The Analysis of Real Data Using a Multiscale Stochastic Volatility Model

In this paper we use filtering and maximum likelihood methods to solve a calibration problem for a multiscale stochastic volatility model. The multiscale stochastic volatility model considered has been introduced in Fatone et al. (2009), generalises the Heston model and describes the dynamics of the asset price using as auxiliary variables two stochastic variances on two different time scales. The aim of this paper is to estimate the parameters of this multiscale model (including the risk premium parameters when necessary) and its two initial stochastic variances from the knowledge, at discrete times, of the asset price and, eventually, of the prices of call and/or put European options on the asset. This problem is translated into a maximum likelihood problem with the likelihood function defined through the solution of a filtering problem. Furthermore we develop a tracking procedure that is able to track the asset price and the values of its two stochastic variances for time values where there are no data available. Numerical examples of the solution of the calibration problem and of the performance of the tracking procedure using high frequency synthetic data and daily real data are presented. The real data studied are two time series of electric power price data taken from the US electricity market and the 2005 data relative to the US S&P 500 index and to the prices of a call and a put European option on the US S&P 500 index. The calibration procedure is applied to these data and the results of the calibration are used in the tracking procedure to forecast the asset and option prices. The forecasts of the asset prices and of the option prices are compared with the prices actually observed. This comparison shows that the forecasts are of very high quality even when we consider ‘spiky’ electric power price data. The website: http://www.econ.univpm.it/recchioni/finance/w9 contains some auxiliary material including animations that help with the understanding of this paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.