Adjusting Black-Scholes Delta for an Implied Volatility Smile
Summary
The document explains how an implied volatility smile can affect the hedge delta of a vanilla option. Within a Black-Scholes framework, it describes a skew-adjusted delta that combines the usual Black-Scholes delta, calculated using the observed implied volatility, with a vega term multiplied by the sensitivity of implied volatility to the underlying spot price. Thus, a spot move can change both the option’s modeled value directly and its implied volatility.
For a pure stochastic volatility model, the response gives a relationship between implied volatility’s spot sensitivity and its strike sensitivity, making the latter observable from the smile. It cautions that this relationship is valid only for stochastic volatility models; local volatility and stochastic-local volatility models require another way to estimate the spot sensitivity. The response supplies a modeling formula and its limitations, but no numerical example or evidence comparing hedge performance. The appropriate delta therefore depends on the model and the assumptions used to describe how the smile moves.
Key ideas
- The appropriate option delta depends on the model used to represent volatility.
- A smile-adjusted Black-Scholes delta adds a vega-weighted implied volatility sensitivity to standard delta.
- In pure stochastic volatility models, spot sensitivity can be related to strike sensitivity.
- That relationship does not generally apply to local volatility or stochastic-local volatility models.
- The document gives no numerical example or hedge performance comparison.
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Full text
# How does the volatility skew/smile relate to hedging/trading vanilla contracts?
# How does the volatility skew/smile relate to hedging/trading vanilla contracts?
I know that obtaining and calibrating the smile is important in the hedging and trading of exotics since we use vanillas to hedge and price exotics. How is the smile important in the hedging and trading of the vanillas themselves given that we are using the standard BS model? Can you give an example of how a hedge or risk management parameter for a vanilla equity option would change under the presence of a smile? I understand that we could switch our model to account for the smile (for example use SABR), but is there a trick/practice that practitioners and traders use when hedging under the presence of a smile?
## Answer by user34971 (score 4, accepted)
https://quant.stackexchange.com/a/47230
The correct (quant) answer is: the delta depends on the model.
If you don't want to calculate the SV or LV or SLV model delta, but like to work within the BSM framework, then the delta to use is
$$ \Delta = \Delta^{BS} (\Sigma) + \nu^{BS}(\Sigma) \frac{\partial \Sigma}{\partial S} $$
where $\Delta$ is the skew adjusted delta, $\Delta^{BS} (\Sigma)$ is the Black-Scholes delta evaluated with the observed implied volatility $\Sigma$, and $\nu^{BS}$ is the Black-Scholes vega.
The hard part is evaluating the sensitivity of the implied vola to the spot price. In a pure stochastic volatility model, for vanilla options,
$$ \frac{\partial \Sigma}{\partial S} = - \frac{K}{S} \frac{\partial \Sigma}{\partial K} $$
so the sensitivity of the option's IV to the spot is directly observable. However, the simple and (almost) model-free formula above is valid only for SV models.
For LV and SLV models you'll need to somehow estimate $\partial\Sigma / \partial S$.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.