<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with <italic>d</italic> risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.</p></abstract>

In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with d risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.

Discrete-time market models from the small investor point of view and the first fundamental-type theorem

Institute of Mathematics, Faculty of Mathematics and Computer Science

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research...

Discrete-time market models from the small investor point of view and the first fundamental-type theorem

Published Online: Jan 27, 2018

Page range: 17 - 40

Received: Mar 14, 2016

Accepted: May 12, 2017

DOI: https://doi.org/10.1515/aupcsm-2017-0002

Keywords
market model, arbitrage strategy, arbitrage opportunity, arbitrege-free market

© 2017 Marek Karaś et al., published by De Gruyter Open

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Discrete-time market models from the small investor point of view and the first fundamental-type theorem

Published Online: Jan 27, 2018

Page range: 17 - 40

Received: Mar 14, 2016

Accepted: May 12, 2017

DOI: https://doi.org/10.1515/aupcsm-2017-0002

Keywordsmarket model, arbitrage strategy, arbitrage opportunity, arbitrege-free market

© 2017 Marek Karaś et al., published by De Gruyter Open

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Keywords
market model, arbitrage strategy, arbitrage opportunity, arbitrege-free market