Shadow Delta as the Volatility-Sensitivity Component of Option Delta

Summary

The document explains shadow delta as the part of an option's spot sensitivity that arises because implied volatility changes when the underlying price moves. Total delta can be separated into the ordinary delta calculated while holding volatility constant and a second contribution equal to vega multiplied by the sensitivity of implied volatility to spot.

The discussion is prompted by the negative shadow delta that can occur for forward-starting products when volatility skew links spot moves to changes in implied volatility. The answer provides the decomposition that identifies the mechanism, but does not derive when the effect must be negative or quantify its size. Its sign and magnitude depend on the volatility surface and the product's vega exposure.

Key ideas

• Option delta can include a component caused by changes in implied volatility as spot moves.
• The shadow delta is the product of option vega and the spot sensitivity of implied volatility.
• Volatility skew can create a link between spot and volatility that affects forward-starting option risk.
• The decomposition identifies the mechanism, while the sign and magnitude depend on the product and volatility surface.

Tags

# Skew and shadow delta


# Skew and shadow delta












> The presence of skew causes a correlation between volatility and spot. This correlation produces a negative shadow delta for all forward starting products (forward starting options have a theoretical delta of zero).

How does this produce a negative shadow delta? The exact mechanism is not clear to me.

## Answer by dm63 (score 17, accepted)

https://quant.stackexchange.com/a/32530

Basically, the author is saying that the delta of an option,

$dC/dS = \frac{\partial C}{\partial S} + \frac{\partial C}{\partial v}\frac{\partial v}{\partial S}$,

where the $\frac{\partial C}{\partial S}$ is the delta assuming constant volatility, the $\frac{\partial C}{\partial v}$ is the vega of the option, and the $\frac{\partial v}{\partial S}$ describes how the implied volatility of the option moves as the spot price moves. This second term is the "shadow delta" being referred to.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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