IKBFU's Vestnik

2021 Issue №1

Portfolio optimization in the case of an exponential utility function and in the presence of an illiquid asset

Pages
73-102

Abstract

We study an optimization problem for a portfolio with a risk-free, a li­quid, and an illiquid risky asset. The illiquid risky asset is sold in an exogenous random moment with a prescribed liquidation time distribution. We assume that the investor chooses an exponential utility function. Study of op­timization problems with three assets including an illiquid asset leads to three-dimensional nonlinear Hamilton — Jacobi — Bellman (HJB) equations.

It is well known that the exponential utility function is connected with the HARA utility function through a limiting procedure if the parameter of the HARA utility function is going to infinity. We show that the optimization problem with the exponential utility function is not connected to the optimization problem with the HARA utility by the limiting procedure and we obtain essentially different results. We provide the Lie group analysis of the corresponding HJB equation.

For the main three-dimensional PDE with the exponential utility function, we obtain the complete set of the nonequivalent Lie group invariant reductions to two-dimensional PDEs according to an optimal system of subalgebras of the admitted Lie algebra. We prove that in just one case the invariant reduction is consistent with the boundary condition. This reduction represents a significant simplification of the original problem.

Reference

1. Bordag L. A. Geometrical properties of differential equations. Applications of Lie group analysis in Financial Mathematics. Singapore, 2015.

2. Bordag L. A., Yamshchikov I. P. Lie group analysis of nonlinear Black-Scholes models // Ehrhardt M., Günther M., ter Maten E. J. W. (eds.). Novel Methods in Computational Finance. Springer, 2017. P. 109—128.

3. Bordag L. A., Yamshchikov I. P. Optimization problem for a portfolio with an illi­quid asset: Lie group analysis // Journal of Mathematical Analysis and Applica­tions. 2017. Vol. 453. P. 668—699.

4. Bordag L. A., Yamshchikov I. P., Zhelezov D. Portfolio optimization in the case of an asset with a given liquidation time distribution // International Journal of En­gineering and Mathematical Modelling. 2015. Vol. 2 (2). P. 31—50.

5. L Bordag L. A., Yamshchikov I. P., Zhelezov D. Optimal allocation-consumption problem for a portfolio with an illiquid asset // International Journal of Computer Mathematics. 2016. Vol. 93 (5). P. 749—760. doi: 10.1080/00207160.2013.877584.

6. Bordag L. A.Optimization problem for a portfolio with an illiquid asset in the case of an exponential utility function // Theory of Probability and its Applications. 2020. Vol. 65 (1). P. 155—157. doi: 10.4213/tvp5367.

7. Bouchard B., Pham H. Wealth-path dependent utility maximization in incom­plete markets // Finance and Stochastics. 2004. Vol. 8 (4). P. 579—603.

8. Duffie D., Fleming W., Soner H. M., Zariphopoulou T. Hedging in incomplete markets with HARA utility // Journal of Economic Dynamics and Control. 1997. Vol. 21. P. 753—782.

9. Diaz A., Esparcia C. Assessing risk aversion from the investor's point of view // Frontiers in Psychology. 2019. 02 July. doi: 10.3389/fpsyg.2019.01490.

10. Karoui N. El, Blanchet-Scalliet C., Jeanblanc M., Martinelli L. Optimal investment decisions when time-horizon is uncertain // Journal of Mathematical Economics. 2008. Vol. 44 (11). P. 1100—1113.

11. Ibragimov N. H. Lie group analysis of differential equations. CRS Press, 1994.

12. Jacod J., Shiryaev A. N. Limit theorems for stochastic processes. (Grundlehren der mathematischen Wissenschaften ; Vol. 288). Springer, 2002.

13. Meleshko S. V. Methods for Constructing Exact Solutions of Partial Differential Equations: Mathematical and Analytical Techniques with Applications to Enginee­ring. Springer, 2005.

14. Meleshko S. V., Moyo S. On the complete group classification of the reaction-dif­fusion equation with a delay // Journal of Mathematical Analysis and Applica­tions. 2008. Vol. 338 (1). P. 448—466.

15. Merton R. Lifetime portfolio selection under uncertainty: The continuous-time case // The Review of Economics and Statistics. 1969. Vol. 51 (3). P. 247—257.

16. Merton R. Optimum consumption and portfolio rules in a continuous time model // Journal of Economic Theory. 1971. Vol. 3. P. 373—413.

17. Monin P. Hedging market risk in optimal liquidation // SSRN Electronic Journal. 2014. Vol. 01.

18. Monin P., Zariphopoulou T. On the optimal wealth process in a log-normal market: Applications to risk management // International Journal of Financial Engineering. 2014. Vol. 01 (02). 1450013.

19. Olver P. J. Applications of Lie groups to differential equations. Springer Scien­ce & Business Media, 2000.

20. Ovsiannikov L. V. Group Analysis of Differential Equations. Academic Press, 1982.

21. Ovsyannikov L. V. The «podmodeli» program. Gas dynamics // Journal of Applied Mathematics and Mechanics. 1994. Vol. 58 (4). P. 601—627.

22. Patera J., Winternitz P. Subalgebras of real three- and four-dimensional Lie algebras // Journal of Mathematical Physics. 1977. Vol. 18 (7). P. 1449—1455.

23. Schied A., Schöneborn T. Risk aversion and the dynamics of optimal liquida­tion strategies in illiquid markets // Finance and Stochastics. 2009. Vol. 13 (2). P. 181—204.

24. Tartakovsky D. M., Dentz M. Diffusion in Porous Media: Phenomena and Mechanisms // Transp. Porous Med. 2019. Vol. 130 (1). P. 105-127. doi: 10.1007/s 11242-019-01262-6.

25. Vazquez J. L. The Porous Medium Equation: Mathematical Theory. Clarendon Press ; Oxford University Press, 2006.

26. Vazquez J. L. Smoothing and Decay Estimates for Nonlinear Diffusion Equa­tions // Equations of Porous Medium Type. (Oxford lecture series in mathematics and its applications ; Vol. 33). Oxford University Press, 2006.