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In Black-Scholes model with constant parameters, a call and a put with the same characteristics have the same vega: https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model#The_Greeks

Using call-put parity yields $\frac{\partial S_t}{\partial \sigma} = 0$. This result is weird because we have: $S_t = S_0.e^{(r-\frac{\sigma^2}{2})t+\sigma W_t}$.

How can we justify this result? Thank you in advance for your answers.

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Vega is the partial derivative of the option price (as a function of parameters -- current stock price $S_t$, strike price $K$, implied volatility $\sigma$, etc.) with respect to $\sigma$ -- holding other parameters fixed:

$$vega = \frac{\partial}{\partial \sigma} V(S_t,K,\tau,r,\sigma) $$

You are confusing the stochastic process with the parameter $S_t$ which is a constant when $t$ is fixed.

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