Estimate Skew-Adjusted Delta with the Local Volatility Slope

Summary

This note explains how to choose the volatility skew slope in a skew-adjusted delta calculation. The formula described adds the product of vega and the volatility skew slope to the Black–Scholes delta. For an option at a given strike, the slope should represent the local change in implied volatility around that strike, rather than a broad average across the whole volatility curve.

The example uses SPY options and recommends estimating the slope from options with strikes nearest to the one being evaluated. If nearby options are illiquid, more distant strikes can be used as a practical fallback, though that makes the estimate less local. The answer appeals to a paper’s mathematical treatment but gives no details on fitting methods, quote selection, smoothing, or empirical tests. A polynomial fit may describe the curve, but the relevant quantity is its slope at the target strike; the document does not specify how to manage noisy or sparse market data.

Key ideas

• Skew-adjusted delta combines Black–Scholes delta with vega multiplied by the volatility skew slope.
• Estimate the slope locally around the option’s strike.
• Nearby strike options are preferred when their quotes are sufficiently liquid.
• More distant strikes can be used when nearby observations are illiquid, with reduced locality.

Tags

# How should I estimate the implied volatility skew term when calculating the skew-adjusted delta?


# How should I estimate the implied volatility skew term when calculating the skew-adjusted delta?












I'm trying to come up with the implied volatility skew adjusted delta for SPY options. I'm working with the following formula:

Skew Adjusted Delta = Black Scholes Delta + Vega * Vol Skew Slope.

I fitted a cubic polynomial to the volatility data, but I am not sure at which points to measure the slope.

## Answer by Tal Fishman (score 6, accepted)

https://quant.stackexchange.com/a/1564

Judging by the math in a paper by Vahamaa (1999), you should measure the slope using the options closest to the strike you are examining. In other words, suppose that you are trying to come up with the skew-adjusted delta of an SPY option with strike of 130. Then the skew slope should be based on the 129 and 131 strike options. If these points happen to be very illiquid, then you can, in principle, estimate the slope based on options further away from the strike of interest, but the skew adjustment is a local adjustment, and so the relevant slope is the slope at that strike.

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.