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The multi-dimensional Stochastic Stefan Financial Model for a portfolio of assets
Antonopoulou, Dimitra Bitsaki, Marina Karali, Georgia D.
Antonopoulou, Dimitra
Bitsaki, Marina
Karali, Georgia D.
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2021-04-01
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Abstract
The financial model proposed in this work involves the liquidation process of a portfolio
of n assets through sell or (and) buy orders placed, in a logarithmic scale, at a (vectorial) price
with volatility. We present the rigorous mathematical formulation of this model in a
financial setting resulting to an n-dimensional outer parabolic Stefan problem with noise. The
moving boundary encloses the areas of zero trading, the so-called solid phase. We will focus on
a case of financial interest when one or more markets are considered. In particular, our aim is to
estimate for a short time period the areas of zero trading, and their diameter which approximates
the minimum of the n spreads of the portfolio assets for orders from the n limit order books of each
asset respectively.
In dimensions n = 3, and for zero volatility, this problem stands as a mean field model for
Ostwald ripening, and has been proposed and analyzed by Niethammer in [25], and in [7] in a more
general setting. There in, when the initial moving boundary consists of well separated spheres, a first
order approximation system of odes had been rigorously derived for the dynamics of the interfaces
and the asymptotic pro le of the solution. In our financial case, we propose a spherical moving
boundaries approach where the zero trading area consists of a union of spherical domains centered
at portfolios various prices, while each sphere may correspond to a different market; the relevant
radii represent the half of the minimum spread. We apply It^o calculus and provide second order
formal asymptotics for the stochastic version dynamics, written as a system of stochastic differential
equations for the radii evolution in time. A second order approximation seems to disconnect the
financial model from the large diffusion assumption for the trading density. Moreover, we solve the
approximating systems numerically.
Citation
Antonopoulou, D. C., Bitsaki, M., & Karali, G. (2022). The multi-dimensional stochastic Stefan financial model for a portfolio of assets. Discrete & Continuous Dynamical Systems - B, 27(4), 1955-1987. https://doi.org/10.3934/dcdsb.2021118
Publisher
American Institute of Mathematical Sciences
Journal
Discrete and Continuous Dynamical Systems B
Research Unit
DOI
10.3934/dcdsb.2021118
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Article
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This is an electronic version of an article published in [Antonopoulou, D. C., Bitsaki, M., & Karali, G. (2022). The multi-dimensional stochastic Stefan financial model for a portfolio of assets. Discrete & Continuous Dynamical Systems - B, 27(4), 1955-1987. https://doi.org/10.3934/dcdsb.2021118]
Series/Report no.
ISSN
1531-3492
EISSN
1553-524X