Computer Construction of Quasi Optimal Portfolio for Stochastic Models with Jumps of Financial Markets

  • Aleksander Janicki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3994)


In the paper we propose a purely computational new method of construction of a quasi–optimal portfolio for stochastic models of a financial market with jumps. Here we present the method in the framework of a Black–Scholes–Merton model of an incomplete market (see, eg. [5], [7]), considering a well known optimal investment and consumption problem with the HARA type optimization functional. Our method is based on the idea to maximize this functional, taking into account only some subsets of possible portfolio and consumption processes. We show how to reduce the main problem to the construction of a portfolio maximizing a deterministic function of a few real valued parameters but under purely stochastic constraints. It is enough to solve several times an indicated system of stochastic differential equations (SDEs) with properly chosen parametrs. This is a generalization of an approach presented in [4] in connection with a well known classical Black–Scholes model.

Results of computer experiments presented here were obtained with the use of the SDE–Solver software package. This is our own professional C++ application to Windows system, designed as a scientific computing tool based on Monte Carlo simulations and serving for numerical and statistical construction of solutions to a wide class of systems of SDEs, including a broad class of diffusions with jumps driven by non-Gaussian random measures (consult [1], [4], [6], [9]).

The approach to construction of approximate optimal portfolio presented here should be useful in a stock market analysis, eg. for evolution based computer methods.


Optimal Portfolio Optimal Investment Incomplete Market Consumption Process Wealth Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Janicki, A., Izydorczyk, A.: Computer Methods in Stochastic Modeling (in Polish). Wydawnictwa Naukowo-Techniczne, Warszawa (2001)Google Scholar
  2. 2.
    Janicki, A., Izydorczyk, A., Gradalski, P.: Computer Simulation of Stochastic Models with SDE–Solver Software Package. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J., Zomaya, A.Y. (eds.) ICCS 2003. LNCS, vol. 2657, pp. 361–370. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Janicki, A., Krajna, L.: Malliavin Calculus in Construction of Hedging Portfolios for the Heston Model of a Financial Market. In: Demonstratio Mathematica XXXIV, pp. 483–495 (2001)Google Scholar
  4. 4.
    Janicki, A., Zwierz, K.: Construction of Quasi Optimal Portfolio for Stochastic Models of Financial Market. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2004. LNCS, vol. 3039, pp. 811–818. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, Berlin (1998)zbMATHGoogle Scholar
  6. 6.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, 3rd edn. Springer, New York (1998)Google Scholar
  7. 7.
    León, J.A., Solé, J.L., Utzet, F., Vives, J.: On Lévy processes, Malliavin calculus, and market models with jumps. Finance and Stochastics 6, 197–225 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ocone, D.L., Karatzas, I.: A generalized Clark representation formula, with application to optimal portfolios. Stochastics and Stochastics Reports 34, 187–220 (1991)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Protter, P.: Stochastic Integration and Differential Equations – A New Approach. Springer, New York (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Aleksander Janicki
    • 1
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland

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