The traditional approach to derivatives pricing consists of dynamic replication of a future liability through trading of the assets on which that liability depends. In such at framework the price is essentially equal to what it costs to manufacture the option payoff. However, the assumption that one can trade the assets is frequently rather restrictive. In some situations one can at best trade another correlated asset. In other, like in the case of basket options even when one can trade the basket components, for the efficiency reasons one may still prefer to use a correlated index for pricing and hedging. Because of the departure from the traditional assumptions of valuation by replication and no arbitrage considerations one needs to review the pricing and hedging methodologies to deal with the above situations. In this paper, in order to determine the price of a claim written on a non-traded asset, we compare the maximal utilities with and without employing the derivative. The price is then determined as the initial investment that would make the investor indifferent between these two strategies. It turns out that such a price can be calculated by a nonlinear transformation, referred to in the future as a distortion, of a solution to a linear parabolic equation associated with a diffusion which in turn is identified by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The diffusion is a drift modified diffusion modelling the dynamics of the non-traded asset. The distortion depends only on the correlation between the traded and non-traded asset and, in particular, does not depend on the risk preferences incorporated in the utility function. The price depends on both; the risk preferences and the distortion.
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