Why Deep Out-of-the-Money Options Can Have Near-Zero Gamma

Summary

The document asks whether a very far out-of-the-money call with little time remaining could have enormous gamma because a large stock-price jump would change its delta from near zero to near one. The reply uses Black–Scholes assumptions to explain why that hypothetical jump is not representative of the model’s local price sensitivity.

For the stated short time to expiry and volatility assumption, the proposed move is extraordinarily many standard deviations away. The option’s probability-weighted sensitivity at its current underlying price is therefore effectively zero, rather than the large gamma suggested by imagining a sudden, extreme jump. This is a narrow, model-based explanation: it does not discuss jump-risk models, changing volatility, or how gamma behaves across strikes in other market conditions.

Key ideas

• Gamma measures the local change in option delta as the underlying price changes.
• A hypothetical extreme price jump does not by itself imply high gamma at the current price.
• Under Black–Scholes assumptions, a far out-of-the-money option with little time remaining can have negligible gamma.
• The explanation relies on a diffusion model and does not address jump-driven price moves.

Tags

# In this scenario could gamma be higher for OTM options?


# In this scenario could gamma be higher for OTM options?












Let's say there is a $1 stock, with say 1 day to expiration. The 1.5 strike call, is probably a 0 delta at this point; however, a 1 point increase would mean the stock would be at trading at 2 dollars; thus the 1.5 strike call would now almost be guaranteed to finish in the money right, and would seemingly now have close to a 100 delta. It seems then, that it's gamma was 100 in this scenario, making it higher than an ATM option. Where am I wrong?

## Answer by mbison (score 1)

https://quant.stackexchange.com/a/20680

under the black scholes assumptions it is nearly impossible to move 50% in 1 day. A 1day move will be of size $\sigma \sqrt{1/252}$ (assuming 252 trading days). So you are talking about something that is about a 40 stdev move (assuming $\sigma$ = 0.2).

that's why your gamma is 0.

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