Indifference Prices and Related Measures

The traditional approach towards derivative pricing consists of dynamically replicating a future liability by trading the assets on which that liability is written. However, the assumption that one can trade the assets is often rather restrictive. In some cases, say of options on commodities or funds, one can at best trade another correlated asset. In others, as in the case of basket options, even when one can trade the basket components, for efficiency reasons one may still prefer to use a correlated index for pricing and hedging. Due to the departure from the traditional assumptions of valuation by replication and no arbitrage considerations, one needs to review the pricing and hedging methodologies to accommodate the above situations. A utility-based approach is developed herein for the specification of indifference price of claims written on non-traded assets. The pricing mechanism is based upon the parity between the maximal utilities, with and without employing the derivative. The residual amount gained from granting the option, which renders the investor impartial towards these two scenarios, is called the indifference price. Under exponential risk preferences such a price can be calculated by a nonlinear transformation of a solution to a linear parabolic equation. The transformation is independent of the risk preferences and only depends on the correlation between the traded and the non-traded risky assets. The equation is associated with a diffusion process whose dynamics are in turn identified by solving a relevant HamiltonJacobi-Bellman equation. The new diffusion turns out to be a drift-modified diffusion of the original one modelling the level of the non-traded asset. The drift modification corresponds to a new measure, referred to as the indifference measure, which depends on the correlation and the Sharpe ratio of the traded risky asset.