How Continuous Delta Hedging Turns Gamma into Expiry P&L

Summary

The document asks whether continuously delta hedging a long option leaves the trader flat at expiry, and how gamma profits can arise if the option is held to expiration. It frames the question with idealized assumptions: no hedge costs or slippage, no theta, and no initial premium. The central issue is how gains and losses from repeated underlying hedges relate to the option’s changing value as the underlying moves.

The text presents a question rather than an answer, so it does not give a hedge rule, derivation, worked example, or evidence resolving the apparent contradiction. It is useful as a prompt to study gamma scalping and the discrete or continuous hedging P&L relationship, but its zero-premium and ignored-theta assumptions are abstractions. A real option has a price, and realized outcomes depend on the underlying path, implied versus realized volatility, hedge timing, financing, and trading costs.

Key ideas

• The document asks whether continuous delta hedging can produce gains from a long gamma position through expiry.
• It assumes zero hedge costs and slippage, ignores theta, and sets the option's entry premium to zero.
• It raises the distinction between repeatedly neutralizing delta and accumulating P&L from hedge trades.
• The text supplies no solution or empirical evidence, so the question remains unresolved within the document.

Tags

# If you continuously delta hedge a long option position will you be flat at expiry regardless of realized vol?


# If you continuously delta hedge a long option position will you be flat at expiry regardless of realized vol?












Learning about gamma and am confused about practical gamma trading strategies and struggling to understand how they can be monetized. Is all gamma just about setting limit orders above and below and hoping they hit? or if you could in theory hedge continuously could you still make money? Even if the option is held to expiry.

I understand that you can continuously delta hedge a long gamma position and gain on mark-to-market of the option - but in cases where you hold to expiry it doesn't make sense to me. It seems to me that continuously hedging delta (in theory) leaves you without a position constantly - and therefore no gains from delta position regardless of how volatile the underlying is. Therefore at expiry would realized PnL not be zero (even if you paid 0 premium to get in?)

Assuming 0 hedge cost/slippage here. Also, let's ignore theta and assume the premium paid to get into the option was 0.

If someone could point me to some literature about this topic I would also appreciate it!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.