# Summarizing the Volatility Skew as a Single Number

Related questions to this topic/subject:

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."