= Finite-horizon exponential-utility portfolio
{title2=$\pi_t^*=\frac{\mu-\rho}{a\sigma^2}e^{-\rho(T-t)}$}
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 $X^*=w_0e^{\rho T}+\vartheta W_T^Q/a$. 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 $(1-\exp(-aw_0e^{\rho T}-\vartheta^2T/2))/a$. A nonnegative-wealth constraint changes this optimizer.
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