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let $\frac{\partial C}{\partial S}=\delta_c$ let $\frac{\partial^2 C}{\partial S^2}=\Gamma_c$ let $\frac{\partial C_0}{\partial S}=\delta_0$ let $\frac{\partial^2 C_0}{\partial S^2}=\Gamma_0$ we want $\frac{\partial V}{\partial S}=\frac{\partial C}{\partial S}=\delta_c$ and $\frac{\partial^2 V}{\partial S^2}=\frac{\partial^2 C}{\partial S^2}=\Gamma_c$ ...


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I tried to use the BPV/delta relashionship $\Delta = \frac{ \frac{\partial Z}{\partial r}-\frac{\partial P(t,T)}{\partial r}\frac{Z}{P(t,T)} } {\frac{\partial P(t,S)}{\partial r}}$ but it doesn't work as well.


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You delta hedge if don't have an opinion of whether the stock will go up or down but think that realized volatility will be substantially different from implied volatility. If you don't delta hedge without having a view on the direction of the stock you are taking unnecessary risk.



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