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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

vandenberg
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  • See discussions in this question and this question. – Gordon Sep 14 '21 at 19:43
  • Why did you delete most of the question's body? – gshpychka Sep 15 '21 at 15:52
  • I rolled-back your edits, please do not vandalize your own question. – Daneel Olivaw Sep 15 '21 at 16:17
  • 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! – vandenberg Sep 15 '21 at 16:29
  • @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: – Daneel Olivaw Sep 15 '21 at 16:38
  • https://quant.stackexchange.com/questions/34535/dynamic-delta-hedging-and-a-self-financing-portfolio/34546#34546 – Daneel Olivaw Sep 15 '21 at 16:38
  • https://quant.stackexchange.com/questions/34027/derivation-of-bs-pde-problem-using-delta-hedging?noredirect=1&lq=1 – Daneel Olivaw Sep 15 '21 at 16:38
  • https://quant.stackexchange.com/questions/32171/black-scholes-differential?noredirect=1&lq=1 – Daneel Olivaw Sep 15 '21 at 16:39
  • Maybe this paper also helps: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2548676 – Daneel Olivaw Sep 15 '21 at 16:43
  • In a nutshell: It does not go back to the self-financing assumption and is generally a common mistake. To be self-financing, we need \begin{align} B_t = \Delta_t^1 S_t + \Delta^2_t C, \end{align} such that \begin{align} \Delta_t^1 = -\frac{\frac{\partial C}{\partial S} B_t}{C_t - \frac{\partial C} {\partial S}S}, \quad \Delta_t^2 =\frac{B_t}{C_t - \frac{\partial C}{\partial S}S}, \tag{2}, \end{align} right? – vandenberg Sep 15 '21 at 17:06
  • However, this then only partially answers my question. How do we get: $$d\Pi_t = dV_t - \Delta dS_t$$ In other words, how do we apply ito's lemma to $$\Pi_t = V_t - \Delta S_t$$ to get $$d\Pi_t = dV_t - \Delta dS_t$$ – vandenberg Sep 15 '21 at 17:11
  • I was thinking about Ito's summe rule; i.e., so let us first denote $\Pi_{V_t}$ as the partial derivate w.r.t. $V_t$

    Then: $\Pi_{V_t} = 1$ and similarly for $\Pi_{S_t} = -\Delta$

    And the second partial derivative is trivially zero, i.e., $\Pi_{V_t V_t} = 0$, similarly for $\Pi_{S_t S_t} = 0$ and also $\Pi_{V_t S_t} = 0.$

    Ito's lemma for two variable is: $$df(X,Y) = f_X dX + f_Y dY + 0.5 f_{XX} dX^2 + 0.5 f_{YY} dY^2 + f_{XY}dXdY.$$ Substituting now, we get: $$ d(\Pi_t) := d(V_t - \Delta S_t) = 1dV_t - \Delta dS_t$$ Do you think that is correct? @DaneelOlivaw

     – vandenberg Sep 15 '21 at 17:13
  • "How do we apply Itô's lemma to [...] to get [...]?" You can not, it's actually a mistake in the original paper by Black and Scholes. The thing is that, due to some terms cancelling downstream, you still can reach the correct answer with the portfolio $V_t-\Delta S_t$. – Daneel Olivaw Sep 15 '21 at 17:21
  • The problem with your last derivation is that $\Delta$ is actually $\Delta(t,S_t)$ so rigorously you also need to differentiate delta. What we do know is that the portfolio $\Pi_t:=\alpha_tS_t+\beta_tB_t$ where $\alpha_t$ is a free parameter and $\beta_t$ is defined as: $$\beta_t:=\frac{\Pi_t-\alpha_tS_t}{B_t}$$ is always self-financing (try to prove it). So I recommend you work with that setting by setting $\alpha_t=\Delta_t$, you'll be sure to be theoretically sound this way. – Daneel Olivaw Sep 15 '21 at 17:24

2 Answers2

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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.

j4bert0
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    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 – Brian B Sep 15 '21 at 17:24
  • "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. – Daneel Olivaw Sep 15 '21 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?

Yoda And Friends
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  • 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. – vandenberg Sep 14 '21 at 08:55