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To my understanding, it is still quite common for market makers of vanilla options to use Black-Scholes greeks. My concern with this is best expressed by Pat Hagan in the original SABR model paper:

" Since different models are being used for different strikes, it is not clear that the delta and vega risks calculated at one strike are consistent with the same risks calculated at other strikes"

Additionally, it seems theoretically ludicrous to derive risk exposures to vol-of-vol or spot vol correlation (volga and vanna) from a model that says nothing about these.

Since we do have alternatives that account for the issues of Black-Scholes (take your pick of a stochastic volatility model), why would anyone relegate themselves to the theoretical inconsistencies involved with using Black-Scholes greeks?

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I do not agree with the answer by @river_rat. SABR greeks (the so-called Bartlett delta and vega) are used by practitioners in Interest Rates trading from my own experience. In general you want your hedges to be consistent with your pricing model, so it really makes sense to use the hedges provided by a "better" (whatever that means) model than a standard Black-Scholes if you use it for pricing your deals. The Black-Scholes greeks are popular as a cheap and dirty model independent approach when you don't want to or can't go to into a depth of things but generally your greeks should be consistent with your pricing model and barely anyone are using a textbook Black-Scholes for pricing nowadays.

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    $\begingroup$ A majority of vanilla option books are priced and hedged using some version of Black76 in my experience. Then comes some version of local vol, then a mixture model or two, then SLV and pure SV for the weird tail stuff. But that main part is Black76 with an exogeneous smile model (sticky strike, sticky delta, correlated riskies etc) $\endgroup$
    – river_rat
    Commented Mar 31 at 20:29
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Pick your poison, what is better? A simple model that is wrong or a complicated model that is also wrong. Add to that computation time on large portfolios and the simplicity of a closed form Black-Scholes greek sensitivity number makes a lot of sense. The nuance is then on the implied volatilities feeding that model and the external factors driving correlation between those implied volatilities and the market.

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    $\begingroup$ SABR Greeks mentioned here also do comes in a closed-form expressions. $\endgroup$
    – Hasek
    Commented Mar 31 at 19:26
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    $\begingroup$ Would not call an asymptotic expansion closed form? $\endgroup$
    – river_rat
    Commented Mar 31 at 20:22
  • $\begingroup$ Well you're right that they are asymptotics rather than real closed form solutions however one of the reasons SABR model is so popular is that people use Hagan et al. approximations and Bartlett greeks as fast analytical formulas instead of doing a Monte Carlo. $\endgroup$
    – Hasek
    Commented Apr 1 at 9:17

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