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

This issue Previous Article A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents Next Article Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

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

Abstract Full Text(HTML) Figure(8) / Table(5) Related Papers Cited by

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.

Keywords:

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

Citation:

Full Text(HTML)

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) $

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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 $

| Show Table

DownLoad: CSV

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 $

| Show Table

DownLoad: CSV

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 $

| Show Table

DownLoad: CSV

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 $

| Show Table

DownLoad: CSV

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 $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.
[2]	N. D. Alikakos and G. Fusco, Ostwald ripening for dilute systems under quasistationary dynamics, Comm. Math. Phys., 238 (2003), 429-479. doi: 10.1007/s00220-003-0833-5.
[3]	N. D. Alikakos, G. Fusco and G. Karali, The effect of the geometry of the particle distribution in Ostwald ripening, Comm. Math. Phys., 238 (2003), 481-488. doi: 10.1007/s00220-003-0834-4.
[4]	N. D. Alikakos, G. Fusco and G. Karali, Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics, J. Differential Equations, 205 (2004), 1-49. doi: 10.1016/j.jde.2004.05.008.
[5]	A. Altarovici, J. Muhle-Karbe and H. M. Soner, Asymptotics for fixed transaction costs, Finance Stoch., 19 (2015), 363-414. doi: 10.1007/s00780-015-0261-3.
[6]	D. C. Antonopoulou, D. Blömker and G. D. Karali, The sharp interface limit for the stochastic Cahn-Hilliard equation, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 280-298. doi: 10.1214/16-AIHP804.
[7]	D. C. Antonopoulou, G. D. Karali and A. N. K. Yip, On the parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force, J. Differential Equations, 252 (2012), 4679-4718. doi: 10.1016/j.jde.2012.01.016.
[8]	British Pound v US Dollar Data, https://www.poundsterlinglive.com.,,
[9]	X. Chen, The Hele-Shaw problem and area-preserving curve shortening motions, Arch. Rational Mech. Anal., 123 (1993), 117-151. doi: 10.1007/BF00695274.
[10]	X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, Journal of Differential Geometry, 44 (1996), 262-311.
[11]	X. Chen, X. Hong and F. Yi, Existence, uniqueness and regularity of classical solutions of Mullins-Sekerka problem, Comm. Partial Differential Equations, 21 (1996), 1705-1727. doi: 10.1080/03605309608821243.
[12]	X. Chen and M. Dai, Characterization of optimal strategy for multiasset investment and consumption with transaction costs, SIAM J. Financial Math., 4 (2013), 857-883. doi: 10.1137/120898991.
[13]	X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362. doi: 10.1016/0022-247X(92)90119-X.
[14]	R. Cont and A. de Larrard, Price dynamics in a Markovian limit order market, SIAM J. Financial. Math., 4 (2013), 1-25. doi: 10.1137/110856605.
[15]	R. Cont, S. Stoikov and R. Talreja, A stochastic model for order book dynamics, Oper. Res., 58 (2010), 549-563. doi: 10.1287/opre.1090.0780.
[16]	E. Ekström, Selected Problems in Financial Mathematics, PhD Thesis, Uppsala Universitet, Sweden, 2004.
[17]	L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.
[18]	T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin. (Engl. Ser.), 15 (1999), 407-438. doi: 10.1007/BF02650735.
[19]	M. D. Gould, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn and S. D. Howison, Limit order books, Quant. Finance, 13 (2013), 1709-1742. doi: 10.1080/14697688.2013.803148.
[20]	V. Henderson, Prospect theory, liquidation, and the disposition effect, Management Science, 58 (2012), 445-460.
[21]	T. Lybek and A. Sarr, Measuring Liquidity in Financial Markets, International Monetary Fund, work-in-progress, No. 02/232, 2002.
[22]	H. M. Markowitz, Portfolio selection: Efficient diversification of investments, John Wiley and Sons, Inc., New York, 1959.
[23]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.
[24]	M. Müller, Stochastic Stefan-type problem under first-order boundary conditions, Ann. Appl. Probab., 28 (2018), 2335-2369. doi: 10.1214/17-AAP1359.
[25]	B. Niethammer, Derivation of the LSW-theory for Ostwald ripening by homogenization methods, Arch. Rational Mech. Anal., 147 (1999), 119-178. doi: 10.1007/s002050050147.
[26]	B. Niethammer, The LSW model for Ostwald ripening with kinetic undercooling, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1337-1361. doi: 10.1017/S0308210500000718.
[27]	W. Ostwald, Blocking of Ostwald ripening allowing long-term stabilization, Z. Phys. Chem., 37 (1901), 385 pp.
[28]	C. Parlour and D. Seppi, Handbook of Financial Intermediation & Banking, North-Holland (imprint of Elsevier), Amsterdam, eds. A. Boot and A. Thakor, 2008.
[29]	Z. Zheng, Stochastic Stefan problems: Existence, uniqueness, and modeling of market limit orders, PhD Thesis, University of Illinois at Urbana-Champaign, 2012.
[30]	G. Zimmerman, 2 Portfolio Protection Strategies That Don't Work - and 2 That Do, Advisors Voices, 2016. https://www.nerdwallet.com/blog/investing/2-portfolio-protection-strategies-dont-work/