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April  2022, 27(4): 1955-1987. doi: 10.3934/dcdsb.2021118

## 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

Received  April 2020 Revised  February 2021 Published  April 2022 Early access  April 2021

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.

Citation: Dimitra C. Antonopoulou, Marina Bitsaki, Georgia Karali. The multi-dimensional stochastic Stefan financial model for a portfolio of assets. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1955-1987. doi: 10.3934/dcdsb.2021118
##### 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/

show all references

##### 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/
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 $\Gamma(0) = \Gamma_1(0)\cup\Gamma_2(0)\cup\Gamma_3(0)$
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
Radii dynamics of $2$ balls at the solid phase
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
100 realizations of $R(t)$, for $t\in[0,15]$, with first order approximation
100 realizations of $R(t)$, for $t = 15$ (first order approximation)
100 realizations of $R(t)$, for $t\in[0,15]$, with second order approximation
100 realizations of $R(t)$, for $t = 15$ (second order approximation)
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 $\bar{spr}_1$ $12.25/5=2.45$ $lspra_1$ $\ln(158.5)-\ln(146.25)=0.080437$ $x_{c1}$ $\ln(152.375/5)=3.417$
 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 $\bar{spr}_1$ $12.25/5=2.45$ $lspra_1$ $\ln(158.5)-\ln(146.25)=0.080437$ $x_{c1}$ $\ln(152.375/5)=3.417$
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 $\bar{spr}_2$ $2.50/5=0.5$ $lspra_2$ $\ln(76.75)-\ln(74.25)=0.03312$ $x_{c2}$ $\ln(75.50/5)=2.715$
 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 $\bar{spr}_2$ $2.50/5=0.5$ $lspra_2$ $\ln(76.75)-\ln(74.25)=0.03312$ $x_{c2}$ $\ln(75.50/5)=2.715$
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 $\bar{spr}_3$ $15.50/5=3.1$ $lspra_3$ $\ln(110.5)-\ln(95)=0.15114$ $x_{c3}$ $\ln(102.75/5)=3.023$
 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 $\bar{spr}_3$ $15.50/5=3.1$ $lspra_3$ $\ln(110.5)-\ln(95)=0.15114$ $x_{c3}$ $\ln(102.75/5)=3.023$
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 $\alpha_{\rm in}=798.438$
 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 $\alpha_{\rm in}=798.438$
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 $\alpha=13337.414$
 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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