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Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 975–991 | Cite as

Variance-Optimal Martingale Measures for Diffusion Processes with Stochastic Coefficients

  • Daniel Hernández–Hernández
Article
  • 39 Downloads

Abstract

In this paper we present the solution of the optimal variance optimal martingale measure for stochastic volatility models, when the noises are correlated. It is proved that the value function of the dual problem is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. The method to develop our results is based on a Bernstein’s type of argument. The dual problem of the quadratic hedging problem is studied analyzing the expression obtained after a change of measure, which corresponds to some class of risk-sensitive control problems.

Keywords

Quadratic hedging Mean variance portfolio Variance-optimal martingale measure Risk sensitive control 

Mathematics Subject Classification (2010)

60G44 91B24 91B70 

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References

  1. 1.
    Bobrovnytska, O., Schweizer, M.: Mean-variance hedging and stochastic control: Beyond the Brownian setting. IEEE Trans. Autumn. Control 49, 1–14 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Delbaen, F., Schachermayer, W.: The variance-optimal martingale measure for continuous processes. Bernoulli 2, 81–105 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ddffie, D., Richardson, H.M.: Mean variance hedging in continuous time. Ann. Appl. Probab. 1, 1–15 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Follmer, H., Sondermann, D.: Hedging of non-redundant contingent claims. In: Mas-Collell, A., Hildenbrand, W. (eds.) Contributions to Mathematical Economics. North Holland, Amsterdam, The Netherlands, pp 205–233 (1986)Google Scholar
  5. 5.
    Gourieroux, C., Laurent, J.P., Pham, H.: Mean variance hedging and numeraire. Math. Finance 3, 179–200 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fleming, W.H., Hernández-Hernández, D.: The tradeoff between consumption and investment in incomplete financial markets. Appl. Math. Optim. 52, 219–235 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fleming, W.H., Sheu, S.J.: Risk sensitive and an optimal investment model II. Ann Appl. Probab. 12, 730–767 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fleming, W.H., Soner, H. M.: Controlled Markov Processes and Viscosity Solutions. Springer Verlag, New York (1993)zbMATHGoogle Scholar
  9. 9.
    Fouque, J-P, Papanicolaou, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press (2000)Google Scholar
  10. 10.
    Hernández-Hernández, D., Sheu, S.J.: Solution of the HJB equations involved in utility-based pricing. In: Mena, R., Pardo, J.C., Rivero, V., Uribe, G. (eds.) XI Symposium on Probability and Stochastic Processes, pp 177–198. Serie Progress in Probability, Birkhauser (2015)CrossRefGoogle Scholar
  11. 11.
    Kaise, H., Nagai, H.: Ergodic type Bellman equations of risk-sensitive control with large parameters and their singular limits. Asymptot. Anal. 20, 279–299 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kaise, H., Sheu, S.J: On the structure of solutions of ergodic type Bellman type equation related to risk sensitive control. Ann. Probab. 34, 284–320 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ladyzenskaya, O.A., Solonikov, V.A., Uralseva, N.N.: Linear and Quasilinear Equations of Parabolic Type, AMS Transl. of Math. Monographs, Providence, RI (1968)Google Scholar
  14. 14.
    Laurent, J.P., Pham, H.: Dynamic programming and mean-variance hedging. Finance and Stochastics 3, 82–110 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lim, A.E.B.: Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market. Math. Oper. Res. 29, 132–161 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lipster, R.S., Shiryayev A.N.: Statistics of Random Processes I: General Theory. Springer-Verlag, New York (1977)Google Scholar
  17. 17.
    Nagai, H.: Bellman equations of risk sensitive control. SIAM J. Control Optim. 34, 74–101 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nagai, H.: Down side risk minimisation via a large deviations approach. Ann. Appl. Probab. 22, 608–669 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pham, H., Rheinlander, T., Schweizer, M.: Mean-Variance Hedging for Continuous Processes: New Proofs and Examples. Finance Stochast. 2, 173–198 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Schweizer, M.: Approximation pricing and the variance-optimal martingale measure. Ann Probab. 64, 206–236 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuatoMéxico

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