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When deriving the Black-Scholes equation, a key step is to construct a portfolio \begin{equation} \Pi = V - \Delta S \end{equation} that contains a long position of the option and a short position of the stock. We then take derivative on both sides and get \begin{equation} d\Pi = dV - \Delta dS. \end{equation} By doing so we implicitly assume that $\Delta$ is a constant so that $d(\Delta S) = \Delta dS$.

However later on we choose $\Delta = \frac{\partial V}{\partial S}$ to eliminate the uncertainty. Since $V$ is a function of $S$ and $t$, so is $\Delta$. In that case, shouldn't we use \begin{equation} d(\Delta S) = \Delta dS + Sd\Delta \end{equation} and then apply Ito's lemma to $d\Delta$?

Xiao Liu
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  • We don't implicitly assume $\Delta$ is constant. It is a function of $t$ and $S$, $\Delta(t, S)$. – user1 Dec 23 '20 at 09:49
  • I believe that the product rule for stochastic processes would create a third term $d[\Delta,S]$ as well. – Riemann Feb 27 '25 at 13:04

2 Answers2

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In the standard derivation of the Black-Scholes model this portfolio is assumed to be self-financing that is there are no inflows or outflows of money. The formal definition of a self-financing portfolio in your case is that

$$d\Pi=dV-\Delta dS.$$

That is the self-financing assumption explains why you do not differentiate w.r.t. $\Delta$. For an intuitive explanation of the self-financing condition, see e.g. here Definition of self-financing strategy.

fes
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To expand on the answer by @fes, I would like to explain how we can make this portfolio self-financing. While the portfolio $V-\Delta S$ is an sich not self-financing, we implicitly assume that $\Pi$ also contains a cash account $M(t)$. So actually, we have $\Pi=V-\Delta S +M$. The buying and selling of shares $S$ is facilitated by money from $M$. For the differential we now have $d \Pi = dV-\Delta dS +rMdt$, where $r$ is the interest rate. Eventually, all terms with $M$ cancel out. Therefore we don't mention $M$ explicitly.

Riemann
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