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We study the "real world" calibration of a partially specified stochastic volatility model, where the analytic expressions of the asset price drift rate and of the stochastic variance drift are not specified. The model is calibrated matching the observed asset log returns and the priors assigned by the investor. No option price data are used in the calibration. The priors chosen for the asset price drift rate and for the stochastic variance drift are those suggested by the Heston model. For this reason, the model presented can be considered as an "enhanced" Heston model. The calibration problem is formulated as a stochastic optimal control problem and solved using the dynamic programming principle. The model presented and the Heston model are calibrated using synthetic and Standard & Poor 500 (S&P500) data. The calibrated models are used to produce 6, 12, and 24 months in the future synthetic and S&P500 forecasts. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A new method for calibrating the Black-Scholes asset price dynamics model is proposed. The data used to test the calibration problem included observations of asset prices over a finite set of (known) equispaced discrete time values. Statistical tests were used to estimate the statistical significance of the two parameters of the Black-Scholes model: the volatility and the drift. The effects of these estimates on the option pricing problem were investigated. In particular, the pricing of an option with uncertain volatility in the Black-Scholes framework was revisited, and a statistical significance was associated with the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments involving synthetic and real data were presented. The real data considered were the daily closing values of the S&P500 index and the associated European call and put option prices in the year 2005. The method proposed here for calibrating the Black-Scholes dynamics model could be extended to other science and engineering models that may be expressed in terms of stochastic dynamical systems. [ABSTRACT FROM AUTHOR]