Rational Hedging and Valuation with Utility-Based Preferences

Abstract

In this thesis, we study stochastic optimization problems in which concave functionals are maximized on spaces of stochastic integrals. Such problems arise in mathematical finance for a risk-averse investor who is faced with valuation, hedging, and optimal investment problems in incomplete financial markets. We are mainly concerned with utility-based methods for the valuation and hedging of non-replicable contingent claims which confront the issuer with some inevitable intrinsic risks. We adopt the perspective of a rational investor who aims to maximize his expected exponential utility. Based on these preferences, the issuer’s valuation process and hedging strategy are defined via utility indifference arguments. In a general semimartingale model, the solution to this problem is characterized by a stochastic representation problem. Solving the problem amounts to finding a martingale measure whose density process can be written in a particular form. We then specialize our analysis to two stochastic models which satisfy further structural assumptions. In a semi-complete product model, the valuation and hedging methods are shown to be additive when applied to an aggregate of “sufficiently independent” individual claims. We study the impact of diversification and derive a computation scheme. For a second model, we set up a Markovian system of stochastic differential equations which describes the dynamics of an Ito process and an additional finite-state process, and permits for various dependencies between both. In the financial market model, the Ito process models the price fluctuations of the risky assets while the second process represents some untradable factors of risk. The solution to the pricing and hedging problem is explicitly described by an interacting system of semi-linear partial differential equations – a so-called reaction-diffusion equation. Using Feynman-Kac results and the Picard-iteration method, we establish existence and uniqueness of a classical solution. In a variation of the basic theme, a similar utility indifference approach is applied to quantify the value of additional investment information. On the mathematical side, this involves a martingale preserving measure transformation and martingale representation results for initially enlarged filtrations. Finally, we show that the so-called numeraire portfolio is related to another utility-based valuation method which relies on a marginal rate of substitution argument and can be seen as a limiting case of the utility indifference method.