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 Option Pricing and Calibration Problems 863Journal of Futures Markets DOI: 10.1002/fut 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 onedimensional 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.univpm.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

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A masking problem in time dependent three dimensional acoustic obstacle scattering is considered. The masking problem consists in making masked a bounded scatterer characterized by an acoustic boundary impedance and immersed in a homogeneous isotropic medium that, when hit by an incident acoustic field, generates a scattered acoustic field. The precise definition of the masking problem is given later. This problem has been formulated as an optimal control problem for the wave equation. The corresponding first order optimality condition is derived and solved with a highly parallelizable numerical method. Some numerical experience on a test problem is shown. (C) 2004 Acoustical Society of America

We investigate the idea of solving calibration problems for stochastic dynamical systems using statistical tests. We consider a specific stochastic dynamical system: the normal SABR model. The SABR model has been introduced in mathematical finance in 2002 by Hagan, Kumar, Lesniewski and Woodward. The model is a system of two stochastic differential equations whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The normal SABR model is a special case of the SABR model. The calibration problem for the normal SABR model is an inverse problem that consists in determining the values of the parameters of the model from a set of data. We consider as set of data two different sets of forward prices/rates and we study the resulting calibration problems. Ad hoc statistical tests are developed to solve these calibration problems. Estimates with statistical significance of the parameters of the model are obtained. Let be a constant, we consider multiple independent trajectories of the normal SABR model associated to given initial conditions assigned at time . In the first calibration problem studied the set of the forward prices/rates observed at time in this set of trajectories is used as data sample of a statistical test. The statistical test used to solve this calibration problem is based on some new formulae for the moments of the forward prices/rates variable of the normal SABR model. The second calibration problem studied uses as data sample the observations of the forward prices/rates made on a discrete set of given time values along a single trajectory of the normal SABR model. The statistical test used to solve this second calibration problem is based on the numerical evaluation of some high-dimensional integrals. The results obtained in the study of the normal SABR model are easily extended from mathematical finance to other contexts in science and engineering where stochastic models involving stochastic volatility or stochastic state space models are used. © 2013 by Walter de Gruyter Berlin Boston 2013

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