The multi-dimensional stochastic Stefan financial model for a portfolio of assets

Dimitra C. Antonopoulou; Marina  Bitsaki; Georgia Karali

doi:10.3934/dcdsb.2021118

Article Contents

2022, Volume 27, Issue 4: 1955-1987. Doi: 10.3934/dcdsb.2021118

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The multi-dimensional stochastic Stefan financial model for a portfolio of assets

1.
Department of Mathematics and Applied Mathematics, University of Crete, GR–714 09 Heraklion, Greece
2.
Institute of Applied and Computational Mathematics, FORTH, GR–711 10 Heraklion, Greece
3.
Computer Science Department, University of Crete, Voutes University Campus, HERAKLION, Crete, GR-70013, Greece
4.
Department of Mathematical and Physical Sciences, University of Chester, Thornton Science Park, CH2 4NU, UK
5.
Institute of Applied and Computational Mathematics, FORTH, GR–711 10 Heraklion, Greece

* Corresponding author: Georgia Karali
* Corresponding author: Georgia Karali

Received: April 2020

Revised: February 2021

Early access: April 2021

Published: April 2022

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Abstract

The financial model proposed involves the liquidation process of a portfolio through sell / buy orders placed at a price $ x\in\mathbb{R}^n $, with volatility. Its rigorous mathematical formulation results to an $ n $-dimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading. We will focus on a case of financial interest when one or more markets are considered. We estimate the areas of zero trading with diameter approximating the minimum of the $ n $ spreads for orders from the limit order books. In dimensions $ n = 3 $, 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]. We propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices with radii representing the half of the minimum spread. We apply Itô calculus and provide second order formal asymptotics for the stochastic dynamics of the spreads that seem to disconnect the financial model from a large diffusion assumption on the liquidity coefficient of the Laplacian that would correspond to an increased trading density. Moreover, we solve the approximating systems numerically.

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Mathematics Subject Classification: Primary: 91G80, 91B70, 60H30, 60H15; Secondary: 65C30.

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Figure 1. Solid phase $ \mathcal{D}(0) $ of $ I = 3 $ initial circular domains (discs) in $ \mathbb{R}^2 $, where $ \mathbb{R}^2-\mathcal{D}(0) $ consists the initial liquid phase, and $ \Gamma(0) = \Gamma_1(0)\cup\Gamma_2(0)\cup\Gamma_3(0) $

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Figure 2. Radii dynamics of $ 4 $ balls at the solid phase at the left, and radii dynamics of $ 100 $ balls at the solid phase at the right

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Figure 3. Radii dynamics of $ 2 $ balls at the solid phase

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Figure 4. Radius dynamics of one ball at the solid phase with relatively large spread at the left, and radius dynamics of one ball at the solid phase with relatively small spread at the right

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Figure 5. 100 realizations of $ R(t) $, for $ t\in[0,15] $, with first order approximation

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Figure 6. 100 realizations of $ R(t) $, for $ t = 15 $ (first order approximation)

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Figure 7. 100 realizations of $ R(t) $, for $ t\in[0,15] $, with second order approximation

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Figure 8. 100 realizations of $ R(t) $, for $ t = 15 $ (second order approximation)

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Table 1. A sample of 5 quotes for asset 1

Time $ t_j $	$ A_1(t_j) $	$ B_1(t_j) $	$ spr_1(t_j) $	$ \frac{A_1(t_j)+B_1(t_j)}{2} $
9:00	30.25	29.75	0.5	30
9:02	30.75	29.50	1.25	30.125
9:04	31.00	29.25	1.75	30.125
9:06	31.50	29.00	2.50	30.25
9:08	35.00	28.75	6.25	31.875
Sum	158.5	146.25	12.25	152.375
$ \bar{spr}_1 $		$ 12.25/5=2.45 $
$ lspra_1 $		$ \ln(158.5)-\ln(146.25)=0.080437 $
$ x_{c1} $		$ \ln(152.375/5)=3.417 $

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Table 2. A sample of 5 quotes for asset 2

Time $ t_j $	$ A_2(t_j) $	$ B_2(t_j) $	$ spr_2(t_j) $	$ \frac{A_2(t_j)+B_2(t_j)}{2} $
9:00	15.00	14.25	0.75	14.625
9:02	15.25	14.25	1.00	14.75
9:04	15.25	15.00	0.25	15.125
9:06	15.50	15.25	0.25	15.375
9:08	15.75	15.50	0.25	15.625
Sum	76.75	74.25	2.50	75.50
$ \bar{spr}_2 $		$ 2.50/5=0.5 $
$ lspra_2 $		$ \ln(76.75)-\ln(74.25)=0.03312 $
$ x_{c2} $		$ \ln(75.50/5)=2.715 $

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Table 3. A sample of 5 quotes for asset 3

Time $ t_j $	$ A_3(t_j) $	$ B_3(t_j) $	$ spr_3(t_j) $	$ \frac{A_3(t_j)+B_3(t_j)}{2} $
9:00	20.75	19.50	1.25	20.125
9:02	21.00	19.50	1.50	20.25
9:04	21.25	19.25	2.00	20.25
9:06	22.00	18.25	3.75	20.125
9:08	25.50	18.50	7.00	22.00
Sum	110.5	95	15.50	102.75
$ \bar{spr}_3 $		$ 15.50/5=3.1 $
$ lspra_3 $		$ \ln(110.5)-\ln(95)=0.15114 $
$ x_{c3} $		$ \ln(102.75/5)=3.023 $

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Table 4. Number of shares sold, and liquidity coefficient

Asset	$ w_i $	$ a_i=w_i/\bar{spr}_i $	$ w_i/w_{\rm tot} $	$ a_i w_i/w_{\rm tot} $
1	550	550/2.45=224.49	550/1600=0.34375	77.168
2	750	750/0.5=1500	750/1600=0.46875	703.125
3	300	300/3.1=96.774	300/1600=0.1875	18.145
Sum	1600			$ \alpha_{\rm in}=798.438 $

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Table 5. Number of shares sold, and liquidity coefficient in logarithmic scale

Asset	$ w_i $	$ w_i/lspr_i $	$ w_i/w_{\rm tot} $	$ \frac{w_i}{lspra_i}\frac{w_i}{w_{\rm tot}} $
1	550	550/0.080437=6837.64	550/1600=0.34375	2350.438
2	750	750/0.03312=22644.92	750/1600=0.46875	10614.806
3	300	300/0.15114=1984.91	300/1600=0.1875	372.170
Sum	1600			$ \alpha=13337.414 $

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