Stochastic Mean-Reverting Trend (SMART) Model in Quantitative Finance

We consider a stochastic model for the return on the price of a certain financial instrument, such that the trend (drift) of the return is itself stochastic—more precisely, it follows the Ornstein–Uhlenbeck process. In financial markets, the price return process is observable whereas the trend is not. For this model (hereafter denoted by the acronym ‘‘SMART’’), we first derive the closed-form transition probability density function (PDF) by solving the corresponding Feynman–Kac equation. The SMART model is affine with constant coefficients, and its transition PDF is bi-variate normal. Then we derive the Bayesian optimal filter for online estimation of unobservable stochastic trend values from the observable price return values. In doing so, we also obtain the closed-form conditional probability density function (PDF) for the observable price return values. This function is subsequently used in the maximum likelihood estimation (MLE) algorithm for calibration of the SMART model parameters. We perform numerical comparisons of our Bayesian optimal filter with the classical model-free trend estimator over a short-term window, and evaluate the applicability of the SMART model to the price dynamics of some financial instruments (CME-traded Futures).