Continuous-time trading and emergence of volatility

TL;DR

This paper demonstrates that in idealized continuous-price financial markets, the strong variation exponent of non-constant price processes must be 2, aligning with properties of continuous martingales, without relying on probabilistic assumptions.

Contribution

It establishes a non-probabilistic proof that the variation exponent of continuous price processes in idealized markets is necessarily 2, revealing fundamental properties of market randomness.

Findings

Strong variation exponent of non-constant processes is 2

Results hold without probabilistic assumptions

Aligns with properties of continuous martingales

Abstract

This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.