Abstract
This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs, and optimal investment strategies are obtained, their rate of convergence is also determined. Stability of utility‐based pricing is studied as an application. Second, a sequence of utilities defined on converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in [Nutz, M. (2012): Risk aversion asymptotics for power utility maximization. Probab. Theory & Relat. Fields 152, 703–749], which establishes the convergence for a sequence of power utilities.
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