In many books and derivations of the Black-Scholes PDE one sees that

$$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF$$

which implicitly assumes that $d\Delta=0$. Somewhere down the road one then deduces that

$$\Delta=\frac{\partial V}{\partial F}$$

to simplify the equation. Doesn't this contradict the initial assumption that $d\Delta=0$? If one performs a full differentiation

$$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF - F d\Delta$$

the rest of the story goes wrong. Isn't it true that $\Delta = \Delta(t, S)$, i.e. is depending on time and the underlying stochastic process and hence has to be differentiated?

  • $\begingroup$ What is $F$ ? also could you provide a link to one of the "many books" you mention ;) - cheers $\endgroup$ Commented Apr 8, 2014 at 6:54
  • $\begingroup$ You are using rather non-standard notation. However, you can look at $\Pi$ as the value of a delta-hedged portfolio (option plus a short position in $\Delta$ underlying). This is why $\Delta$ is not differentiated. In this context, $\Delta$ is a given quantity, by definition equal as the partial derivative of V wrt F. $\endgroup$ Commented Apr 8, 2014 at 7:34
  • $\begingroup$ @selfTaught as I see it this is an actual answer to the question at hand - don't be afraid to post it as such ;) $\endgroup$ Commented Apr 8, 2014 at 8:16
  • $\begingroup$ An undergraduate introduction to financial mathematics, J.Buchanan, Chapter 7. The mathematics of derivatives, R.Navin, Chapter 5. $\endgroup$ Commented Apr 8, 2014 at 8:32
  • $\begingroup$ Whether given or derived, the fact that $\Delta$ is the partial derivative of the (stochastic) option $V$ means it's stochastic and time dependent. Trying to deduce that it's constant (i.e. $d\Delta=0$) also fails. $\endgroup$ Commented Apr 8, 2014 at 8:36

2 Answers 2


$\Pi$ is the value of a delta-hedged portfolio (option plus a short position in Δ underlying). The notation for $\Delta$ is overloaded. Here it represents the number of underlying contracts (f.ex shares) in your delta hedged portfolio, equal to the greek $\Delta$ when the portfolio is created. Therefore in the calculation of $d \Pi$, $\Delta$ (the number of shares in your portfolio) is treated as a constant.

Yes, the greek $\Delta$ evolves as the option approaches maturity and wrt $F$ and you will have to rebalance your portfolio. But this is not contemplated in the infinitesimal $d \Pi$.

  • $\begingroup$ I think that the misunderstanding is in the way things are formulated. The $\Delta$ is indeed constant but in the calculation people forget to tell that the 'simplification' $\Delta=\frac{\partial V}{\partial F}$ is not defining $\Delta$ but is a (first order) constraint on $V$. Later on the constraint is inserted into the derivation and forgotten about. It suggests to me that some sort of Lagrangian or path-integral formulation would be more appropriate, but I haven't looked into it. $\endgroup$ Commented Apr 8, 2014 at 14:01
  • $\begingroup$ Is there a derivation with non-constant $\Delta$ of Black-Scholes? Does it make semantically sense? $\endgroup$ Commented Apr 8, 2014 at 14:04
  • $\begingroup$ The are different other derivations of the B&S PDE (see for example the "martingale method" and the "CAPM method"). However, AFAIK, they all draw from the concept that delta-hedging results in a (locally) riskless portfolio hence the existence of a risk-neutral measure. $\endgroup$ Commented Apr 8, 2014 at 14:49
  • $\begingroup$ This may interest you: emanuelderman.com/media/smile-lecture2.pdf $\endgroup$ Commented Apr 8, 2014 at 14:52

The contradiction is true. See Question V in Peter Carr's FAQ's in Option Pricing Theory (1999).


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.