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In my course, I have been asked to prove a number of statements about "the Greeks" from the Black-Scholes model for pricing a European call option with no dividends and a strike price of $K$. One of the problems was to prove that $\nu$ (Vega) was positive. To prove the fact, I began by differentiating the Black-Scholes model: $$ C_t = S_t\Phi(d_1(t)) - K e^{-r(T - t)}\Phi(d_2(t)), $$ where all of the variables are defined as is done typically, and $S_t$ is equal to $$ S_t = S_0\exp\left(\left(b - \frac 1 2 \sigma^2\right)t + \sigma B_t\right), $$ with $B_t$ a Brownian motion. Now, when I take the derivative w.r.t. $\sigma$ of $C_t$, I find that $$ \frac{\partial C_t}{\partial\sigma} = \frac{\partial S_t}{\partial \sigma}\Phi(d_1(t)) + S_t\Phi'(d_1(t))\frac{\partial d_1(t)}{\partial \sigma} - K e^{-r(T - t)}\Phi'(d_2(t))\frac{\partial d_2(t)}{\partial \sigma}. $$ Using the fact that $S_t\Phi'(d_1(t)) = K e^{-r(T - t)}\Phi'(d_2(t))$, and that $$ \frac{\partial d_1}{\partial \sigma} - \frac{\partial d_2}{\partial \sigma} = \sqrt{T - t}, $$ I find that $$ \frac{\partial C_t}{\partial\sigma} = \frac{\partial S_t}{\partial \sigma}\Phi(d_1(t)) + \sqrt{T - t} K e^{-r(T - t)}\Phi'(d_2(t))\frac{\partial d_2(t)}{\partial \sigma}. $$

Obviously, I now need to solve $\frac{\partial S_t}{\partial \sigma}$. In so doing, I find that $$ \frac{\partial S_t}{\partial \sigma} = (B_t - \sigma) S_t, $$ which makes my final answer that $$ \frac{\partial C_t}{\partial\sigma} = (B_t - \sigma) S_t\Phi(d_1(t)) + \sqrt{T - t} K e^{-r(T - t)}\Phi'(d_2(t))\frac{\partial d_2(t)}{\partial \sigma}. $$ However, all the sources that I can find (e.g. here, here and here) report that Vega is just equal to $$ \frac{\partial C_t}{\partial\sigma} = \sqrt{T - t} K e^{-r(T - t)}\Phi'(d_2(t))\frac{\partial d_2(t)}{\partial \sigma}. $$ Is there something that I am doing wrong? I suspect that I am doing something wrong w.r.t. the inclusion of $B_t$, as that seems strange. Any help would be appreciated.

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  • $\begingroup$ How does stock price change with respect to volatility? $\endgroup$ – Calvin Lin Mar 29 '14 at 5:18
  • $\begingroup$ Isn't the change in stock price w.r.t. volatility given by the $\frac{\partial S_t}{\partial \sigma} = (B_t - \sigma)S_t$ term? $\endgroup$ – fbt Mar 29 '14 at 16:44
  • $\begingroup$ $S_t$ is the stock valuation's price at time $t$, which is a given / constant. You do not model it as the expected effect of Brownian motion. $\endgroup$ – Calvin Lin Mar 29 '14 at 16:59
  • $\begingroup$ Oh- that makes sense. We would only model its evolution over time as the expected effect of Brownian motion? $\endgroup$ – fbt Mar 29 '14 at 17:22

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