The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the assumption that the specified portfolio is self-financing.

We have a two asset market:

$dB_t = B_t r dt $

$dS_t = s_t (\mu dt + \sigma dW_t)$

We introduce a European option with price $v(t,S_t)$ at time $t$. We now consider a portfolio consisting of one option and -$\frac{\partial v}{ \partial S}$ stocks. Therefore if $X_t$ is our wealth at time $t$, we must have $X_t = v(t,S_t) - \frac{\partial v}{ \partial S} S_t$.

It is then claimed that we have $dX_t = dv(t,S_t) -  \frac{\partial v}{ \partial S} dS_t$, as the portfolio is self-financing.

However, it seems to me that if we have a constant holding of 1 option in our portfolio, then the only way to make the overall portfolio self-financing is to have a constant holding of stock too (otherwise, if we increase/decrease our holding in stock, where do the extra funds for this come from?).

Typically, the way I have seen self-financing portfolios constructed is that the holding in 1 asset (for example, the risk-free asset) is not explicitly specified, and is determined by the self-financing condition (i.e. the condition that $X_t = \pi_t \cdot P_t$ and $dX_t = \pi_t \cdot dP_t$, where $\pi$ is the portfolio and $P_t$ is the price process - this gives a linear equation for the unspecified holding).

Based on the above, it seems that in order to have a self-financing portfolio where we hold a constant 1 option and $-  \frac{\partial v}{ \partial S}$ shares, we must also have a dynamic holding in the risk-free asset which allows us to ensure that we can always have $-  \frac{\partial v}{ \partial S}$ shares in our portfolio without injecting external funds (and thus breaking the self-financing condition). However, if we do have this holding of the risk-free asset in our portfolio as well, then our equation for the wealth process ($X_t = v(t,S_t) - \frac{\partial v}{ \partial S} S_t$) becomes incorrect, as we are not taking into account our holding in the risk-free asset.

In summary, I don't believe that the portfolio of 1 option and $-\frac{\partial v}{ \partial S}$ shares specified in the wikipedia derivation of the Black-Scholes equation is self-financing, but the derivation makes use of the fact that it [i]is[/i] self-financing. Am I missing something?