= Binomial exponential-utility terminal wealth
{title2=$H^*=R^nw_0+(\mathbb E_Q\log Z-\log Z)/a$}
For unrestricted final <portfolio wealth> in an $n$-period <binomial market>, write $R$ for the bank growth factor, $a>0$ for absolute risk aversion, and $Z=dQ/dP$ for the <binomial-market probability density>. The displayed optimizer has the <state-price budget constraint> $\mathbb E_QH^*=R^nw_0$. Its <exponential utility> derivative satisfies $e^{-aH^*}=\lambda Z$ for a constant $\lambda>0$. The <concave supporting-tangent inequality> bounds any competing <expected utility> by this value, because the expected linear error is $\lambda\mathbb E_Q(H-H^*)=0$. Strict <concavity> proves uniqueness, and <market completeness> supplies replication. The formula is altered by a constraint requiring nonnegative final wealth. With physical up probability $p$ and <risk-neutral probability> $q$, $\mathbb E_Q\log Z=n[q\log(q/p)+(1-q)\log((1-q)/(1-p))]$.
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