Optimal positioning in derivative securities in incomplete markets

Figure 1.  For the model considered in Example 8, we plot $ f^*(\log s) $ (solid line) and $ h(\log s,1/ \lambda) $ (dashed line) as a function of $ s $. Model parameters in the plot are as follows: $ \lambda = 1/(0.1)^2 $, $ \nu = 1/(0.15)^2 $, $ \mu = 0,1 $ and $ T = 1.0 $. For the case of exponential utility, we set $ \gamma = 2 $ and the cost constraint equal to the price of the option sold $ c = \widetilde{\mathbb{E}} h(X_T,Y_T) $. In the case of mean-variance maximization, we set $ \gamma = 0 $, and the cost constraint is given by $ c = \int \mathrm{d} x \, \widetilde{p}_X(x) \int \mathrm{d} y \, \frac{ p_{X,Y}(x,y) }{ p_X(x) } h(x,y) + \gamma - \Big| \int \mathrm{d} x \int \mathrm{d} y \, \widetilde{p}_{X,Y}(x,y) h(x,y) \Big| , $ which guarantees that $ f^* $ is given by (22)

Figure 2.  The yellow dot is the vector $ h $. The blue plane is the space spanned by $ X_T $ and $ Y_T $, which are represented by the blue and yellow dots respectively. The purple dots represent the sequence of portfolios obtained by iterating the operator $ \mathcal{H} $. When the algorithm stops, the coordinates of the last purple dot are in the proximity of the origin $ (0,0,0) $, showing that a perfect hedge is achieved

Figure 3.  Figure (a) and Figure (b) show, respectively, the optimal $ f^* $ and $ g^* $ for Problem 2 with exponential utility and different values of $ \gamma $, and assuming that $ X_T $ and $ Y_T $ are jointly normally distributed with $ \mu $ and $ \Sigma $ as in equation (50) and $ \rho = -0.1 $. Figure (c) and Figure (d) depict the optimal $ f^* $ and $ g^* $ for different values of the correlation $ \rho $ between $ X_T $ and $ Y_T $, assuming that they are jointly normally distributed with $ \mu $ and $ \Sigma $ as in equation (50) and that utility is exponential with $ \gamma = 0.01 $