Why Delta Hedging Leaves Gamma-Related P&L

Summary

The document considers a short call hedged with stock. As the underlying price changes, the option’s delta changes because an option’s value is nonlinear in the stock price. A stock position sized to offset delta at the initial price therefore may no longer offset the option’s exposure after the move, leaving residual profit or loss.

The answer identifies gamma hedging as a way to address this effect: adding options with suitable gamma can offset the position’s gamma, helping stabilize delta as the underlying moves. The numerical example is meant to illustrate the residual from convexity, but it does not fully establish the exact mark-to-market result for a return trip to the starting price. The explanation also omits practical details such as hedge sizing, rebalancing frequency, volatility changes, and transaction costs, so it is a conceptual account rather than a complete hedging procedure.

Key ideas

• Delta hedging offsets an option’s local sensitivity to the underlying price.
• An option’s delta changes as its underlying price moves because the payoff value is nonlinear.
• A delta hedge can therefore leave residual P&L when the underlying moves.
• A position in other options can be used to offset gamma and reduce changes in delta.

Tags

# Delta heding & PnL


# Delta heding & PnL












Sorry if it's a duplicate but i didn't find an answer to my simple question in the other posts.

Let say we short a call option on a stock. $K = 100$, $C = 1$, $S = 100$ and $\Delta = 0.5$. No dividends or transaction fees. We buy 0.5 stock then our portfolio $\Pi = -1 + 50 = 49$. If the stock goes to 101, the call worths theoretically 1.5 so we have $\Pi = -1.5 + 50.5$. Also, $\Delta = 0.6$ now. But due to the convexity, C = 1.6 and $\Pi = -1.6 + 50.5 = 48.9$ our $PnL = -0.1$

If the stock goes by to 100 and the call to 1, is the only way to "erase" the PnL of -0.1 not to hedge by buying 0.6 stock at 101 ? Or even if we don't hedge, the PnL will be realized also at 100 ? If the latter, why ?

Thanks

## Answer by Valometrics.com (score 1)

https://quant.stackexchange.com/a/51060

In order to erase the remaining P&L, you should gamma hedge your options. Actually, this is due to the non linearity of the options price. This can be done by going long or short options with important gamma to make the gamma of your hedge portfolio equal to your options one. This way, the change of your delta will be hedged. If you want to know how many options you should purchase, please have a look on this document: http://banach.millersville.edu/~bob/math472/GammaHedging.pdf

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