Calibration in the "real world" of a partially specified stochastic volatility model

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.