Asset price bubbles in markets with transaction costs

Francesca Biagini; Thomas Reitsam

doi:10.3934/fmf.2022002

Article Contents

2022, Volume 1, Issue 3: 397-424. Doi: 10.3934/fmf.2022002 open access

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Asset price bubbles in markets with transaction costs

Francesca Biagini and
Thomas Reitsam^,

Department of Mathematics, Ludwig-Maximilians Universität, Theresienstraße 39, 80333 Munich, Germany

^*Corresponding author: Thomas Reitsam
^*Corresponding author: Thomas Reitsam

Received: January 2020

Revised: November 2021

Early access: July 2022

Published: September 2022

The financial support of the Verein zur Versicherungswissenschaft München e.V. is gratefully acknowledged.

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Abstract

We study asset price bubbles in market models with proportional transaction costs $ \lambda\in (0, 1) $ and finite time horizon $ T $ in the setting of [49]. By following [29], we define the fundamental value $ F $ of a risky asset $ S $ as the price of a super-replicating portfolio for a position terminating in one unit of the asset and zero cash. We then obtain a dual representation for the fundamental value by using the super-replication theorem of [50]. We say that an asset price has a bubble if its fundamental value differs from the ask-price $ (1+\lambda)S $. We investigate the impact of transaction costs on asset price bubbles and show that our model intrinsically includes the birth of a bubble.

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Mathematics Subject Classification: Primary: 91G99, 91B70; Secondary: 60G44.

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Figure 1. Example 4: Simulation of $ 253 $ days of $ S $, defined in 60 with $ \mu = 0.3 $, $ \sigma_0 = 0.4 $ and the starting time $ \gamma $ of the bubble being uniformly distributed on $ (0, 1) $. Before $ \gamma $ the fundamental value coincides with $ (1+\lambda)S $. When $ \gamma $ occurs, the fundamental value drops to $ 0 $

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