Why an Option’s Dividend Sensitivity Matches Its Delta

Summary

The document explains the link between an option’s sensitivity to spot price and its sensitivity to the present value of dividends. It assumes the forward price equals spot less the present value of dividends, grown at the risk-free rate to maturity. For a European claim, the chain rule relates changes in its value to changes in the forward price, spot, and dividends.

Under this setup, the option’s spot delta is the negative of its sensitivity to the present value of dividends: a rise in expected dividends lowers the forward price in the same way that a fall in spot does. The explanation is an algebraic relationship rather than empirical evidence. It depends on the stated forward-price representation and applies to European claims within that setup; the document does not establish that the relationship holds unchanged under different dividend, pricing, or product conventions. It raises whether the observation is specific to quanto options but does not develop quanto pricing further.

Key ideas

• The forward price is modeled using spot less the present value of dividends, compounded at the risk-free rate.
• The chain rule connects a claim’s sensitivity to spot with its sensitivity to the forward price.
• Within the stated setup, the spot delta equals the negative sensitivity to dividend value.
• The derivation concerns European claims and does not separately analyze quanto-specific effects.

Tags

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


# 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.

## Answer by LocalVolatility (score 2, accepted)

https://quant.stackexchange.com/a/32886

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.