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Portfolio Optimization for Assets with Stochastic Yields and Stochastic Volatility

  • Tao PangEmail author
  • Katherine Varga
Article
  • 51 Downloads

Abstract

In this paper, we consider a stochastic portfolio optimization model for investment on a risky asset with stochastic yields and stochastic volatility. The problem is formulated as a stochastic control problem, and the goal is to choose the optimal investment and consumption controls to maximize the investor’s expected total discounted utility. The Hamilton–Jacobi–Bellman equation is derived by virtue of the dynamic programming principle, which is a second-order nonlinear equation. Using the subsolution–supersolution method, we establish the existence result of the classical solution of the equation. Finally, we verify that the solution is equal to the value function and derive and verify the optimal investment and consumption controls.

Keywords

Portfolio optimization Stochastic volatility Stochastic yield HJB equation Subsolution Supersolution 

Mathematics Subject Classification

93E20 91B70 49L20 60H30 97M30 62P05 

Notes

References

  1. 1.
    Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51(3), 247–257 (1969)Google Scholar
  2. 2.
    Fouque, J.-P., Papanicolaou, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  3. 3.
    Lorig, M., Sircar, R.: Portfolio optimization under local-stochastic volatility: coefficient Taylor series approximations and implied sharpe ratio. SIAM. J. Financ. Math. 7, 418–447 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: Calibration of a multi-scale stochastic volatility model using European option prices. Math. Methods Econ. Finance 3(1), 49–61 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: An explicitly solvable multi-scale stochastic volatility model: option pricing and calibration problems. J. Futures Mark 29(9), 862–893 (2009)Google Scholar
  6. 6.
    Zariphopoulou, T.: A solution approach to valuation with un-hedgeable risks. Finance Stoch. 5(1), 61–82 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Fleming, W.H., Hernández-Hernández, D.: An optimal consumption model with stochastic volatility. Finance Stoch. 7, 245–262 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fouque, J.-P., Han, C.-H.: Pricing Asian options with stochastic volatility. Quant. Finance 3(5), 352–362 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fouque, J.-P., Sircar, R., Zariphopoulou, T.: Portfolio optimization and stochastic volatility asymptotics. Math. Finance 27(3), 704–745 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fleming, W.H., Pang, T.: An application of stochastic control theory to financial economics. SIAM J. Control Optim. 43(2), 502–531 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Pang, T.: Portfolio optimization models on infinite time horizon. J. Optim. Theory Appl. 122(3), 573–597 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Pang, T.: Stochastic portfolio optimization with log utility. Int. J. Theor. Appl. Finance 9(6), 869–887 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Goel, M., Kumar, K.S.: Risk-sensitive portfolio optimization problems with fixed income securities. J. Optim. Theory Appl. 142(1), 67–84 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fleming, W.H., Hernández-Hernández, D.: The tradeoff between consumption and investment in incomplete financial markets. Appl. Math. Optim. 52(2), 219–235 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Nagai, H.: H–J–B equations of optimal consumption-investment and verification theorems. Appl. Math. Optim. 71(2), 279–311 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Noh, E.J., Kim, J.H.: An optimal portfolio model with stochastic volatility and stochastic interest rate. J. Math. Anal. Appl. 375(2), 510–522 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hata, H., Sheu, S.J.: On the Hamilton–Jacobi–Bellman equation for an optimal consumption problem: I. Existence of solution. SIAM J. Control Optim 50(4), 2373–2400 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hata, H., Sheu, S.J.: On the Hamilton–Jacobi–Bellman equation for an optimal consumption problem: II. Verification theorem. SIAM J. Control Optim. 50(4), 2401–2430 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kaise, H., Sheu, S.J.: On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control. Ann. Probab. 34(1), 284–320 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Tunaru, R.: Dividend derivatives. Quant. Finance 18(1), 1–19 (2018)MathSciNetGoogle Scholar
  21. 21.
    Geske, R.: The pricing of options with stochastic dividend yield. J. Finance 33(2), 617–625 (1978)Google Scholar
  22. 22.
    Lioui, A.: Black–Scholes–Merton revisited under stochastic dividend yields. J. Futures Mark. 26(7), 703732 (2006)Google Scholar
  23. 23.
    Pang, T., Varga, K.: Optimal investment and consumption for a portfolio with stochastic dividends. J. Res. Finance Manag. 1(2), 1–22 (2015)Google Scholar
  24. 24.
    Chevalier, E., Vath, V.L., Scotti, S.: An optimal dividend and investment control problem under debt constraints. SIAM J. Financ. Math. 4, 297–326 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fleming, W.H., Pang, T.: A stochastic control model of investment, production and consumption. Q. Appl. Math. 63, 71–87 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, Berlin (1993)zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.CoBankGreenwood VillageUSA

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