While the option implied distribution obtained by the maximum entropy method is quite efficient, it has a drawback that the distribution may not exist in exotic options. Therefore we propose a modified option implied distribution and apply to binary options, European options and multi-asset options. Also we suggest a method that find the Lagrange multipliers sequentially without finding the Jacobian matrix.
Furthermore the maximum entropy method which is one of non-parametric methods is used to estimate the probability of default. From binary option prices as the constraints, we explicitly find it while Capuano(2008) suggests numerical computation from European call option prices.