 Finite-horizon exponential-utility portfolio

For exponential utility $(1-e^{-ax})/a$ on unrestricted terminal portfolio wealth in the Black-Scholes model, the optimal dollar holding in the stock is the displayed deterministic amount. Set $\vartheta =(\mu -\rho )/\sigma$ and let $W^Q=W+\vartheta t$ be the Brownian motion under the risk-neutral measure. The terminal optimizer is $X^{\ast }=w_0{e}^{\rho T}+\vartheta W_T^Q/a$. Its marginal utility is proportional to the state-price density, and the concave supporting-tangent inequality proves global optimality under the state-price budget constraint. The expected utility at the optimum is $(1-\exp (-a{w}_0{e}^{\rho T}-{\vartheta }^{2}T/2))/a$. A nonnegative-wealth constraint changes this optimizer.