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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.

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Assume that the time $t$ forward for the maturity $T > t$ is given by

\begin{equation} F_t(T) = \left( S_t - D_t(T) \right) e^{r (T - t)}, \end{equation}

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

\begin{equation} \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)}. \end{equation}

I.e. the derivative of the option price w.r.t. to the spot is minus that of the option price w.r.t. changes in the present value of dividends.

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    $\begingroup$ It is intuitive if you consider that Delta is the sensitivity to the Stock price and that when the stock goes ex dividend the price will drop by (approximately) the amount of the dividend. $\endgroup$ – noob2 Mar 8 '17 at 13:09

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