility Models

In this paper we develop and implement a method for maximum simulated likelihood estimation of the continuous time stochastic volatility model with the constant elasticity of volatility. The approach do not require observations on option prices nor volatility. To integrate out latent volatility from the joint density of return and volatility, a modified efficient importance sampling technique is used after the continuous time model is approximated using the Euler-Maruyama scheme. The Monte Carlo studies show that the method works well and the empirical applications illustrate usefulness of the method. Empirical results provide strong evidence against the Heston model.Efficient importance sampler; Constant elasticity of volatility

TMB: Automatic Differentiation and Laplace Approximation

TMB is an open source R package that enables quick implementation of complex nonlinear random effect (latent variable) models in a manner similar to the established AD Model Builder package (ADMB, admb-project.org). In addition, it offers easy access to parallel computations. The user defines the joint likelihood for the data and the random effects as a C++ template function, while all the other operations are done in R; e.g., reading in the data. The package evaluates and maximizes the Laplace approximation of the marginal likelihood where the random effects are automatically integrated out. This approximation, and its derivatives, are obtained using automatic differentiation (up to order three) of the joint likelihood. The computations are designed to be fast for problems with many random effects (~10^6) and parameters (~10^3). Computation times using ADMB and TMB are compared on a suite of examples ranging from simple models to large spatial models where the random effects are a Gaussian random field. Speedups ranging from 1.5 to about 100 are obtained with increasing gains for large problems. The package and examples are available at http://tmb-project.org

The Sibling Distribution for Multivariate Life Time Data

A flexible class of multivariate distributions for continuous lifetimes is proposed. The distribution is defined in terms of the age-at-death of m siblings. The expression for the joint density is derived using classical results from mathematical demography. The parameters of the distribution are the age-specific birth and death rates, in addition to a vector of relative death times for the m siblings. For the case of constant birth and death rates we are able to derive an explicit expression for the bivariate sibling density, which is proven to be MTP2, and hence has positive dependence. Further, we show that a special case of the sibling distribution belongs to the Block-Basu class of multivariate distribution. In the general case, with age-dependent birth and death rates, evaluation of the density involves numerical integration, but is still feasible.publishedVersio