Time-Series Heston Model Calibration Using a Trinomial Tree

Abstract

In this work a trinomial tree representing the Heston model variance process is used to estimate the parameters for the Heston stochastic volatility model using historical daily observations of the asset.

The results include estimates for all Heston model parameters as well as an estimated most likely path for the latent variance process.

The variance tree is constructed using a somewhat novel approach that uses a non-uniform, recombining grid, and some analysis is included to justify the approach.

The probability of the observed asset path is computed as an average over all variance paths using this tree, with the asset return observations approximated as normal conditional on the values of the variance at the start and end of each observation.

Calibration is achieved using a reasonably generic multidimensional optimization algorithm, with the negative of the logarithm of the asset path probability computed using the trinomial tree as the objective function.

Bounds are imposed on all parameters based on the ability of the tree to estimate the parameters accurately, in particular noting that with increasing the volatility of variance the grid becomes coarser, and there is therefore a limit to how large this parameter can be in order for the tree to accurately resolve the variance distribution.

Reasonable convergence with increasing number of asset return observations is demonstrated for all parameters as well as the latent variance estimation, using paths simulated from the Heston model.

Results from the calibration to four foreign exchange processes are also provided, showing that the results are reasonably stable.