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Your volatility also depends on your forward level, so does the value of your derivatives so a more accurate definition of your delta under a variable volatility is: $$ \dfrac{\partial V(F,\sigma(F))}{\partial F } = \dfrac{\partial V(F,\sigma)}{\partial F } + \dfrac{\partial V}{\partial \sigma}\dfrac{\partial \sigma(F)}{\partial F} $$ This is because, ...


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The relationship between the two models is described in details in Implied Volatility Formulas for Heston Models by Hagan et al. In particular an expansion of the implied volatility under the Heston model that matches the one of a SABR model is described. It gives an explicit correspondence between the parameters of each model.


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As @XiaotianDeng mentioned, the simple at-the-money approximation you mention does not always hold: it works only if you assume that $\alpha^2 T, \nu^2 T$ are small, typically $o(1)$. I wanted to add that there is really no need for such an approximation, except, possibly, to do calculations in your head, or for understanding the scale of $\alpha$ against $\...


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Using a cubic spline or worse, SVI is overkill to find the at-the-money (ATM) volatility when it is not quoted by the market: both approaches are global in the sense that a small change of one of the quotes far from the money will have a not so small impact on the at-the-money implied vol. Yes, one solution is to truncate the range of option strikes ...


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