Estimation of asset distributions from option prices: Analysis and regularization

We analyze the stability of a method for estimating the risk-neutral density (RND) for the price of an asset from option prices. The method first applies the principle of maximum entropy, where the maximum entropy solution (MES) corresponds to the estimated RND. Next, it provides an effective characterization of the constraint qualification (CQ) under which the MES can be computed by solving the dual problem, where an explicit function in finitely many variables is minimized. In our analysis, we show that the MES is stable under parameter perturbation, but the parameters are unstable under data perturbation. When noisy data are used, we show how to project the data so that the CQ is satisfied and the method can be used. To stabilize the method, we use Tikhonov regularization and choose the penalty parameter via the L-curve method. We demonstrate with numerical examples that the method then becomes much more stable to perturbation in data. Accordingly, we perform a convergence analysis of the regularized solution.