Modelling Bounded Stochastic Motion

Abstract

In this paper a simple stochastic approach has been presented for modelling financial observables which are constrained to lie between two positive bounds. While the proposed stochastic process has an inaccessible upper boundary, the lower boundary is quasi-bounded, implying that the lower boundary can be breached if the probability leakage condition is met. The quasi-boundedness of the process at the lower boundary can thus provide us an indicator of possible downside risk of the corresponding observables. Empirical calibration of model parameters of the proposed process for different financial observables can also be easily performed due to the availability of an analytically tractable probability density function. Hence, in terms of the calibrated model parameters, making predictions of future movements of these observables becomes feasible.

Keywords: Constrained stochastic motion, quasi-bounded process, time-varying boundaries, crash risk