Expiry Break-Even for a Static Delta-Hedged Option
Summary
The document asks how to estimate the expiry stock price at which an option position breaks even after an initial delta hedge that is left unchanged. The reply assumes no stock financing or repo costs and no subsequent trading of the option’s gamma. Under those assumptions, it gives call and put break-even expressions based on the option premium, strike, and initial hedge delta.
An at-the-money call example shows the intuition: the static short-stock hedge offsets part of the option’s exposure, leaving only the unhedged fraction to recover the premium at expiry. The discussion does not derive the formulas or account for changing delta, volatility, early exercise, transaction costs, or hedge financing. Although the question raises possible asymmetry from charm, the reply does not analyze it, so the expressions should be treated as a simplified expiry approximation.
Key ideas
- The proposed break-even is calculated for expiry with the initial stock hedge held constant.
- The reply assumes financing and repo effects are ignored and gamma is not actively traded.
- For a call, the unhedged portion of the position must recover the premium above the strike.
- For a put, the analogous simplified expression places break-even below the strike.
- The response does not address the question’s proposed charm-driven asymmetry.
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Full text
# Breakeven of a delta-hedged option
# Breakeven of a delta-hedged option
Basic question to which I surprisingly did not find an answer on here.
What's the best approximation to the break-even (with respect to stock price) for an option that was hedged fully at point of trade (and not adjusted later). (by fully I mean however many deltas the option had - not 100 deltas per option)
I have seen the approximation Premium/Delta and $\sigma * (\Delta t)^{0.5}$.
Are these model specific?
Am I incorrect to have thought that break-evens should be slightly asymmetric (the effect of charm will change the delta of the option leg, but the static stock hedge will not adjust)?
Any sources about delta hedged options are also appreciated.
## Answer by luckylwk (score 3)
https://quant.stackexchange.com/a/9149
Lets give it a rough go then.
Two assumptions. (1) We disregard repo (to lend the stock you may want to short) or financing on your hedged position. And (2) We assume no trading of the gamma on the option.
Then I would assume the break-even is equal to the expiry should be equal to... (CALL) paidPremium/(1-hedgedDelta) + callStrike (PUT) putStrike - paidPremium/(1-hedgedDelta)
Assume you pay 2.5 USD for a ATM CALL that is hedged ATM with delta 52%. You only make money if on expiry the spot it above your strike and you then only make money on (100-52) 48% delta that you have run. So you need to make 2.5 USD with 48% delta (is 5.21 USD per 'unhedged' delta). Which puts the breakeven-spot at 105.21?
The other way around you can find it by dividing the premium by the hedged delta and take the hedge-level minus that value (for a CALL). The your short hedge has paid for the premium.
(sorry, i am not familiar with including formula's)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.