Properly Discounted Asset Prices Are Semimartingales

Abstract

We study general undiscounted asset price processes, which are only assumed to be non-negative, adapted and RCLL (but not a priority semimartingales). Traders are allowed to use simple (piecewise constant) strategies. We prove that under a discounting-invariant condition of absence of arbitrage, the original prices discounted by the value of any simple strategy with positive wealth must follow semimartingales. As a side result, we establish two corresponding versions of the fundamental theorem of asset pricing that involve supermartingale discounters with some additional strict positivity property.

Keywords: semimartingales, discounting, dynamic share viability, simple strategies, noshort-sales constraints, NA1 for simple strategies, supermartingale discounter

JEL Classification: C00, G10