# Gamma smoothing of vanilla options

I want to ask a question about the answer provided here: https://quant.stackexchange.com/a/35211/61083. I'm wondering if there is mathematical proof as to why it is working. Meaning if I reprice a vanilla option of strike K, with a stripe of vanillas of strikes ranging from K1 to KN why the gamma would be capped when the option is ATM and close to expiry and not explode.

• The link you have provided opens up a question, not an answer. Aug 30, 2022 at 10:00
• @Alper, I'm talking about the second answer that's been provided as a response to the question. In fact, I want to know if there's a mathematical proof as to why the gamma of the strip of vanillas is capped. Aug 31, 2022 at 9:50
• You can provide direct link to an answer using the share buttton below an answer. “The second answer” is not a clear reference because answers can be sorted in more than one way in Stack Exhange sites. Hope you get a good answer. Aug 31, 2022 at 10:11
• @Alper, thank you I will do so. Aug 31, 2022 at 10:42

The gamma of an option as it approaches the expiry date becomes ill defined at $$S_T = K$$. However, if you approximate your option sitting at $$K$$ as a set of options with strikes ranging from $$K-\Delta_K$$ to $$K+\Delta_K$$, what you're doing is limiting the spike sitting at $$K$$ and replacing that whole gamma for a wider one sitting along all those strikes.
Here's just a toy example on how that looks when you replace the gamma of an option with strike $$K$$ to ten option of strikes from 95% to 105% and 1/10 notional. You can just plug in the BS formulas (for price and gamma) and reproduce it in python easily.