Utility-based pricing of stochastic income - Angus, IJ (2012)
In this thesis we investigate an optimal investment problem for an investor who also receives an unhedegable stochastic income stream. We extend the log-Brownian model of Henderson to a stochastic volatility model where the underlying market is incomplete and the income is a function of the stock volatility. With exponential preferences, explicit solutions to the primal and dual problems are possible using extensions of the distortion power solution of Zariphopoulou and Henderson. We derive expressions for the utility indifference price and the associated marginal utility-based price of the stochastic income stream. Even in the absence of stochastic income the market is incomplete and the optimal dual measure is the minimal entropy measure ℚE. We show how this horizon dependent measure serves as a pricing measure in the utility indifference valuation scheme. This is in contrast to Henderson [18] where the underlying market is a Black-Scholes model and the optimal dual measure, without the income, is the minimal martingale measure ℚM. The horizon dependence of the pricing kernel in the stochastic volatility model can potentially lead to inconsistencies in the pricing of claims with multiple payment dates. We carefully construct a forward utility function which restores the minimal martingale measure as the dual minimiser for the model without the claim, thus removing the horizon dependence.