Options on a traded account: symmetric treatment of the underlying assets

An option on a traded account is a type of a contract that insures an actively traded portfolio. A holder of the option is free to trade in two or more assets subject to constraints defined by the contract. He keeps resulting trading profits, but he is forgiven any trading loss. Previously studied contracts, such as passport options written on two assets, impose a trading limit on the first asset to be between $[-1,1] $[−1,1] (a short and a long position) and the residual wealth is invested in the second asset. For passport options, one of the positions in the underlying assets is typically short, making the contract expensive in relationship to the client's wealth as it insures a leveraged portfolio. Our paper presents a version of an insurance of a traded account that treats both underlying assets in a symmetric way. In our approach, we impose a natural symmetric limit in which the agent can fully invest in any underlying asset up to his current wealth and without shorting any asset. This makes the proposed contract relatively cheap and attractive in comparison to the client's initial wealth. In order to preserve the asset symmetry, we use a reference asset that treats both assets equally. We choose a static index that starts with equal weights in the underlying assets. We find an optimal strategy that maximizes the expected payoff of this contract for the two asset case. This strategy has the largest volatility with respect to the index. In order to prove the optimality, we need to generalize Hajek's comparison theorem to the situation of multivariate equations with univariate payoff with an imposed finite boundary condition. The optimal solution leads to a well known stop-loss strategy when all wealth is invested in one asset only, in this case it is the cheaper asset of the two. This trading strategy is interesting on its own as it creates a portfolio with the largest volatility with respect to the index.