Volatility-penalized terminal fee ($y_T=a{w}_T\exp (-\epsilon {\int }_0^T\Vert {\sigma }^T{\pi }_s{\Vert }^{2}ds/2)$)

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 $A=\sigma {\sigma }^T$, $b=\mu -r1$ and assume $A$ is invertible. The fee process has drift $r+b^T\pi -\epsilon {\pi }^{T}A\pi /2$ and volatility ${\sigma }^T\pi$. Its Hamilton-Jacobi-Bellman equation therefore maximizes $b^T\pi -(R+\epsilon ){\pi }^{T}A\pi /2$, giving ${\pi }^{\ast }=A^{-1}b/(R+\epsilon )$. The fee increases effective relative risk aversion from $R$ to $R+\epsilon$, independently of its positive scale $a$.

There is also a direct global bound. Put $p=1-R$, $v^{\ast }={\sigma }^T{\pi }^{\ast }$, and tilt probability by $\exp (p{v}^{\ast }\cdot W_T-p^2|v^{\ast }{|}^{2}T/2)$. Under this probability let $Z$ be the stochastic exponential of $(1+\epsilon )\int ({\sigma }^T\pi -v^{\ast })\cdot d{W}^{\ast }$. The terminal fee ratio satisfies $(y_T/y_T^{\ast }{)}^p=Z_T^{p/(1+\epsilon )}$. 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.