Under the usual notations,

In most textbooks on Quantative Finance, for deriving the Black-Scholes solution I find that authors, while setting up the riskless portfolio, assume that,

$$\text{d} (\frac{\partial V}{\partial S} S_t) = \frac{\partial V}{\partial S} \text{d} S_t $$

At least can we prove this post facto, as in, does this equation hold true for famous Black Scholes equation

The same issue is also pointed out here.

  • $\begingroup$ Uhm, so you're saying your question is a duplicate? I see that the question you link to also has an answer. Where do you require further clarification? $\endgroup$ – Bob Jansen Aug 20 '17 at 20:17
  • $\begingroup$ The answers to that question are not accepted; Also, while the question is similar, what I am asking is a slightly more different one. If someone were to answer this question, probably a part of the linked question would be answered. $\endgroup$ – honeybadger Aug 20 '17 at 20:21
  • $\begingroup$ Yes, that is precisely my question!!! $\endgroup$ – honeybadger Aug 20 '17 at 20:27

This is not true. Note that $\frac{\partial C}{\partial S_t} = N(d_1)$. Then \begin{align*} d\left(\frac{\partial C}{\partial S_t}S_t\right) &= \underbrace{S_t dN(d_1) + d\langle N(d_1), S\rangle_t} + N(d_1) dS_t\\ &\ne N(d_1) S_t. \end{align*} That is, \begin{align*} d\left(\frac{\partial C}{\partial S_t}S_t\right)\ne \frac{\partial C}{\partial S_t}dS_t. \end{align*}

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  • $\begingroup$ $${S_t dN(d_1) + d\langle N(d_1), S\rangle_t}$$ can this be negligible ? Why I'm asking is that textbooks like Hull are rarely wrong in these matters. I hope I am making myself clear. $\endgroup$ – honeybadger Aug 21 '17 at 3:13
  • $\begingroup$ It is not negligible, as there is diffusion term which can take any values. Textbooks like Hull's did made a mistake here. See discussions in this question and also this question. $\endgroup$ – Gordon Aug 21 '17 at 13:01

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