I've always been confused with Delta hedging. It is well-known that for a (smooth enough) function of $(S,t)$ we have, due to Ito's lemma, that: \begin{eqnarray*} dC = \left(\frac{\partial C}{\partial t} (S,t) + \mu S \frac{\partial C}{\partial S} (S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t)\right)dt + \sigma S \frac{\partial C}{\partial S} (S,t) dX \end{eqnarray*} (source here, where $X$ is the standard Brownian motion). The author then proceeds to show that if we choose $\Delta = -\frac{\partial C}{\partial S} (S,t)$, then we will have \begin{align*} d(C+\Delta S) &= \left(\frac{\partial C}{\partial t} (S,t) + \mu S \frac{\partial C}{\partial S} (S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t) + \Delta \mu S\right) dt\\ &+ \sigma S \left(\frac{\partial C}{\partial S}+\Delta\right) dX\\ &=\left(\frac{\partial C}{\partial t}(S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t)\right)dt \end{align*} (I've corrected the author's typo and replaced the term $ \Delta S (\frac{\partial C}{\partial S}+\Delta) dX$ in the formula above with the correct one $\sigma S (\frac{\partial C}{\partial S}+\Delta) dX$ )
I understand the usual procedure of $\Delta$, but I never understand how $\Delta$ can be defined as $-\frac{\partial C}{\partial S} (S,t)$. By definition a hedging strategy must be a predictable process, is there any justification that $\Delta$ given as above is predictable, not merely adapted, to the filtration generated by the stock process?
Even if we accept the definition of $\Delta$, the author also doesn't explain how to compute $d(C+\Delta S)$. But, as far as I know, if $\Delta$ is really the symbol for $-\frac{\partial C}{\partial S} (S,t)$, then we will have $$d(\Delta S)=Sd(\Delta)+\Delta dS+(dS)\cdot(d\Delta).$$ And as for $d\Delta$, I think it's just replacing $C$ by $\Delta$ in the expression of $dC$, and the resulting $d(\Delta S)$ will be super-complicated with $\partial^3 C/\partial S^3$ present. What I obtained in the end was a horribe mess, the risky term $dX$ wasn't eliminated and the non-random term $dt$ doesn't match that in the BS PDE.
Wherein lies the problem?
Answer
This question has been asked many times and some clarifications appear needed.
As pointed out in an answer to this question, the portfolio \begin{align*} \Delta_t^1 S_t + \Delta^2_t C, \end{align*} where $\Delta_t^1 = -\frac{\partial C}{\partial S}$ and $\Delta_t^2 =1$, is, generally, neither self-financing nor locally risk-free.
To derive the Black-Scholes' PDE, we seek a portfolio $\Delta_t^1 S_t + \Delta^2_t C$ such that it is self-financing and locally risk-free. As is shown in answer to the above question, we derived the PDE \begin{align*} \frac{\partial C}{\partial t} + r S_t \frac{\partial C}{\partial S} + \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \sigma^2S_t^2 -rC = 0. \tag{1} \end{align*} and the portfolio weights \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*} where $B_t=e^{rt}$ is the money-market account value. Note that \begin{align*} \Delta_t^1 S_t + \Delta^2_t C = B_t. \end{align*}
Based on PDE $(1)$ and the portfolio weights give by $(2)$, it is easy to verify that \begin{align*} \Delta_t^1 dS_t + \Delta^2_t dC = dB_t. \end{align*} That is, the portfolio is indeed self-financing and locally risk-free.
Though messy, based on PDE $(1)$ and the portfolio weights give by $(2)$, it is possible to show directly that \begin{align*} d\left(\Delta_t^1 S_t + \Delta^2_t C \right) = \Delta_t^1 dS_t + \Delta^2_t dC \end{align*} using Ito's lemma. Here, we assume that the respective partial derivatives, such as $\frac{\partial^2 C}{\partial t\partial S}$ and $\frac{\partial^3 C}{\partial S^3}$, exist.
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Below, we show the self-financing property using Ito's lemma, that is, \begin{align*} Sd\Delta_t^1 + C d\Delta_t^2 + d\langle \Delta_t^1,\, S \rangle_t +d\langle \Delta_t^2,\, C \rangle_t =0. \end{align*}
For weight $\Delta_t^1$, \begin{align*} \frac{\partial \Delta_t^1}{\partial t} &= -\frac{\left(\frac{\partial^2 C}{\partial t \partial S} B_t + r\frac{\partial C}{\partial S}B_t\right) \left(C_t - \frac{\partial C}{\partial S}S \right) - \left(\frac{\partial C}{\partial t} - \frac{\partial^2 C}{\partial t \partial S}S \right)\frac{\partial C}{\partial S} B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\\ &=-\frac{\frac{\partial^2 C}{\partial t \partial S} B_t C_t + r\frac{\partial C}{\partial S}B_t C_t - r \left(\frac{\partial C}{\partial S} \right)^2B_t S_t - \frac{\partial C}{\partial t}\frac{\partial C}{\partial S} B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\\ &=-\frac{\frac{\partial^2 C}{\partial t \partial S} B_t C_t + \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \frac{\partial C}{\partial S}\sigma^2S_t^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2} \qquad\qquad\qquad\qquad\qquad\qquad \mbox{(From Eqn $(1)$)}\\ &=-\frac{\frac{\partial^2 C}{\partial t \partial S} B_t C_t^2 + \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \frac{\partial C}{\partial S}\sigma^2S_t^2 B_tC_t - \frac{\partial^2 C}{\partial t \partial S}\frac{\partial C}{\partial S} B_t C_t S - \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \left(\frac{\partial C}{\partial S}\right)^2\sigma^2S_t^3 B_t}{\left(C_t - \frac{\partial C}{\partial S}S\right)^3},\\ \frac{\partial \Delta_t^1}{\partial S} &=-\frac{\frac{\partial^2 C}{\partial S^2} B_t \left(C_t - \frac{\partial C}{\partial S}S \right) - \left(\frac{\partial C}{\partial S} - \frac{\partial^2 C}{\partial S^2}S - \frac{\partial C}{\partial S}\right)\frac{\partial C}{\partial S} B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\\ &=-\frac{\frac{\partial^2 C}{\partial S^2} B_t C_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2},\\ \frac{\partial^2 \Delta_t^1}{\partial S^2} &=-\frac{\left(\frac{\partial^3 C}{\partial S^3} B_tC_t + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t \right) \left(C_t - \frac{\partial C}{\partial S}S \right)^2 +2\left(C_t - \frac{\partial C}{\partial S}S\right) \left(\frac{\partial^2 C}{\partial S^2}\right)^2S B_t C_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^4}\\ &=-\frac{\frac{\partial^3 C}{\partial S^3} B_tC_t^2 - \frac{\partial^3 C}{\partial S^3}\frac{\partial C}{\partial S} B_tC_t S + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t C_t - \frac{\partial^2 C}{\partial S^2}\left(\frac{\partial C}{\partial S}\right)^2 B_t S + 2\left(\frac{\partial^2 C}{\partial S^2}\right)^2S B_t C_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^3} \end{align*}
For weight $\Delta_t^2$, \begin{align*} \frac{\partial \Delta_t^2}{\partial t} &= \frac{rB_t \left(C_t - \frac{\partial C}{\partial S}S \right) - \left(\frac{\partial C}{\partial t} - \frac{\partial^2 C}{\partial t \partial S}S \right) B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\\ &=\frac{ rB_t C_t - r\frac{\partial C}{\partial S} B_t S - B_t \frac{\partial C}{\partial t} +\frac{\partial^2 C}{\partial t \partial S}S B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\\ &=\frac{\frac{1}{2}\frac{\partial^2 C}{\partial S^2} \sigma^2S_t^2 B_t +\frac{\partial^2 C}{\partial t \partial S}S B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2} \qquad\qquad\qquad\qquad\qquad\qquad \mbox{(From Eqn $(1)$)}\\ &=\frac{\frac{1}{2}\frac{\partial^2 C}{\partial S^2} \sigma^2S_t^2 B_tC_t +\frac{\partial^2 C}{\partial t \partial S}S B_tC_t - \frac{1}{2}\frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} \sigma^2S_t^3 B_t -\frac{\partial^2 C}{\partial t \partial S}\frac{\partial C}{\partial S}S^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2},\\ \frac{\partial \Delta_t^2}{\partial S} &=\frac{ - \left(\frac{\partial C}{\partial S} - \frac{\partial^2 C}{\partial S^2}S - \frac{\partial C}{\partial S}\right) B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\\ &=\frac{\frac{\partial^2 C}{\partial S^2} B_t S }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2},\\ \frac{\partial^2 \Delta_t^2}{\partial S^2} &=\frac{\left(\frac{\partial^3 C}{\partial S^3} B_t S + \frac{\partial^2 C}{\partial S^2} B_t \right) \left(C_t - \frac{\partial C}{\partial S}S \right)^2 +2\left(C_t - \frac{\partial C}{\partial S}S\right) \left(\frac{\partial^2 C}{\partial S^2}\right)^2S^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^4}\\ &=\frac{\frac{\partial^3 C}{\partial S^3} B_tC_t S - \frac{\partial^3 C}{\partial S^3}\frac{\partial C}{\partial S} B_t S^2 + \frac{\partial^2 C}{\partial S^2} B_t C_t - \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t S + 2\left(\frac{\partial^2 C}{\partial S^2}\right)^2S^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^3}. \end{align*}
Then \begin{align*} &\ d\langle \Delta_t^1,\, S \rangle_t +d\langle \Delta_t^2,\, C \rangle_t \\ =& \left(-\frac{\frac{\partial^2 C}{\partial S^2} B_t C_t \sigma^2 S^2}{\left(C_t - \frac{\partial C}{\partial S}S\right)^2} +\frac{\frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S}\sigma^2 B_t S^3 }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\right)dt\\ =& \left(\frac{-\frac{\partial^2 C}{\partial S^2} B_t C_t^2 \sigma^2 S^2 + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t C_t \sigma^2 S^3 + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S}\sigma^2 B_t S^3 C_t - \frac{\partial^2 C}{\partial S^2}\left(\frac{\partial C}{\partial S}\right)^2\sigma^2 B_t S^4}{\left(C_t - \frac{\partial C}{\partial S}S\right)^3} \right)dt. \end{align*} Moreover, \begin{align*} Sd\Delta_t^1 + C d\Delta_t^2 &= S\left(\frac{\partial \Delta_t^1}{\partial t}dt + \frac{1}{2}\frac{\partial^2 \Delta_t^1}{\partial S^2}\sigma^2 S^2 dt + \frac{\partial \Delta_t^1}{\partial S}dS \right) \\ &\qquad + C\left(\frac{\partial \Delta_t^2}{\partial t}dt + \frac{1}{2}\frac{\partial^2 \Delta_t^2}{\partial S^2}\sigma^2 S^2 dt + \frac{\partial \Delta_t^2}{\partial S}dS \right)\\ &=S\left(\frac{\partial \Delta_t^1}{\partial t} + \frac{1}{2}\frac{\partial^2 \Delta_t^1}{\partial S^2}\sigma^2 S^2\right)dt + C\left(\frac{\partial \Delta_t^2}{\partial t} + \frac{1}{2}\frac{\partial^2 \Delta_t^2}{\partial S^2}\sigma^2 S^2 \right)dt. \end{align*} By combining all terms together, we can show that \begin{align*} Sd\Delta_t^1 + C d\Delta_t^2 + d\langle \Delta_t^1,\, S \rangle_t +d\langle \Delta_t^2,\, C \rangle_t =0. \end{align*} Then \begin{align*} d\left(\Delta_t^1 S_t + \Delta_t^2 C_t \right) = \Delta_t^1 dS_t + \Delta_t^2 dC_t, \end{align*} that is, $\left(\Delta_t^1, \, \Delta_t^2\right)$ constitutes a self-financing strategy.
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