# An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

*Lorella Fatone*,
*Francesca Mariani*,
*Maria Recchioni* and
*Francesco Zirilli*

*Journal of Futures Markets*, 2009, vol. 29, issue 9, 862-893

**Abstract:**
We introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univ pm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm. it/recchioni/finance . © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009

**Date:** 2009

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