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
    Commented Aug 20, 2017 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. $\endgroup$ Commented Aug 20, 2017 at 20:21
  • $\begingroup$ Yes, that is precisely my question!!! $\endgroup$ Commented Aug 20, 2017 at 20:27

1 Answer 1


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*}

  • $\begingroup$ $${S_t dN(d_1) + d\langle N(d_1), S\rangle_t}$$ 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. $\endgroup$ Commented Aug 21, 2017 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
    Commented Aug 21, 2017 at 13:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.