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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021118
References:
[1]

N. D. AlikakosP. 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.  Google Scholar

[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.  Google Scholar

[3]

N. D. AlikakosG. 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.  Google Scholar

[4]

N. D. AlikakosG. 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.  Google Scholar

[5]

A. AltaroviciJ. Muhle-Karbe and H. M. Soner, Asymptotics for fixed transaction costs, Finance Stoch., 19 (2015), 363-414.  doi: 10.1007/s00780-015-0261-3.  Google Scholar

[6]

D. C. AntonopoulouD. 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.  Google Scholar

[7]

D. C. AntonopoulouG. 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.  Google Scholar

[8]

British Pound v US Dollar Data, https://www.poundsterlinglive.com.,, Google Scholar

[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.  Google Scholar

[10]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, Journal of Differential Geometry, 44 (1996), 262-311.   Google Scholar

[11]

X. ChenX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. ContS. Stoikov and R. Talreja, A stochastic model for order book dynamics, Oper. Res., 58 (2010), 549-563.  doi: 10.1287/opre.1090.0780.  Google Scholar

[16]

E. Ekström, Selected Problems in Financial Mathematics, PhD Thesis, Uppsala Universitet, Sweden, 2004.  Google Scholar

[17]

L. C. EvansH. 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.  Google Scholar

[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.  Google Scholar

[19]

M. D. GouldM. A. PorterS. WilliamsM. McDonaldD. J. Fenn and S. D. Howison, Limit order books, Quant. Finance, 13 (2013), 1709-1742.  doi: 10.1080/14697688.2013.803148.  Google Scholar

[20]

V. Henderson, Prospect theory, liquidation, and the disposition effect, Management Science, 58 (2012), 445-460.   Google Scholar

[21]

T. Lybek and A. Sarr, Measuring Liquidity in Financial Markets, International Monetary Fund, work-in-progress, No. 02/232, 2002. Google Scholar

[22]

H. M. Markowitz, Portfolio selection: Efficient diversification of investments, John Wiley and Sons, Inc., New York, 1959.  Google Scholar

[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.  Google Scholar

[24]

M. Müller, Stochastic Stefan-type problem under first-order boundary conditions, Ann. Appl. Probab., 28 (2018), 2335-2369.  doi: 10.1214/17-AAP1359.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Ostwald, Blocking of Ostwald ripening allowing long-term stabilization, Z. Phys. Chem., 37 (1901), 385 pp. Google Scholar

[28]

C. Parlour and D. Seppi, Handbook of Financial Intermediation & Banking, North-Holland (imprint of Elsevier), Amsterdam, eds. A. Boot and A. Thakor, 2008. Google Scholar

[29]

Z. Zheng, Stochastic Stefan problems: Existence, uniqueness, and modeling of market limit orders, PhD Thesis, University of Illinois at Urbana-Champaign, 2012.  Google Scholar

[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/ Google Scholar

show all references

References:
[1]

N. D. AlikakosP. 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.  Google Scholar

[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.  Google Scholar

[3]

N. D. AlikakosG. 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.  Google Scholar

[4]

N. D. AlikakosG. 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.  Google Scholar

[5]

A. AltaroviciJ. Muhle-Karbe and H. M. Soner, Asymptotics for fixed transaction costs, Finance Stoch., 19 (2015), 363-414.  doi: 10.1007/s00780-015-0261-3.  Google Scholar

[6]

D. C. AntonopoulouD. 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.  Google Scholar

[7]

D. C. AntonopoulouG. 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.  Google Scholar

[8]

British Pound v US Dollar Data, https://www.poundsterlinglive.com.,, Google Scholar

[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.  Google Scholar

[10]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, Journal of Differential Geometry, 44 (1996), 262-311.   Google Scholar

[11]

X. ChenX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. ContS. Stoikov and R. Talreja, A stochastic model for order book dynamics, Oper. Res., 58 (2010), 549-563.  doi: 10.1287/opre.1090.0780.  Google Scholar

[16]

E. Ekström, Selected Problems in Financial Mathematics, PhD Thesis, Uppsala Universitet, Sweden, 2004.  Google Scholar

[17]

L. C. EvansH. 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.  Google Scholar

[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.  Google Scholar

[19]

M. D. GouldM. A. PorterS. WilliamsM. McDonaldD. J. Fenn and S. D. Howison, Limit order books, Quant. Finance, 13 (2013), 1709-1742.  doi: 10.1080/14697688.2013.803148.  Google Scholar

[20]

V. Henderson, Prospect theory, liquidation, and the disposition effect, Management Science, 58 (2012), 445-460.   Google Scholar

[21]

T. Lybek and A. Sarr, Measuring Liquidity in Financial Markets, International Monetary Fund, work-in-progress, No. 02/232, 2002. Google Scholar

[22]

H. M. Markowitz, Portfolio selection: Efficient diversification of investments, John Wiley and Sons, Inc., New York, 1959.  Google Scholar

[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.  Google Scholar

[24]

M. Müller, Stochastic Stefan-type problem under first-order boundary conditions, Ann. Appl. Probab., 28 (2018), 2335-2369.  doi: 10.1214/17-AAP1359.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Ostwald, Blocking of Ostwald ripening allowing long-term stabilization, Z. Phys. Chem., 37 (1901), 385 pp. Google Scholar

[28]

C. Parlour and D. Seppi, Handbook of Financial Intermediation & Banking, North-Holland (imprint of Elsevier), Amsterdam, eds. A. Boot and A. Thakor, 2008. Google Scholar

[29]

Z. Zheng, Stochastic Stefan problems: Existence, uniqueness, and modeling of market limit orders, PhD Thesis, University of Illinois at Urbana-Champaign, 2012.  Google Scholar

[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/ Google Scholar

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) $
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
Figure 3.  Radii dynamics of $ 2 $ balls at the solid phase
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
Figure 5.  100 realizations of $ R(t) $, for $ t\in[0,15] $, with first order approximation
Figure 6.  100 realizations of $ R(t) $, for $ t = 15 $ (first order approximation)
Figure 7.  100 realizations of $ R(t) $, for $ t\in[0,15] $, with second order approximation
Figure 8.  100 realizations of $ R(t) $, for $ t = 15 $ (second order approximation)
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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 $
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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