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The famous Black-Scholes PDE is:

$ \frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}=rC$

Then, how should I find another PDE for $\frac{\partial C}{\partial \sigma}$ from this PDE ? I tried chain rule, but it seems it didn't work straightly.

Thanks for helping me out!

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I don't think you need the chain rule, because changes in $\sigma$ affect $C$ directly, not indirectly through other variables like $S$.

You can differentiate both sides with respect to $\sigma$:

$$ C_{t\sigma} + rS C_{S\sigma} + \sigma S^2 C_S + \frac{1}{2}\sigma^2 S^2 C_{SS\sigma} = rC_\sigma $$

where the subscripts denote partial derivatives (for example, $C_{t\sigma} = \frac{\partial^2 C}{\partial t \partial \sigma}$).

You can verify the formula above by plugging the Black-Scholes solution for $C$ into it.

If you're just trying to calculate $C_\sigma$, an easier way would probably be to derive the Black-Scholes solution for $C$ (by guess and check) and then simply differentiate it with respect to $\sigma$.

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