No Arbitrage in Continuous Financial Markets

TL;DR

This paper establishes integral tests to determine the presence or absence of arbitrage in continuous financial markets with a single risky asset modeled via stochastic processes, including Markov switching models.

Contribution

It introduces new integral criteria for arbitrage detection and characterizes the minimal martingale measure in complex stochastic models with switching mechanisms.

Findings

Derived conditions for the existence of the minimal martingale measure.

Showed that minimal martingale measure preserves noise independence in Markov switching models.

Developed new criteria for martingale and strict local martingale properties of stochastic exponentials.

Abstract

We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Ito process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We also show that for Markov switching models the minimal martingale measure preserves the independence of the noise and we study how the minimal martingale measure can be modified to change the structure of the switching mechanism. Our main mathematical tools are new criteria for the martingale and strict local martingale property of certain stochastic exponentials.