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For Black-Scholes, $\Delta_C=\partial_{S} C=N(d_1)$, $d_1= \frac{\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)}{\sigma\sqrt{T - t}}$ You may fit the volatility $\sigma$ to this term by $$\Delta_C({\hat{\sigma}})=0.25$$Note that $\Delta_P=1-\Delta_C$ by Put-Call-Parity.


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You need to use log of prices, because log of returns are normally distributed. So or where x is return- $$ x=-\frac{1}{\tau} ln(\frac{S_{t+\tau}}{S_{t}}) $$ The annualized standard deviation can be scaled as +/-$ n\frac{\sigma}{\sqrt{\tau}} $ where n is your multiple. You can either ignore or estimate drift. or look at it another way, S refers to the index ...


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If you want to calculate the change of a greek, lets say Delta, from a change in the volatility, you would need Vanna: $$Vanna=\partial_\sigma\Delta=\partial_\sigma\partial_S C$$, which under Black-Scholes becomes: $$ Vanna=\left(\sqrt{T-t+\frac{1}{\sigma}}\right)\phi\left(d_1\right) $$ where $\phi\left(\cdot\right)$ is the standardnormal density and $$ ...



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