How Skew and Spot Moves Affect Delta-Hedged Index Option P&L
Summary
The document considers the volatility exposure of a long, delta-hedged at-the-money SPX call after the index falls. It explains that the option’s implied volatility can change because its fixed strike moves along the implied volatility skew, while at-the-money volatility may also rise or fall with the market. These effects can offset or reinforce one another, so the direction of the option’s volatility-related profit and loss cannot be inferred from the spot move alone.
The discussion introduces skew or shadow delta: when volatility varies by strike, an option’s effective exposure can differ from the Black–Scholes delta used for a hedge. Residual results also depend on changes in the whole volatility surface, spot-volatility correlation, and whether the market behaves more like sticky strike or sticky delta. A second answer suggests that the relative effect of skew and at-the-money volatility can vary by volatility regime, while noting exceptions such as very high volatility and melt-up conditions. The material is qualitative and does not provide a calibrated model or a numerical P&L example.
Key ideas
- A falling index can lower the implied volatility of a fixed-strike call as it moves along the skew.
- At-the-money implied volatility may respond separately to the market decline, with the size and direction varying by regime.
- Delta hedging under a skewed volatility surface can leave residual exposure beyond standard Black–Scholes delta.
- Option P&L depends on realized volatility, volatility-surface changes, and the hedging path.
- Spot-volatility correlation and sticky-strike or sticky-delta behavior affect the resulting exposure.
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Full text
# You are long a hedged ATM SPX Call and the market moves down. Do you gain or lose in volatility terms? # You are long a hedged ATM SPX Call and the market moves down. Do you gain or lose in volatility terms? The shape of the volatility curve in index options trading typically shows that the 'just' OTM Calls (ITM Puts) options have the lowest implied volatility. If you are long an ATM Call and the market moves down I see two results; - That Call should now be approaching the bottom of the curve and its implied volatility will decrease - Volatility as a whole should increase due to the move to the downside So my question is, which effect will dominate? I feel I may be overstating the effect of result 2 above and that we would need a pretty violent move for volatility to be materially bid. ## Answer by user46424 (score 1) https://quant.stackexchange.com/a/69709 So firstly, only ITM puts have a low implied volatility relative to ITM calls. ITM puts and OTM calls must have the same IV due to Put-Call Parity. Negative IV skew exists in indices such as the S&P500, NASDAQ, etc where OTM puts and therefore ITM calls (Put-Call Parity) have a greatly higher IV relative to ATM IV and to the IV of OTM calls and ITM puts. Although each product you look at will have a different shape of its respective implied volatility curve, so that must be taken into consideration when determinting residual PnL. If you are long an ATM delta-hedged SPX call, your PnL will actually not just be dependent on whether the underlying realizes more volatility than the implied volatility you bought it at. In other words, your profits AND losses will not just be made up from the profits and losses from continuously delta-hedging (gamma profits outpacing losses from theta). This is due to the existence of "skew delta" or "shadow delta". Under Black-Scholes, implied volatility should be constant across all strikes. But obviously that is not the case in the real-world due to existence of skew. If you go long an ATM call or put in general, and then delta-hedge to 0 right after, you are actually (most of the time) either long or short some extra deltas. Why? Because if the market slides down, your fixed-strike call will have slid and moved to the right of the implied vol curve where it picks up an implied vol of the OTM calls, which even lower than ATM vol. You will lose the change in fixed strike vol * vega. Now add in another factor. If there is an increase of X vol points to the entire floating vol curve of your tenor (aka your call option along with every other option on the chain gains 1 or 2 vol pts) the (vega) profits from that may have outpaced your fixed strike vol losses. You will have residual PnL from being long/short extra deltas than you should when your Black-Scholes delta would tell you otherwise. The spot and implied vol correlation, and whether or not the market is following the sticky delta or sticky strike dynamics, must be accounted for nowadays or you will be wondering about where you extra profit, or in most cases, losses came from due to residual Greeks e.g. shadow delta, shadow gamma and so on. ## Answer by MainCom (score 0) https://quant.stackexchange.com/a/69700 Item 1 is the vol move due to skew and item 2 is the atm vol move. Historically, in low vol regime (SPX making new highs), item 1 dominates item 2. The reason is simple, in a low vol regime there is not much room for the atm vol to be lower. The atm vol could still gradually decrease but incomparable to the skew. However, if spot is down a lot, then it enters the high vol regime, in which item 2 dominates. However I should mention that it is not always the case that atm will increase when spot drops. There are mainly two scenarios where the opposite could happen. One is in an extremely high vol regime, say the implied vol is already 50, a not large enough drop of the spot could lead to a DECREASE of the vol. The other scenario is the so called "melt-up". In this case vol up when spot up, in other words vol down as spot down.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.