Abstract
In this chapter we discuss the problem of financial illiquidity and give an overview of different modeling approaches to this problem. We focus on one of the approaches to an optimization problem of a portfolio with an illiquid asset sold in an exogenous random moment of time. We formulate the problem in a mathematically rigorous way and apply it to the problems of liquidity in such context for the first time to our knowledge. We provide a compact summary of achieved results. We have demonstrated the uniqueness and existence of the viscosity solution for this problem under certain conditions. The formulation of such problem gives rise to a number of three dimensional nonlinear partial differential equations (PDEs) of Black-Scholes type. Such equations rather challenging for further studies with analytical or numerical methods. One of the standard techniques to reduce the complexity of the problem is to find an inner symmetry of the equation with a help of Lie group analysis. We carried out a complete Lie group analysis of PDEs describing value function as well as investment and consumption strategies for a portfolio with an illiquid asset that is sold in an exogenous random moment of time with a prescribed liquidation time distribution. The admitted Lie algebra of the studied PDEs and the optimal system of subalgebras of this algebra provides a complete set of different invariant reductions of three dimensional PDEs to lower dimensional ones. We provide two examples of such reductions for the case of a logarithmic utility function.
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References
Bangia, A., Diebold, F.X., Schuermann, T., Stroughair, J.D.: Modeling liquidity risk, with implications for traditional market risk measurement and management. Technical Report, Stern School of Business (1998). http://www.ssc.upenn.edu/~fdiebold/papers/paper25/bds.pdf
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)
Bordag, L.A.: Geometrical properties of differential equations. Applications of Lie group analysis in Financial Mathematics. World Scientific, Singapore (2015)
Bordag, L.A., Yamshchikov, I.P.: Optimization problem for a portfolio with an illiquid asset: Lie group analysis. J. Math. Anal. Appl. 453(2), 668–699 (2017). http://dx.doi.org/10.1016/j.jmaa.2017.04.014
Bordag, L.A., Yamshchikov, I.P., Zhelezov, D.: Portfolio optimization in the case of an asset with a given liquidation time distribution. Int. J. Eng. Math. Modell. 2(2), 31–50 (2015)
Bordag, L.A., Yamshchikov, I.P., Zhelezov, D.: Optimal allocation-consumption problem for a portfolio with an illiquid asset. Int. J. Comput. Math. 93(5), 749–760 (2016)
Cauley, S.D., Pavlov, A.D., Schwartz, E.S.: Homeownership as a constraint on asset allocation. J. Real Estate Financ. Econ. 34(3), 283–311 (2007)
Coppejans, M., Domowitz, I., Madhavan, A.: Liquidity in an automated auction. Technical Report, Pennsylvania State University (2000). http://ssrn.com/abstract=295765
Dick-Nielsen, J., Feldhütter, P., Lando, D.: Corporate bond liquidity before and after the onset of the subprime crisis. J. Financ. Econ. 103(3), 471–492 (2012)
Duffie, D., Zariphopoulou, T.: Optimal investment with undiversifiable income risk. Math. Financ. 3, 135–148 (1993)
Duffie, D., Fleming, W., Soner, H.M., Zariphopoulou, T.: Hedging in incomplete markets with HARA utility. J. Econ. Dyn. Control. 21, 753–782 (1997)
Ekeland, I., Pirvu, T.A.: On a non-standard stochastic control problem. Technical Report, Department of Mathematics, The University of British Columbia (2008). http://arxiv.org/abs/0806.4026v1
Ekeland, I., Pirvu, T.A.: Investment and consumption without commitment. Math. Finan. Econ. 2(1), 57–86 (2008)
Goyenko, R., Sarkissian, S.: Treasury bond illiquidity and global equity returns. J. Financ. Quant. Anal. 49, 1227–1253 (2014)
Grossman, S.J., Laroque, G.: Asset pricing and optimal portfolio choice in the presence of illiquid durable consumption goods, National Bureau of Economic Research; w2369
Hooker, M.A., Kohn, M.: An empirical measure of asset liquidity. Technical Report, Dartmouth College Department of Economics (1994). http://ssrn.com/abstract=5544
Ibragimov, N.H.: Lie Group Analysis of Differential Equations. CRC Press, Boca Raton (1994)
Keynes, J.: A Treatise on Money. Macmillan, London (1930)
Lie, S.: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Stuttgart (1891)
Lippman, S.A., McCall, J.J.: An operational measure of liquidity. Am. Econ. Rev. 76(1), 43–55 (1986)
Lykina, V., Pickenhain, S., Wagner, M.: On a resource allocation model with infinite horizon. Appl. Math. Comput. 204(2), 595–601 (2008)
Merton, R.: Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373–413 (1971)
Miller, B.L.: Optimal consumption with a stochastic income stream. Econometrica 42, 253–266 (1974)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (2000)
Pickenhain, S.: Sufficiency conditions for weak local minima in multidimensional optimal control problems with mixed control-state restrictions. Z. Anal. Anwend. 11(4), 559–568 (1992)
Pickenhain, S.: Infinite horizon optimal control problems in the light of convex analysis in Hilbert spaces. Set Valued Var. Anal. 23(1), 169–189 (2015)
Schwartz, E.S., Tebaldi, C.: Illiquid assets and optimal portfolio choice. Technical Report, NBER Working Paper Series (2006). http://www.nber.org/papers/w12633
Scott, D.: Wall Street Words: An A to Z Guide to Investment Terms for Today’s Investor. Houghton Mifflin Corporation, Boston (2003)
Skiba, P.M., Tobacman, J.: Payday loans, uncertainty and discounting: explaining patterns of borrowing, repayment, and default. Vanderbilt Law and Economics Research Paper No. 08-33 (2008)
Spicer, A., McDermott, G.A., Kogut, B.: Entrepreneurship and privatization in central Europe: The tenuous balance between destruction and creation. Acad. Manage. Rev. 25(3), 630–649 (2000)
Yamshchikov, I.P., Zhelezov, D.: Liquidity and optimal consumption with random income. Project Report IDE1149, Technical Report, Department of Mathematics, Physics and Electrical Engineering, IDE, Halmstad University, Sweden (2011). http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-16108
Zariphopoulou, T.: Optimal investment and consumption models with non-linear stock dynamics. Math. Meth. Oper. Res. 50, 271–296 (1999)
Zeldes, S.P.: Optimal consumption with stochastic income: deviations from certainty equivalence. Q. J. Econ. 104, 275–298 (1989)
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Bordag, L.A., Yamshchikov, I.P. (2017). Lie Group Analysis of Nonlinear Black-Scholes Models. In: Ehrhardt, M., Günther, M., ter Maten, E. (eds) Novel Methods in Computational Finance. Mathematics in Industry(), vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-61282-9_7
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