Asymptotic Approximation of Optimal Portfolio for Small Time Horizons
Pages 755 - 774
Abstract
We consider the problem of portfolio optimization in a simple incomplete market and under a general utility function. By working with the associated Hamilton--Jacobi--Bellman partial differential equation (HJB PDE), we obtain a closed-form formula for a trading strategy which approximates the optimal trading strategy when the time horizon is small. This strategy is generated by a first order approximation to the value function. The approximate value function is obtained by constructing classical sub- and super-solutions to the HJB PDE using a formal expansion in powers of horizon time. Martingale inequalities are used to sandwich the true value function between the constructed sub- and super-solutions. A rigorous proof of the accuracy of the approximation formulas is given. We end with a heuristic scheme for extending our small-time approximating formulas to approximating formulas in a finite time horizon.
References
[1]
G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, Rev. Financ. Stud., 18 (2005), pp. 1369--1402.
[2]
M. G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1--67.
[3]
J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2 (1992), pp. 767--818.
[4]
J.-P. Fouque, R. Sircar, and T. Zariphopoulou, Portfolio optimization and stochastic volatility asymptotics, Math. Finance, 27 (2015), pp. 704--745.
[5]
I. Karatzas, J. P. Lehoczky, S. E. Shreve, and G.-L. Xu, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim., 29 (1991), pp. 702--730.
[6]
D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), pp. 904--950.
[7]
M. Lorig and R. Sircar, Portfolio optimization under local-stochastic volatility: Coefficient Taylor series approximations and implied Sharpe ratio, SIAM J. Financial Math., 7 (2016), pp. 418--447.
[8]
R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Rev. Econ. Stat., 51 (1969), pp. 247--257.
[9]
R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), pp. 373--413.
[10]
S. Nadtochiy and T. Zariphopoulou, An approximation scheme for solution to the optimal investment problem in incomplete markets, SIAM J. Financial Math., 4 (2013), pp. 494--538.
[11]
T. Pang, Portfolio optimization models on infinite-time horizon, J. Optim. Theory Appl., 122 (2004), pp. 573--597.
[12]
T. Pang, Stochastic portfolio optimization with log utility, Int. J. Theor. Appl. Finance, 9 (2006), pp. 869--887.
[13]
W. Schachermayer, Optimal investment in incomplete markets when wealth may become negative, Ann. Appl. Probab., 11 (2001), pp. 694--734.
[14]
M. Tehranchi, Explicit solutions of some utility maximization problems in incomplete markets, Stochastic Process. Appl., 114 (2004), pp. 109--125.
[15]
T. Zariphopoulou, A solution approach to valuation with unhedgeable risks, Finance Stoch., 5 (2001), pp. 61--82.
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© 2018, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2018
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