= Volatility-penalized terminal fee
{title2=$y_T=aw_T\exp(-\varepsilon\int_0^T\lVert\sigma^T\pi_s\rVert^2ds/2)$}
A fee proportional to terminal <portfolio wealth> but reduced by accumulated portfolio <quadratic variation> changes the optimal risk exposure of a manager with <constant relative risk aversion utility>. Write $A=\sigma\sigma^T$, $b=\mu-r\mathbf1$ and assume $A$ is invertible. The fee process has drift $r+b^T\pi-\varepsilon\pi^TA\pi/2$ and volatility $\sigma^T\pi$. Its <Hamilton-Jacobi-Bellman equation> therefore maximizes $b^T\pi-(R+\varepsilon)\pi^TA\pi/2$, giving $\pi^*=A^{-1}b/(R+\varepsilon)$. The fee increases effective relative risk aversion from $R$ to $R+\varepsilon$, independently of its positive scale $a$.
There is also a direct global bound. Put $p=1-R$, $v^*=\sigma^T\pi^*$, and tilt probability by $\exp(pv^*\cdot W_T-p^2|v^*|^2T/2)$. Under this probability let $Z$ be the <stochastic exponential> of $(1+\varepsilon)\int(\sigma^T\pi-v^*)\cdot dW^*$. The terminal fee ratio satisfies $(y_T/y_T^*)^p=Z_T^{p/(1+\varepsilon)}$. A nonnegative <local martingale> has expectation at most one. The <Jensen inequality>, applied to a concave positive power when $p>0$ and a convex negative power when $p<0$, proves that the constant exposure maximizes expected <CRRA utility>. Effective risk aversion one uses logarithmic utility.
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