# Why is the dividend risk of an option equal to its delta?

In this document, https://www.eurexgroup.com/blob/2435406/f1b0086a8c6d05954c58a8dc24308c81/data/20160304_Colin-Bennent-Trading-Volatility-.pdf, it states that

"This is because the dividend risk of an option is equal to its delta, and the dividend used in quanto pricing increases as correlation increases. "

Why is the dividend risk of an option equal to its delta? I am also not sure whether this is just for quanto options.

Assume that the time $t$ forward for the maturity $T > t$ is given by
$$F_t(T) = \left( S_t - D_t(T) \right) e^{r (T - t)},$$
where $D_t(T)$ is the time $t$ value of all dividends paid over $(t, T]$. Consider a European contingent claim with time $t$ value $V_t$. Then
$$\frac{\partial V_t}{\partial S_t} = \frac{\partial V_t}{\partial F_t(T)} \frac{\partial F_t(T)}{\partial S_t} = -\frac{\partial V_t}{\partial F_t(T)} \frac{\partial F_t(T)}{\partial D_t(T)} = -\frac{\partial V_t}{\partial D_t(T)}.$$