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

• Volume: 16, Issue: 1, page 17-40
• ISSN: 2300-133X

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## Abstract

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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.

## How to cite

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Marek Karaś, and Anna Serwatka. "Discrete-time market models from the small investor point of view and the first fundamental-type theorem." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 16.1 (2017): 17-40. <http://eudml.org/doc/288362>.

@article{MarekKaraś2017,
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.},
author = {Marek Karaś, Anna Serwatka},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
keywords = {market model; arbitrage strategy; arbitrage opportunity; arbitrege-free market},
language = {eng},
number = {1},
pages = {17-40},
title = {Discrete-time market models from the small investor point of view and the first fundamental-type theorem},
url = {http://eudml.org/doc/288362},
volume = {16},
year = {2017},
}

TY - JOUR
AU - Marek Karaś
AU - Anna Serwatka
TI - Discrete-time market models from the small investor point of view and the first fundamental-type theorem
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2017
VL - 16
IS - 1
SP - 17
EP - 40
AB - 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.
LA - eng
KW - market model; arbitrage strategy; arbitrage opportunity; arbitrege-free market
UR - http://eudml.org/doc/288362
ER -

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