Growth Optimal Portfolio Insurance in Continuous and Discrete Time

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.