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.
Gamma for vanilla options are always capped, the delta of an option can only move from 0 to 1, so these numbers are capped especially with the underlying having a minimum tick size in reality. If you’re asking why the Gamma is smaller for spread-out strikes then it is that as time approaches expiry for an option, only one strike can really be in play at any one time so only the Gamma from that strike near the current underlying price will be most relevant.
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.