Dividend-yield sensitivity of the perpetual put trigger ($\frac{d{S}^{\ast }}{d\delta }=\frac{X{\beta }_{-}}{\Delta ({\beta }_{-}-1{)}^2}<0$)

Here $\Delta =\sqrt{(r-\delta -{\sigma }^2/2{)}^2+2r{\sigma }^2}$. Implicit differentiation of the negative-root equation gives $d{\beta }_{-}/d\delta =-{\beta }_{-}/\Delta >0$. Differentiating $S^{\ast }=X{\beta }_{-}/({\beta }_{-}-1)$ gives the displayed negative derivative. A higher proportional dividend yield lowers risk-neutral stock growth and makes waiting for lower stock prices more attractive to the put holder.