On the utility maximization of the discrepancy between a perceived and market implied risk neutral distribution

A method is developed to determine the portfolio that maximizes the expected utility of an agent that trades the difference between a perceived future price distribution of an asset and the associated market implied risk neutral density. Exact results to construct and price such a portfolio are presented under the assumption that the underlying asset price evolves according to a geometric Brownian motion. Integer programming optimization techniques are applied to the general case where one first calibrates the asset price risk neutral density directly from option market data using Gatheral's SVI parameterization. Several numerical examples approximating the optimal payoff function with liquid securities are given.