Indifference valuation in non-reduced incomplete models with a stochastic risk factor
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This work contributes to the methodology of valuation of financial derivative contracts in an incomplete market. It focuses on a special type of incompleteness caused by the presence of a non-traded stochastic risk factor, affecting the value of the contract. The non-traded risk factor may only appear in the payoff of the contract or, in addition, may enter the dynamics of the traded asset. We consider both cases. We suggest a discrete time discrete space binomial model for the traded stock and the non-traded risk factor. We work in the utility maximization framework with dynamically changing agent's preferences. We present a discrete time multi-period analog of the forward and backward utility processes recently developed in continuous time. We use methods of stochastic control and provide the indifference valuation algorithm with both the forward and backward dynamic utilities. We compare the two approaches and provide conditions under which they assign the same value to the contract. We show that unlike the backward dynamic utility, the forward dynamic utility yields prices that do not depend on the end of the investment horizon. We pay attention to the choice of the equivalent martingale measure used for valuation (i.e., the minimal martingale measure and the minimal entropy measure for the forward and the backward utility processes correspondingly). We explicitly characterize both measures and give conditions under which they coincide. We extend our algorithm to the case of American and partial exercise contracts. We illustrate our work with numerical examples, showing that in an incomplete market, a call option on a non-traded risk factor may optimally be exercised early, and that it may be optimal to exercise only a fraction of the total number of contracts held, if partial exercise is allowed. In continuous time we extend the existing results to the case of American contracts with both the backward and the forward utilities. We emphasize the similarities between our discrete time valuation algorithm and the continuous time valuation. The two approaches use the same pricing measures, yield prices through nonlinear functionals of similar form, exhibit a similar relationship between the backward and forward prices, and a similar structure for the aggregate minimal entropy. We believe that our work makes a contribution by exposing the two above mentioned ways of dependence on the non-traded risk factor, and by providing a new dynamic indifference pricing algorithm that allows consistent valuation across different investment horizons.