Indifference Pricing Under Epstein-Zin Utility: Maximum Principle

Abstract

The problem of dealing with claims on asset which is not traded has attracted a lot of interests recently. Naturally the approach consist of choosing a related traded asset or index to use for hedging purposes. In this thesis, we consider a model for which the non-traded asset is driven by a Ornstein- Uhlenbeck process. We introduce it into a consumption-investment problem with factor model under recursive utility of Epstein-Zin type. Due to the second Brownian motion, we are working in an incomplete market in which the objective of an agent is pricing and hedging this random payoff. Making use of the maximum principle method, we solve our forward-backward system and find the optimal consumption and investment strategies and a relation given the indifference price. Since a closed form formula for the indifference price is not obtained, a finite difference method is applied to estimate its value. For numerical purpose, we consider a one period model. We perform some numerical analysis on the optimal investment in presence of a claim and on the indifference price. In general, we observe that, for the parameters specification considered, the optimal investment becomes an increasing function with regard to initial wealth of the agent so as to be higher than its value in the no claim case. However, it is rather a decreasing function with respect to the correlation between non-traded and traded assets and is always net off the investment with zero claim. Regarding the indifference price, we observe that it increases when the traded asset becomes more and more correlated to the non-traded one. Then, analysing also the dependency of the indifference price to the risk aversion, we obtain that an agent is willing to pay less for the non-traded asset when he/she becomes less tolerant of risk. Finally, we notice from the indifference price versus the initial wealth that an agent is less willing to take on more risk.