In the simplified derivation of [[Black-Scholes PDE]], we assumed that $q$ were constant. However in reality, it must be updated continuously because $\frac{\partial V}{\partial S_t}$ itself changes as $t$ and $S_t$ evolve. Therefore we need to treat $q$ as a function, to maintain our fully hedged portfolio position.
$ q=\frac{\partial V}{\partial S_t}$
## Self-Financing Hedge Portfolio
To handle this continuous rebalancing without injecting outside funds, we introduce a *bond* (or “cash”) account earning the risk‐free rate $r$. This ensures the portfolio is self‐financing, i.e. any purchase or sale of shares is funded by (or credited to) the bond holding.
We defined the portfolio $X$, which combines both instruments with an initial value of zero.
$ X_t= qS_t+cM_t=0$
| Notation | Description |
| -------- | ------------------------------ |
| $q$ | Quantity of shares |
| $S_t$ | Value of the share price |
| $c$ | Quantity of the risk-free bond |
| $M$ | Value of the bond |
## Portfolio Rebalancing
The rebalancing at time $t$ simply exchanges the risk-free instrument for stocks of equal value. Hence, it does not change the portfolio value.
$ X_t^{\text{post}}- X_t^{\text{pre}} = S_t(q_t -q_{t-1})+M_t(c_t -c_{t-1})=0 $
If $(q_t > q_{t-1})$, we buy $(q_t - q_{t-1})$ number of shares. We finance this order by debiting the cash account with quantity $(c_t -c_{t-1})$. We can expand the equation to separate quantity and price changes.
$
\begin{align}
X_t
&=S_{t-1}(q_t-q_{t-1})+M_{t-1}(c_t -c_{t-1})\\[2pt]
&+(S_t-S_{t-1})(q_t-q_{t-1}) \\[2pt]
&+(M_t-M_{t-1})(c_t -c_{t-1})=0
\end{align}
$
In continuous‐time notation, this becomes:
$
\begin{align}
X_t^{\text{post}}- X_t^{\text{pre}} &= S \,dq+M \,dc=0\\[2pt]
&=S\,dq + M\,dc + dS\,dq + dM\,dc=0
\end{align}
$
## Incorporating the Financial Derivative
We add the financial derivative $V$ to the stocks and bond portfolio.
$ \pi = V + \overbrace{qS + cM}^X$
Hence the portfolio's differential $d \pi$ (i.e. change in the portfolio value) is:
$\begin{align}
d\pi &= dV + d(qS+cM)\\
d\pi &= dV + d(qS)+d(cM)\\
\end{align}$
To solve $d(qS)$ we acknowledge that both $S$ and $q$ are time-dependent processes (since $q$ is not a constant but $\frac{\partial V}{\partial S}$, it inherits the stochastic behavior of $S$). Therefore we need the Itô product rule, which is similar the regular [[Differentiation Rules#Product Rule|Product Rule]], with an extra differential term.
$
\begin{align}
d(qS) &= q\, dS + S\, dq+ \overbrace{ds\,dq}^{\text{Itô term}} \\[2pt]
d(cM)&= c \, dM+M \, dc+ dc \, dM
\end{align}
$
We substitute terms and recognize the rebalancing terms, which have a net zero value.
$
\begin{align}
d\pi &= dV +
\overbrace{(q\, dS + S\, dq+ ds\,dq)}^{d \, qS} +
\overbrace{(c \, dM+M \, dc+ dc \, dM)}^{d \, cM} \\[2pt]
&= dV+ q \, dS+c \, dM+
\underbrace{(S\, dq +M \, dc+ ds\,dq + dc \, dM)}_{\text{rebalancing}}
\end{align}
$
Thus the following remains:
$
\begin{align}
d\pi &= dV +q \, dS + c \, dM \tag{1}\\
&= dV +q \, dS + rcM \, dt \tag{2}\\
&= dV+q \, dS + r \underbrace{(\pi -V-qS)}_{cM} \, dt \tag{3}
\end{align}
$
where:
- (2) Express the change in money market bond value $dM$ (at fixed quantity $c$) as the risk-free rate of return $r$ times the value $M$ over period $dt$, hence $dM = rM \,dt$.
- (3) Express $CM$ as the remainder of the portfolio.
## Applying Itô's Lemma
Next we roll out [[Itô Processes and Itô's Lemma#Itô's Lemma for $F(t, X_t)$|Itô's Lemma]] for $dV$:
$
\begin{align}
d \pi &= \left ( \frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial S}dS+ \frac{1}{2}\frac{\partial^2V}{\partial S^2} b^2 dt\right) + q\, dS+r(\pi-V-qS) \, dt \\[2pt]
&= \left ( \frac{\partial V}{\partial t} + \frac{1}{2}\frac{\partial^2V}{\partial S^2} (\sigma S)^2 \right)dt + \left(\frac{\partial V}{\partial S}+q\right) dS+r(\pi-V-qS) \, dt
\end{align}
$
We insert $(\sigma S)^2$ for $b^2$, which comes from the stochasticity of the [[Geometric Brownian Motion]] from $S$. Also we group terms with $dS$ together.
## Delta Hedging
Now we look for $q$ that dynamically hedges the risk in the portfolio continuously. We can expect to find such $q$ since there is only one stochastic driver that imports both the stock and the derivative.
We achieve such a solution by setting $q$ of be the negative of the partial derivative.
$ q = - \frac{\partial V}{\partial S}$
This cancels all terms with $dS$:
$ d \pi = \left ( \frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2V}{\partial S^2} \right)dt +r(\pi-V-qS) \, dt$
## Classical Black-Scholes PDE
We acknowledge that this dynamic choice of $q$ makes the portfolio risk-free. Since the initial portfolio value was also $0$, it must remain zero at all time, to avoid arbitrage opportunities.
- Hence $r\pi=0$
- We replace $q$ with its dynamic function $-\frac{\partial V}{\partial S}$
$ d \pi = \left ( \frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2V}{\partial S^2} +rS\frac{\partial V}{\partial S} -rV\right)dt=0$
This again (but this time without shortcuts) leads to the Black Scholes PDE:
$ \boxed{\frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2V}{\partial S^2} +rS\frac{\partial V}{\partial S} -rV=0}$