Choosing an Implied Volatility Skew Model for Option Delta

Summary

The document considers how to calculate option deltas across strikes when implied volatility varies with strike. It frames the key issue as selecting the implied volatility input to Black–Scholes when the volatility surface is skewed, especially for hedging an options position.

The answer compares approaches such as sticky strike and sticky delta through their assumptions about how at-the-money volatility and skew change when the underlying moves. It says practitioners assess these assumptions against residual profit and loss and may find different approaches work better at different times. The response does not provide a specific calculation, empirical comparison, or recommendation among Black–Scholes adjustments and stochastic volatility models such as SABR or Heston. It also leaves open whether position delta should be represented as a range with confidence limits. Its central caveat is that no single method is accepted as best; the choice depends on how the volatility structure behaves in the market and over the period being hedged.

Key ideas

• Black–Scholes delta depends on the implied volatility assigned to each option strike.
• Sticky strike and sticky delta represent different assumptions about how the volatility surface responds to an underlying price move.
• A suitable skew adjustment may vary across market periods.
• Practitioners can compare methods by how well they explain residual profit and loss.
• The document identifies no universally accepted best method or specific preferred model.

Tags

# Black Scholes - how to calculate delta with a vol skew


# Black Scholes - how to calculate delta with a vol skew












I am trying to calculate the delta of an option at different strike prices where the underlying has a pronounced implied volatility skew in order to correctly hedge an options strategy.

Researching on the net and previous questions on this site imply that BS can be used, but input of the correct IV is the hard part. Tags like "the wrong number in the wrong formula to get the right price", "sticky delta vs sticky strike", "skew adjusted delta" and Derman's work are the solutions I have found so far.

Can anyone tell me if these are the latest or best methods, or is a stochastic vol model like SABR or Heston better? Is calculating one value for the position delta too optimistic - should the position delta actually be a range with associated confidence limits?

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/21895

There's no best method. The question is : what is the behavior of the volatility structure (atm and skew) when the underlying moves? Each method assumes something different. In the real market, one method might work well for a period of time (in the sense that it minimizes residual p/l), but then another method might take over as best. Practitioners tend to experiment with different methods until they get comfortable which is best (as they see it). However, there's no accepted best method.

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