Asset Trading in Continuous Time: A Cautionary Tale

TL;DR

This paper questions the foundational assumption that continuous-time asset trading models can be approximated by discrete models, highlighting potential discrepancies when constraints are strictly enforced.

Contribution

It demonstrates that, under realistic constraints, the continuous-time model may not be the limit of discrete trading models, challenging a core assumption in modern finance.

Findings

Continuous models may not be limits of discrete models under certain constraints.

Terminal wealth and self-financing constraints are critical in this discrepancy.

Raises foundational questions about the validity of continuous-time models in finance.

Abstract

The continuous time model of dynamic asset trading is the central model of modern finance. Because trading cannot in fact take place at every moment of time, it would seem desirable to show that the continuous time model can be viewed as the limit of models in which trading can occur only at (many) discrete moments of time. This paper demonstrates that, if we take terminal wealth constraints and self-financing constraints as seriously in the discrete model as in the continuous model, then the continuous trading model need not be the limit of discrete trading models. This raises serious foundational questions about the continuous time model.