Growth Optimal Portfolio Insurance in Continuous and Discrete Time

  • Sergey Sosnovskiy
Conference paper
Part of the Operations Research Proceedings book series (ORP)


We propose simple solution for the problem of insurance of the Growth Optimal Portfolio (GOP). Our approach comprise two layers, where essentially we combine OBPI and CPPI for portfolio protection. Firstly, using martingale convex duality methods, we obtain terminal payoffs of non-protected and insured portfolios. We represent terminal value of the insured GOP as a payoff of a call option written on non-protected portfolio. Using the relationship between options pricing and portfolio protection we apply Black-Scholes formula to derive simple closed form solution of optimal investment strategy. Secondly, we show that classic CPPI method provides an upper bound of investment fraction in discrete time and in a sense is an extreme investment rule. However, combined with the developed continuous time protection method it allows to handle gap risk and dynamic risk constraints (VaR, CVaR). Moreover, it maintains strategy performance against volatility misspecification. Numerical simulations show robustness of the proposed algorithm and that the developed method allows to capture market recovery, whereas CPPI method often fails to achieve that.


Option Price Trading Strategy Risky Asset Hedging Strategy Portfolio Insurance 
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.


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  1. 1.
    S. Basak and A. Shapiro. Value-at-risk-based risk management: optimal policies and asset prices. Review of Financial Studies, 14(2):371, 2001. ISSN 0893-9454.Google Scholar
  2. 2.
    D. Cuoco, H. He, and S. Isaenko. Optimal Dynamic Trading Strategies with Risk Limits. Operations Research, 56(2):358–368, 2008. ISSN 0030-364X.Google Scholar
  3. 3.
    J. Cvitanic and I. Karatzas. On portfolio optimization under” drawdown” constraints. IMA volumes in mathematics and its applications, 65:35–35, 1995. ISSN 0940-6573.Google Scholar
  4. 4.
    S.J. Grossman and J.L. Vila. Portfolio insurance in complete markets: A note. The Journal of Business, 62(4):473–476, 1989. ISSN 0021-9398.Google Scholar
  5. 5.
    S.J. Grossman and Z. Zhou. Optimal investment strategies for controlling drawdowns. Mathematical Finance, 3(3):241–276, 1993. ISSN 1467-9965.Google Scholar
  6. 6.
    S.J. Grossman and Z. Zhou. Equilibrium analysis of portfolio insurance. Journal of Finance, 51(4):1379–1403, 1996. ISSN 0022-1082.Google Scholar
  7. 7.
    N.H. Hakansson and W.T. Ziemba. Capital growth theory. Handbooks in Operations Research and Management Science, 9:65–86, 1995. ISSN 0927-0507.Google Scholar
  8. 8.
    J.L. Kelly. A new interpretation of information rate. Information Theory, IRE Transactions on, 2(3):185–189, 1956. ISSN 0096-1000.Google Scholar
  9. 9.
    T.A. Pirvu and G. ˇZitkovi´c. Maximizing the growth rate under risk constraints. Mathematical Finance, 19(3):423–455, 2009. ISSN 1467-9965.Google Scholar
  10. 10.
    S. Sosnovskiy, U. Walther. Insurance of the Growth Optimal Portfolio. In Frankfurt School - Working Papers Series.Google Scholar
  11. 11.
    E.O. Thorp. Portfolio choice and the Kelly criterion. Stochastic models in finance, pages 599–619, 1971.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Frankfurt School of Finance and ManagementFrankfurt am MainGermany

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