Measuring Volatility Skew Curvature with Finite Differences
Summary
The document examines whether the curvature of an implied-volatility skew can be summarized with a second derivative estimated from discrete option strikes. It presents a central finite-difference approximation around at-the-money and considers summing such estimates across strikes to represent the whole skew. The author asks how that aggregate measure changes when a fitted model supplies a denser, smoother curve.
The discussion identifies two complications: smoothing may change local curvature, and adding evaluation points changes the number of terms in the proposed sum. It also cites a practical trade-off: smaller strike spacing can amplify errors in measured implied volatility, while wider spacing averages behavior over a broader region. No model, empirical data, or definitive definition of a whole-skew curvature measure is supplied. Consequently, the proposed sum cannot be assumed to remain comparable as spacing changes; the document frames this as an open measurement question rather than offering a validated statistic.
Key ideas
- A central finite difference approximates local volatility-skew curvature from observations at neighboring strikes.
- Summing local second-derivative estimates is proposed as a single measure of whole-skew curvature.
- Changing strike spacing or fitting a smoother curve can change both the estimates and the number of summed terms.
- Smaller spacing can magnify implied-volatility measurement errors, while wider spacing averages over a broader interval.
- The document does not establish a scale-stable aggregate curvature measure or test the proposal empirically.
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Full text
# Summarizing the Volatility Skew as a Single Number
# Summarizing the Volatility Skew as a Single Number
Related questions to this topic/subject:
- Expressing Volatility Smile as One Number
- Volatility skew and how to capture it?
In both posts, the authors/respondents recommend using the second derivative to capture the curvature of the volatility skew:
$$\frac{f(x+h)-2f(x)+f(x-h)}{h^2} = f^{''}(K)\approx\frac{f(K_{ATM}+h)-2f(K_{ATM})+f(K_{ATM}-h)}{h^2}$$
I understand that this is merely an approximation as options are offered at discrete intervals, which does not meet the theoretical requirements of the second derivative that:
$$h \rightarrow 0$$
Furthermore, there could be instances where $f^{''}(K)$ could be convex (concave) with positive (negative) values of $f^{''}(K)$ depending on the options price.
My question is - if we choose to express the curvature of the entire volatility skew as:
$$Curvature = \sum_i{f^{''}(K_i)}$$
How would $Curvature$ be affected if we reduced $h$ by fitting a volatility skew model to the market-implied volatility skew (making the volatility skew "more continuous")?
Some of my thoughts would be:
- Smoothing of the volatility skew removes concave parts of the market-implied volatility skew, increasing the overall curvature measure.
- As we reduce $h$, the number of second derivative components $f^{''}(K_i)$ would increase as there are more points on the volatility skew. However, the individual contributions of $f^{''}(K_i)$ are ambiguous as we are unsure how they would turn out as the $f(K_{ATM}+h)-2f(K_{ATM})+f(K_{ATM}-h)$ decreases as well.
-- EDIT --
- One answer from a user in another post "Moreover, as h becomes small, small errors in the measurement of the sigma of the calls and puts are magnified. Choosing a large h reduces this noise in the calculations at the expense of approximating the average slope over a wider interval."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.