Some no-arbitrage rules under short-sales constraints, and applications to converging asset prices

Abstract

Under short-sales prohibitions, no free lunch with vanishing risk (NFLVRS) is known to be equivalent to the existence of an equivalent supermartingale measure for the price process (Pulido in Ann. Appl. Probab. 24:54–75, 2014). We give a necessary condition for the drift of a price process to satisfy NFLVRS. For two given price processes, we introduce the concept of fundamental supermartingale measure, and when a certain condition necessary for the construction of this fundamental supermartingale measure is not fulfilled, we provide the corresponding arbitrage portfolios. The motivation of our study lies in understanding the particular case of converging prices, i.e., two prices that coincide at a bounded random time.

Appendix A: Some recalls on measures and increasing processes

For the reader’s convenience, we gather here some elementary results that are used in the paper.

Theorem A.1

Let$\mu ^{1}$and$\mu ^{2}$be two finite (possibly signed) measures.

(a) Assume that both$\mu ^{1}$and$\mu ^{2}$are positive measures. Then$\mu ^{1}-\mu ^{2}$is a positive measure only if$\mu ^{2}$is absolutely continuous with respect to$\mu ^{1}$.

(b) Assume that$\mu ^{1}$is singular with respect to$\mu ^{2}$and furthermore$\mu ^{1}+\mu ^{2}$is a positive measure. Then both$\mu ^{1}$and$\mu ^{2}$are positive measures.

Given a filtered probability space, we call an increasing process a process that is positive, adapted and whose paths are increasing and càdlàg. A process which can be written as the difference of two increasing processes is called a process of finite variation.

An increasing process can be seen as a random measure $dA_{t} (\omega )$ on $\mathbb{R}_{+}$ whose distribution function is $A_{\cdot }( \omega )$. Similarly, a process of finite variation can be seen as a signed random measure, since it can be written as the difference of two increasing processes.

Proposition A.2

([14, Proposition 3.13])Let$A$, $B$be finite variation processes (resp. increasing processes) such that$dB\ll dA$. Then there exists an optional (resp. positive) process$H$such that$B=\int HdA$up to an evanescent set. If moreover$A$and$B$are predictable, one may choose$H$to be predictable.

Proposition A.3

([7])

Let$A$, $B$be càdlàg, predictable processes of finite variation, with$B$being increasing. Then there exist a predictable process$\varphi$and a predictable subset$N$of$\Omega \times \mathbb{R}_{+}$such that

$$A=\int \phi dB+\int {\mathbb{1}}_{N}dA$$

and

$${\int }_{{\mathbb{R}}_{+}}{\mathbb{1}}_N(u)d{B}_u=0.$$