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Hasek
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Posts: 9
Joined: October 2nd, 2021, 9:53 am

Statistical metric to measure how well does the volatility surface fit the market

December 9th, 2021, 2:21 pm

Suppose that I have a model for implied volatility surface and want to figure out required recalibration frequency based on historical quotes. Since I have a large range of strikes and tenors over a long period of time I need to somehow automate this process, i.e. I need a computable metric rather than "ahh it seems pretty close to market".

What kind of statistical metric can I use for that purpose? I'm thinking about the mean of percentage differences between market and model quotes, i.e. the mean value of

$$100\cdot\frac{\sigma^{market}-\sigma^{model}}{\sigma^{market}}$$

over the entire volatility surface, however the mean over the entire surface can be quite misleading as it will not capture large single outliers on a big enough surface and will cancel out differences with similar magnitude but opposite signs. Nevertheless I can't see a better single metric to assess an overall surface fit.

How much sense does an average percentage difference over the entire surface make to assess the quality of a fit? Is there a better metric?
 
Hasek
Topic Author
Posts: 9
Joined: October 2nd, 2021, 9:53 am

Re: Statistical metric to measure how well does the volatility surface fit the market

December 9th, 2021, 3:16 pm

UPD: Does it make more sense to pick a squared sum of differences across all tenors and strikes as a metric?

$$\rho = \sum\limits_{K,T}(\sigma^{market}-\sigma^{model})^2$$
 
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Alan
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Re: Statistical metric to measure how well does the volatility surface fit the market

December 9th, 2021, 5:16 pm

There are lots of possibilities. One I like is eqn (18) here