Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

OPTIMAL PORTFOLIO SELECTION UNDER STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES

  • Received : 2015.09.30
  • Accepted : 2015.11.27
  • Published : 2015.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

Although, in general, the random fluctuation of interest rates gives a limited impact on portfolio optimization, their stochastic nature may exert a significant influence on the process of selecting the proportions of various assets to be held in a given portfolio when the stochastic volatility of risky assets is considered. The stochastic volatility covers a variety of known models to fit in with diverse economic environments. In this paper, an optimal strategy for portfolio selection as well as the smoothness properties of the relevant value function are studied with the dynamic programming method under a market model of both stochastic volatility and stochastic interest rates.

Keywords

References

  1. R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous case, The Review of Economics and Statistics, 51(3) (1969), 247-257. https://doi.org/10.2307/1926560
  2. R.C. Merton, Optimal consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3 (1971), 373-413. https://doi.org/10.1016/0022-0531(71)90038-X
  3. W.H Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, New York, Springer, 1993.
  4. R.C. Merton, Continuous-Time Finance, Blackwell, 2005.
  5. A. Lioui and P. Poncet, On optimal portfolio choice under stochastic interest rates, Journal of Economic Dynamics and Control, 25(11) (2001), 1841-1865. https://doi.org/10.1016/S0165-1889(00)00005-1
  6. R. Korn and H. Kraft, A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40(4) (2002), 1250-1269. https://doi.org/10.1137/S0363012900377791
  7. W.H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal on Control and Optimization, 43(2) (2004), 502-531. https://doi.org/10.1137/S0363012902419060
  8. J. Detemple and M. Rindisbacher, Closed form solutions for optimal portfolio election with stochastic interest rate and investment constraints, Mathematical finance, 15(4) (2005), 539-568. https://doi.org/10.1111/j.1467-9965.2005.00250.x
  9. J. Liu, Portfolio selection in stochastic environments, The Review of Financial Studies, 20(1) (2007), 1-39. https://doi.org/10.1093/rfs/hhl001
  10. J.Z. Li and R. Wu, Optimal investment problem with stochastic interest rate and stochastic volatility: maximizing a power utility, Applied Stochastic Models in Business and Industry, 25(3) (2009), 407-420. https://doi.org/10.1002/asmb.759
  11. E.-J. Noh and J.-H. Kim, An optimal portfolio model with stochastic volatility and stochastic interest rate, Journal of Mathematical Analysis and Applications, 375(2) (2011), 510-522. https://doi.org/10.1016/j.jmaa.2010.09.055
  12. H. Chang and K. Chang, Dynamic portfolio selection with stochastic interest rates for quadratic utility maximizing, Chinese Journal of Applied Probability and Statistics, 28(3) (2012), 301-310.
  13. Y. Shen and T.K. Siu, Asset allocation under stochastic interest rate with regime switching, Economic Modelling, 29(4) (2012), 1126-1136. https://doi.org/10.1016/j.econmod.2012.03.024
  14. T. Zariphopoulou, Optimal investment and consumption models with nonlinear stock dynamics, Mathematical Methods of Operations Research, 50 (1999), 271-296. https://doi.org/10.1007/s001860050098
  15. W.H. Fleming and D. Hernandez-Hernandez, An optimal consumption model with stochastic volatility, Finance and Stochastics, 7(2) (2003), 245-262. https://doi.org/10.1007/s007800200083
  16. G. Chacko and L.M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, The Review of Financial Studies, 18(4) (2005), 1369-1402. https://doi.org/10.1093/rfs/hhi035
  17. M. Fukasawa, Asymptotic analysis for stochastic volatility: martingale expansion, Finance and Stochastics, 15(4) (2011), 635-654. https://doi.org/10.1007/s00780-010-0136-6
  18. R.W. Lee, Implied and local volatilities under stochastic volatility, International Journal of Theoretical and Applied Finance, 4(1) (2001), 45-89.
  19. A.L. Lewis, Option Valuation under Stochastic Volatility with Mathematica Code, Finance Press, Newport Beach, 2000.
  20. O. Vasicek, An equilibrium characterisation of the term structure, Journal of Financial Economics, 5(2) (1977), 177-188. https://doi.org/10.1016/0304-405X(77)90016-2
  21. K. Ito, Stochastic Integral, Proc. Imperial Acad. Tokyo, 20 (1944), 519-524. https://doi.org/10.3792/pia/1195572786
  22. B. Oksendal, Stochastic Differential Equations, Springer, New York, 2003.
  23. R.E. Bellman, Dynamic programming and a new formalism in the calculus of variations, Proc. Nat. Acad. Sci, 40(4) (1954), 231-235. https://doi.org/10.1073/pnas.40.4.231
  24. J.L. Kelley, General topology, Springer-Verlag, 1991.
  25. T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, The Annals of Mathematics, 20 (1919), 292-296. https://doi.org/10.2307/1967124
  26. W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975.