Globally Optimal Parameter Estimates for Non-Linear Diffusions

Abstract

This paper studies an approximation method for the log likelihood function of a non-linear diffusion process using the bridge of the diffusion. The main result (Theorem 1) shows that this approximation converges uniformly to the unknown likelihood function and can therefore be used efficiently with any algorithm for sampling from the law of the bridge. We also introduce an expected maximum likelihood (EML) algorithm for inferring the parameters of discretely observed diffusion processes. The approach is applicable to a subclass of non-linear SDEs with constant volatility and drift that is linear in the model parameters. In this setting globally optimal parameters are obtained in a single step by solving a square linear system whose dimension equals the number of parameters in the model. Simulation studies to test the EML algorithm show that it performs well when compared with algorithms based on the exact maximum likelihood as well as closed-form likelihood expansions.

Empirical Asset Pricing with Nonlinear Risk Premia

“…We approximate the law of the true diffusion bridge Q x,y M with the law of a Brownian bridge W x,y M . It is shown in Mijatović and Schneider (2009) that Q x,y M is absolutely continuous with respect to W x,y M , and that there is in fact very little deviation between the two even for long time intervals. Exact draws from the Brownian bridge are obtained from the stochastic difference equation (Stramer and Yan, 2007)…”

“…The transition densities for the transformed variance processes ( 11) and ( 12) that arise in the loglikelihood ( 19) are not available in closed form. To overcome this issue and that of the supposedly flat likelihood function we apply expected maximum likelihood (EML) estimation algorithm from Mijatović and Schneider (2009). This technique makes use of the closeness of the law of the Brownian bridge to the true law of the diffusion bridge, and of the Euler scheme approximation for the transition density when the time interval between observations is small.…”

“…( 15) the observed data implies a time series of Y . An estimate of the parameters of the transformed variance process Y can now be obtained by means of EML (Mijatović and Schneider, 2009). For this purpose we introduce functions f and g from difference equations ( 20) and ( 21) which are displayed in Table 2.…”

“…The most obvious application is a state variable formulation that entails (semi-)analytic pricing under the risk-neutral measure, which leaves flexibility for the dynamics under the physical measure similar to that of the discrete-time approach considered in Dai et al (2006) and Bertholon et al (2008). Recent advances in estimating the parameters of nonlinear diffusions such as the algorithms introduced in Aït-Sahalia (2001), Beskos et al (2006) and Mijatović and Schneider (2009) ensure that reliable parameter inference can be made without explicit formulae for transition densities. An empirical application based on the joint time series of the S&P 100 and the VXO implied volatility indices reveals that our framework offers statistically significant advantages out of sample over extant model specifications in predicting implied as well as realized variance over several forecasting horizons.…”