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On weak notions of no-arbitrage in a 1D general diffusion market with interest rates

Alexis Anagnostakis, David Criens, Mikhail Urusov

TL;DR

This paper characterizes three weak no-arbitrage notions for a one-dimensional general diffusion with a constant interest rate $r$ using deterministic diffusion characteristics $(\mathfrak{s},\mathfrak{m},\mathfrak{q})$. By translating FTAP-style structure conditions into boundary-sensitive relations among the second-derivative measures of $\mathfrak{q}$ and the speed measure, it provides explicit criteria for NIP, NSA, and NUPBR, including how boundary types (absorbing vs reflecting) and the interest rate influence regularity and arbitrage viability. Notably, NIP can co-exist with reflecting boundaries at $r=0$ and does not force $\mathfrak{q}$ to be $C^1$, while NSA becomes the minimal regime enforcing $\mathfrak{q}$ to be $C^1$ and boundary-regular; with $r\neq 0$, even NUPBR can hold despite reflecting boundaries or lower diffusion regularity, due to compensating effects between stickiness and skewness. The results illuminate surprising interactions between boundary behavior, scale-function regularity, and interest rates, offering new guidance for diffusion-based asset-pricing models and the structural understanding of weak no-arbitrage conditions.

Abstract

We establish deterministic necessary and sufficient conditions for the no-arbitrage notions "no increasing profit" (NIP), "no strong arbitrage" (NSA) and "no unbounded profit with bounded risk" (NUPBR) in one-dimensional general diffusion markets. These are markets with one risky asset, which is modeled as a regular continuous strong Markov process that is also a semimartingale, and a riskless asset that grows exponentially at a constant rate $r\in \mathbb{R}$. All deterministic criteria are provided in terms of the scale function and the speed measure of the risky asset process. Our study reveals a variety of surprising effects. For instance, irrespective of the interest rate, NIP is not excluded by reflecting boundaries or an irregular scale function. In the case of non-zero interest rates, it is even possible that NUPBR holds in the presence of reflecting boundaries and/or skew thresholds. In the zero interest rate regime, we also identify NSA as the minimal no arbitrage notion that excludes reflecting boundaries and that forces the scale function to be continuously differentiable with strictly positive absolutely continuous derivative, meaning that it is of the same form as for a stochastic differential equation.

On weak notions of no-arbitrage in a 1D general diffusion market with interest rates

TL;DR

This paper characterizes three weak no-arbitrage notions for a one-dimensional general diffusion with a constant interest rate using deterministic diffusion characteristics . By translating FTAP-style structure conditions into boundary-sensitive relations among the second-derivative measures of and the speed measure, it provides explicit criteria for NIP, NSA, and NUPBR, including how boundary types (absorbing vs reflecting) and the interest rate influence regularity and arbitrage viability. Notably, NIP can co-exist with reflecting boundaries at and does not force to be , while NSA becomes the minimal regime enforcing to be and boundary-regular; with , even NUPBR can hold despite reflecting boundaries or lower diffusion regularity, due to compensating effects between stickiness and skewness. The results illuminate surprising interactions between boundary behavior, scale-function regularity, and interest rates, offering new guidance for diffusion-based asset-pricing models and the structural understanding of weak no-arbitrage conditions.

Abstract

We establish deterministic necessary and sufficient conditions for the no-arbitrage notions "no increasing profit" (NIP), "no strong arbitrage" (NSA) and "no unbounded profit with bounded risk" (NUPBR) in one-dimensional general diffusion markets. These are markets with one risky asset, which is modeled as a regular continuous strong Markov process that is also a semimartingale, and a riskless asset that grows exponentially at a constant rate . All deterministic criteria are provided in terms of the scale function and the speed measure of the risky asset process. Our study reveals a variety of surprising effects. For instance, irrespective of the interest rate, NIP is not excluded by reflecting boundaries or an irregular scale function. In the case of non-zero interest rates, it is even possible that NUPBR holds in the presence of reflecting boundaries and/or skew thresholds. In the zero interest rate regime, we also identify NSA as the minimal no arbitrage notion that excludes reflecting boundaries and that forces the scale function to be continuously differentiable with strictly positive absolutely continuous derivative, meaning that it is of the same form as for a stochastic differential equation.

Paper Structure

This paper contains 19 sections, 17 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. The Financial Market and Weak Notions of No-Arbitrage
  3. The Financial Market
  4. Increasing Profits, Strong Arbitrage and Unbounded Profits with Bounded Risk
  5. A Deterministic Characterization of NIP
  6. A Deterministic Characterization of NSA
  7. A Deterministic Characterization of NUPBR
  8. Discussions and Examples
  9. Influence of Reflecting Boundaries
  10. The Zero Interest Rate Regime
  11. The Non-Zero Interest Rate Regime
  12. Summary
  13. Implications for the Regularity of the Scale Function
  14. The Zero Interest Rate Regime
  15. The Non-Zero Interest Rate Regime
  16. ...and 4 more sections

Key Result

Lemma 2.2

Assume that $Y$ is a semimartingale. Then, the inverse scale function $\mathfrak{q} \triangleq \mathfrak{s}^{-1}$ is the difference of two convex functions on the interior $\mathfrak{s} (J^\circ)$. Furthermore, in case $J = [\alpha, \infty)$ and $\alpha$ is absorbing for $Y$, it holds In case $J = [\alpha, \infty)$ and $\alpha$ is reflecting for $Y$, the second derivative measure $\mathfrak{q}" (

Theorems & Definitions (43)

  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4: FTAP for NIP
  • Definition 2.5
  • Theorem 2.6: FTAP for NSA
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9: FTAP for NUPBR
  • Remark 2.10
  • ...and 33 more