Complete-market terminal utility optimizer ($X^{\ast }=I(y\zeta )$)

For an increasing strictly concave utility function satisfying the Inada conditions, put $I=(U^{\prime }{)}^{-1}$ and let $\zeta$ be the positive terminal state-price density. In a finite complete market, select $y>0$ so that $\mathbb{E}[\zeta I(y\zeta )]$ equals initial wealth. The displayed payoff is optimal under that budget: the supporting-line inequality for utility has marginal slope $y\zeta$, and its expected linear error vanishes on the budget constraint. Market completeness supplies its replicating strategy.

In a complete $n$-period binomial market with riskless gross return $R$, pricing density $Z$ and utility function $U(x)=\gamma x^{1/\gamma }$, put $\eta =\gamma /(\gamma -1)$. The optimal terminal payoff is $X^{\ast }=R^n{w}_0{Z}^{-\eta }/\mathbb{E}[Z^{1-\eta }]$. Its marginal utility is proportional to $Z$; the supporting-line inequality for the strictly concave utility function proves optimality using the pricing budget constraint.