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In the black-scholes model, the hedging portfolio is given (in some textbooks) by $$\Pi_t = V_t - \Delta S_t,$$ i.e., the portfolio consits of a long position in the option $V$ and $\Delta$ units of short positions in the stock $S_t$.

How does we get the change in the portfolio value, i.e., how does we get $$d\Pi_t = dV_t - \Delta dS_t$$ In some textbook they argue it is due to Ito's lemma. I know Ito's lemma, but I don't know how to apply it to get the above.

My Idea: In some textbooks they start with a (replicating) portfolio given by $$\Pi_t = \alpha S_t + \beta B_t,$$ where $B_t$ is the risk-free asset. Then, by assuming that the replicating portfolio is self-financing, we have $$d\Pi_t = \alpha dS_t + \beta dB_t,$$ i.e., the change in portfolio value is due to changes in market conditions and not to either infusion or extraction of cash.

Does we have $$d\Pi_t = dV_t - \Delta dS_t$$ because of the self-financing assumption? But why do some books say that we get it by Ito's lemma? How do we apply Ito's lemma to $$\Pi_t = V_t - \Delta S_t \qquad ?$$

Kind regards

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  • $\begingroup$ See discussions in this question and this question. $\endgroup$
    – Gordon
    Sep 14, 2021 at 19:43
  • $\begingroup$ Why did you delete most of the question's body? $\endgroup$
    – gshpychka
    Sep 15, 2021 at 15:52
  • $\begingroup$ I rolled-back your edits, please do not vandalize your own question. $\endgroup$
    – Daneel Olivaw
    Sep 15, 2021 at 16:17
  • $\begingroup$ first of all, i wanted to add my answer in the "question". however, I had to go right away and thought I would do it later. however, it is funny that people like you directly writes something when I take something out of my! question but you write nothing to the question! $\endgroup$
    – vandenberg
    Sep 15, 2021 at 16:29
  • $\begingroup$ @vandenberg it is standard practice in the site to roll-back "vandalized" (i.e. deleted) questions $-$ they're actually quite common for some reason $-$ but it seems my assumption was wrong in this case, apologies for the miscommunication. As an old user, my role is also to ensure the site remains in good form, for example avoiding emptied questions $-$ because the answers are useless if you don't know what the question was! As to the question itself, I think the link by j4bert0 is quite helpful. Otherwise I recommend you look at the following questions: $\endgroup$
    – Daneel Olivaw
    Sep 15, 2021 at 16:38

2 Answers 2

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The "hedging portfolio" argumentation is problematic. In this paper Peter Carr comments to "Is the Hedging Argument Given in the Black-Scholes Paper Correct?". I believe this answers to your question.

The other way of arguing the BS equation, using the "replicating portfolio", can be found, for example, from Shreve II. As you have written, you can assume the self-financing condition. It can be subsequently shown, that the BS equation implies the existence of self-financing portfolio. Ideas on how to do this can be found from this post.

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    $\begingroup$ This is the beginning of a good answer and I upvoted, but it would be improved by including synopses of some of Carr's key points $\endgroup$
    – Brian B
    Sep 15, 2021 at 17:24
  • $\begingroup$ "The BS equation implies the existence of [a] self-financing portfolio": this is actually a very interesting (and powerful) result which somehow got lost in time, see "A Characterization of Self-Financing Portfolio Strategies" (1981) by Bergman. $\endgroup$
    – Daneel Olivaw
    Sep 15, 2021 at 17:43
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I am not sure I have fully understood your question.

Anyway, consider your financial derivative as a function of time and stock: $$V_t = v(t, \ S).$$ Using Ito's lemma, we can recover: $$dV_t = \frac{\partial v}{\partial t}dt + \frac{\partial v}{\partial S}dS_t + \frac{1}{2}\frac{\partial^2 v}{\partial S^2}d<S>_t.$$ Being $\frac{\partial v}{\partial S} = \Delta$ by definition, you get: $$d\Pi_t = \frac{\partial v}{\partial t}dt + \frac{1}{2}\frac{\partial^2 v}{\partial S^2}d<S>_t.$$ Note a perfect hedge is not possible due to the diffusion of $S_t$.

Is that what you are looking for?

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  • $\begingroup$ Thank you for your contribution. No, my question was: how we get $d\Pi_t = dV_t - \Delta dS_t$, i.e, the dynamics of the portfolio. $\endgroup$
    – vandenberg
    Sep 14, 2021 at 8:55

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