Sensitivity analysis for expected utility maximization in incomplete Brownian market models

TL;DR

This paper investigates how small changes in model parameters affect utility maximization in incomplete Brownian markets, introducing a weak perturbation method that decouples the wealth equation from parameter variations and deriving explicit sensitivity formulas.

Contribution

It introduces a weak perturbation approach for sensitivity analysis in incomplete markets and proves Hadamard directional differentiability of the value function with explicit derivatives.

Findings

Weak perturbations yield different sensitivities than strong perturbations.

Explicit formulas for directional derivatives of the value function are derived.

Sensitivity results hold under stability conditions on volatility matrices.

Abstract

We examine the issue of sensitivity with respect to model parameters for the problem of utility maximization from final wealth in an incomplete Samuelson model and mainly, but not exclusively, for utility functions of positive power-type. The method consists in moving the parameters through change of measure, which we call a weak perturbation, decoupling the usual wealth equation from the varying parameters. By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the work of Backhoff and Fontbona (SIFIN 2016), the previous formulation let us prove the Hadamard directional differentiability of the value function w.r.t. the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives. We contrast our proposed weak perturbations against what we call strong perturbations, where the wealth equation is directly influenced by the changing parameters. Contrary to conventional wisdom, we find that both points of view generally yield different sensitivities unless e.g. if initial parameters and their perturbations are deterministic.