Gamma Scalping, Implied Volatility, and Discrete Hedging
Summary
The document explains how a long option’s gamma profit and loss relates to realized volatility versus the implied volatility used to price it. Under the stated continuous hedging formula, the gamma component is positive when realized variance exceeds implied variance and negative when it falls short. The answer accepts that implied volatility often exceeds realized volatility, so this component is typically negative.
It then explains why traders may still value positive gamma: real hedges happen at discrete intervals. As prices move, a long gamma position adjusts its delta hedge by selling after rises and buying after falls. The answer argues that this timing can improve execution relative to continuous hedging. It offers an intuition rather than a full derivation or quantitative evidence, and does not address transaction costs, changing implied volatility, or how discrete hedge performance varies with the rebalancing schedule.
Key ideas
- The continuous hedging formula ties long gamma profit and loss to realized variance minus implied variance.
- The gamma component is negative when realized variance is below the implied variance paid for the option.
- Discrete hedging can benefit a long gamma trader by allowing hedge adjustments after price moves.
- The explanation is intuitive and does not quantify the effects of transaction costs or hedge frequency.
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Full text
# Realized vol, implied vol, and gamma scalping
# Realized vol, implied vol, and gamma scalping
First of all, apologies for my lack of knowledge in derivatives trading. So I've been spending a lot of time, in the past few days, trying to understand gamma scalping... and got really confused.
Here is what I think I understand:
- I went thru the derivation of the following calculations $$ P\&L_{[0,T]} = \int_0^T \frac{1}{2} \Gamma(t,S_t,\sigma^2_{t,\text{impl.}})S_t^2( \sigma^2_{t,\text{real.}} - \sigma^2_{t,\text{impl.}})\,dt$$
Here is what I think I don't understand:
- From equation above, gamma scalping will only make sense if $\sigma^2_{t,\text{real.}} > \sigma^2_{t,\text{impl.}}$ which is a little counter intuitive to me because I thought implied vol is always higher than realized vol? Or, it should really be the expected realized vol in the future?
- I also read about this post (which is super helpful): Delta Hedging with fixed Implied Volatility to get rid of vega?. In this post, the answer explained the difference between whether m2m implied vol. So if I only want to look at pure gamma pnl, I assume I should ignore m2m implied vol, hence a 0 pnl from vega, correct? And I'll be always looking at the implied vol I paid (at purchase) and compare that to the actual realized vol for gamma pnl?
Thanks!
## Answer by Kurt G. (score 0, accepted)
https://quant.stackexchange.com/a/80566
In the end I decided to write a brief answer:
- Let's accept that typically implied vol (IV) is larger then realized vol (RV). This is another good answer making this plausible.
- Your PnL formula is correct. Here is a proof. The PnL should therefore typically be negative.
- This begs the question why there is the consensus that this trading strategy (hedging a position that has positive gamma) "makes money".
- The answer is that there is another principle at work: in practice we cannot hedge continuously but instead only at discrete times. When we are long gamma we have bought the option. Therefore, when the stock price rises we must sell $N(d_1)$ shares. If we do this discretely there is a time lag and we sell at a higher price than the guy that aims to hedge continuously. Therefore we are better off than that other guy. Similar arguments apply when the stock price falls. This explains intuitively what traders like when they hedge: positive gamma. They abhor nothing more than negative gamma.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.