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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?

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  • $\begingroup$ As long as you use the actual IV in the marketplace for each option when calculating the Deltas of the options, your hedging automatically takes into account the smile without having to eplicitly model it. $\endgroup$ – noob2 Aug 19 at 20:20
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    $\begingroup$ So the process would be: 1. Sell an option for price P. 2. Back out IV using P. 3. Use implied vol in Delta calculation. 4. Repeat as P changes on the market. ? $\endgroup$ – roz Aug 19 at 20:54
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    $\begingroup$ @noob2 Even when you use the option's IV you can still have different deltas. For example, is your delta the BS delta with the IV,or is your delta the skew adjusted delta using the IV? And if you calculate the skew adjusted delta, what stickiness ratio are you going to use, etc. $\endgroup$ – ilovevolatility Aug 20 at 14:54
  • $\begingroup$ I am just curious what practitioners usually do (specifically in the case of equity derivs). Do they change their model to something like SABR and get different formulas for their delta or do they tend to use the black formula with IV and some adjustment like skew adjustment (with black vega) as you mentioned? $\endgroup$ – roz Aug 20 at 16:18
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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 $\partial\Sigma / \partial S$.

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