Abstract This paper considers a worst-case investment optimization problem with delay for a fund manager who is in a crash-threatened financial market. Driven by existing of capital inflow/outflow related to history performance, we investigate the optimal investment strategies under the worst-case scenario and the stochastic control framework with delay. The financial market is assumed to be either in a normal state (crash-free) or in a crash state. In the normal state the prices of risky assets behave as geometric Brownian motion, and in the crash state the prices of risky assets suddenly drop by a certain relative amount, which induces to a dropping of the total wealth relative to that of crash-free state. We obtain the ordinary differential equations satisfied by the optimal investment strategies and the optimal value functions under the power and exponential utilities, respectively. Finally, a numerical simulation is provided to illustrate the sensitivity of the optimal strategies with respective to the model parameters.
Fund: Supported by the National Natural Science Foundation of China (71501050), Startup Foundation for Doctors of ZhaoQing University (611-612282) and the National Science Foundation of Guangdong Province of China (2017A030310660)
Cite this article:
Chunxiang A,Yi SHAO. Worst-Case Investment Strategy with Delay[J]. Journal of Systems Science and Information, 2018, 6(1): 35-57.
Markowitz H. Portfolio selection. Journal of Finance, 1952, 7:77-98.
Merton R C. Lifetime portfolio selection under uncertainty:The continuous time case. Review Economic Statistics, 1969, 51(3):247-257.
Merton R C. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 1971, 3(4):373-413.
Björk T, Murgoci A. A general theory of markovian time inconsistent stochastic control problems. Working Paper, Stockholm School of Economics, 2009.
Björk T, Murgoci A. A theory of markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics, 2014, 18(3):545-592.
Björk T, Murgoci A, Zhou X Y. Mean-variance portfolio optimization with state-dependent risk aversion. Mathematical Finance, 2014, 24:1-24.
Liu J, Longstaff F A, Pan J. Dynamic asset allocation with event risk. Journal of Finance, 2003, 58:231-259.
Cont R, Tankov P. Financial model with jump process. Chapman Hall/CRC, Printed in the United States of America, 2004.
Korn R, Wilmott P. Optimal portfolios under the threat of a crash. International Journal of Theory Applied Finance, 2002, 5:171-187.
Korn R, Menkens O. Worst-case scenario portfolio optimization:A new stochastic control approach. Mathematical Methods of Operations Research, 2005, 62:123-140.
Seifried T. Optimal investment for worst-case crash scenarios:A martingale approach. Mathematics of Operations Research, 2010, 35(3):559-579.
Korn R, Steffensen M. On worst-case portfolio optimization. SIAM Journal on Control and Optimization, 2007, 46(6):2013-2030.
Belak C, Menkens O, Sass J. Worst-case portfolio optimization with proportional transaction costs. Working Paper, 2013.
Desmettre S, Korn R, Ruckdeschel P, et al. Robust worst-case optimal investment. Working Paper, 2013.
Goetzmann, William N, Ingersoll J, et al. High-water marks and hedge fund management contracts. Journal of Finance, 2003, 58:1685-1717.
Panageas S, Westerfield M. High-water marks:High risk appetites? Convex compensation, long horizons, and portfolio choice. Journal of Finance, 2009, 64(1):1-36.
Guasoni P, Obłój J. The incentives of hedge fund fees and high-water marks. Mathematical Finance, 2013. In press. Available at http://onlinelibrary.wiley.com/doi/10.1111/mafi.12057/pdf.
Elsanosi I, Øksendal B, Sulem A. Some solvable stochastic control problems with delay. Stochastics and Stochastic Reports, 2000, 71(1):69-89.
Øksendal B, Sulem A. A maximum principle for optimal control of stochastic systems with delay, with applications to finance. Optimal Control and Partial Differential Equations-Innovations and Applications. Eds. by Menaldi J M, Rofman E, Sulem A. IOS Press, Amsterdam, The Netherlands, 2000.
Elsanousi L, Larssen B. Optimal consumption under partial observations for a stochastic system with delay. Preprint 9, University of Oslo, Oslo, Norway, 2001.
Chang M H, Pang T, Yang Y P. A stochastic portfolio optimization model with bounded memory. Mathematics of Operations Research, 2011, 36(4):604-619.
Federico S. A stochastic control problem with delay arising in a pension fund model. Finance and Stochastics, 2011, 15:421-459.
Shen Y, Zeng Y. Optimal investment-reinsurance with delay for mean-variance insurers:A maximum principle approach. Insurance:Mathematics and Economics, 2014, 57:1-12.
A C X, Li Z F. Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model. Insurance:Mathematics and Economics, 2015, 61:181-196.
Lee M K, Kim J H, Kim J. A delay financial model with stochastic volatility:Martingale method. Physica A, 2011, 390:2909-2919.
Mao X R, Sabanis S. Delay geometric Brownian motion in financial option valuation. Stochastics:An International Journal of Probability and Stochastic Processes, 2013, 85(2):295-320.
Shen Y, Meng Q, Shi P. Maximum principle for jump-diffusion mean-field stochastic delay differential equations and its application to finance. Automatica, 2014, 50:1565-1579.