No arbitrage in continuous financial markets

Abstract

We study no arbitrage conditions for financial models in which either the stocks itself or its log returns are continuous Itô processes. More precisely, we derive deterministic conditions for the existence and nonexistence of equivalent (local) martingale measures and strict martingale densities. For models with a random switching mechanism we also study the set of equivalent (local) martingale measures which are structure preserving. In particular, for one dimensional Markov switching models we provide sufficient and necessary conditions for the existence of structure preserving equivalent (local) martingale measures. Mathematically, our proofs are based on local changes of measures and existence and uniqueness conditions.

Deterministic Criteria for the Absence and Existence of Arbitrage in Multi-Dimensional Diffusion Markets

“…Contrary, there are many papers on the characterization of NFLVR and the existence of EMMs. The articles of Criens [6,7], Delbaen and Shirakawa [11] and Mijatović and Urusov [29] appear to be closest to ours in the sense that they also aim for deterministic conditions. In all of these papers, the asset price process is some sort of Itô process with non-vanishing volatility, which excludes, for instance, sticky points in the interior of the state space.…”

On Absolute Continuity and Singularity of Multidimensional Diffusions

“…For general one-dimensional diffusion models with finite and infinite time horizon, analytic conditions for a local martingale measure to be a martingale measure were given in [25]. Conditions for one-and multidimensional diffusion models with finite time horizon were proven in [7,8,35]. We extend part of these results to a multidimensional setting with infinite time horizon.…”

On absolute continuity and singularity of multidimensional diffusions

“…The integral tests in [9,29] for absolute continuity, singularity and the UI martingale property in one-dimensional frameworks follow from our result and Feller's test for explosion under additional assumptions on the coefficients. The existence of E(L)MMs for one-and multidimensional diffusion models with finite time horizon has, e.g., been studied in [10,11,30,39]. Certain one-dimensional diffusion models with infinite time horizon have been studied in [30].…”

“…On an intuitive level, any solution to the MP (a, b, x) on (Σ, A, A) should coincide locally with P x and consequently, explosion should happen in the same manner for both problems. We now make this intuition precise.The following local uniqueness property of well-posed martingale problems can be proven similar to[11, Lemma 9.1], cf. [40, Exercise 11.5.1] and [17, Theorem 4.6.1].Lemma A.3.…”