How Spreading Option Strikes Smooths Gamma Near Expiry
Summary
The discussion examines why replacing a vanilla option at one strike with a strip of options across nearby strikes can reduce the sharp gamma concentration near expiry. It offers two related explanations: a strip distributes exposure across strikes, and as expiry approaches, gamma is most relevant at the strike closest to the underlying price.
One answer describes gamma as a limit on delta’s change, while another characterizes the single-strike gamma at the terminal strike as ill defined and explains the strip as replacing a narrow spike with a wider profile. A Black–Scholes toy example is suggested for illustrating the effect, but no numerical results or rigorous proof are included. The replies provide intuition rather than a complete derivation, and the final response only names a hedging technique and points elsewhere for supporting mathematics.
Key ideas
- A strip of options across strikes spreads gamma exposure over a wider range.
- As expiry approaches, the strike nearest the underlying price contributes the most relevant gamma.
- The replies explain smoothing intuitively but do not provide a rigorous proof.
- A Black–Scholes example is proposed as a way to illustrate the effect.
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Full text
# Gamma smoothing of vanilla options # 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. ## Answer by Cloudman88 (score 0) https://quant.stackexchange.com/a/72120 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. ## Answer by KT8 (score 0) https://quant.stackexchange.com/a/75723 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. ## Answer by Eric Huang (score -2) https://quant.stackexchange.com/a/80451 This is called strike overhedge technique. I just wrote a doc on this topic with rigorous math proofs. Hope this could help. Doc link: https://connecthkuhk-my.sharepoint.com/:b:/g/personal/huangdch_connect_hku_hk/Ef8-k1sQ0IhJvMyCCO-vG2ABg2I0lgyNxo1xtnunZJMh6A?e=hHcHRi
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