Vanila Option self financing under Stock as numeraire

Question

I am trying to see how the vanilla call option can be seen as self financing using the Stock as the numeraire. The case with the Bond as numeraire is quite simple and can be found in Wilmott's FAQ for instance.

We assume that under the real world probability measure

$$ dS=S(\mu dt + \sigma dW)\\ dB=B rdt $$

It is well known that under the Stock as as numeraire we can get the call option as

$$ \frac{C_t}{S_t} = E_{\mathbb S,t} [ \frac{S_T-K}{S_T} \Theta(S_T-K) ] $$

where under measure $\mathbb S$ we have

$$ dS= S( (r+ \sigma^2) dt + \sigma dW_S) \\ dB= B r dt $$ which gives $dW_S = dW + \frac{\mu - r- \sigma^2}{\sigma} dt$. Specifically under this measure $\frac{B}{S}$ is a Martingale

$$ d(\frac{B}{S})= \sigma \frac{B}{S} dW_S $$

My question is that I should be able to see this as a self financing portfolio but I cannot. I am sure that I am making a mistake but cannot find it. I do the following (I suppress the time subscript in the following)

From the Martingale representation theorem we have $$ d(\frac{C}{S})=\alpha d( \frac{B}{S}) $$ for some pre-visible process $\alpha$.

This can be written as

$$ dC = \alpha dB + \frac{ C-\alpha B}{S} dS - \frac{ C-\alpha B}{S^2} dS^2 $$

It's the last piece that I don't understand. Without it we would have a self-financed portfolio. I could try to write it in terms of $dB$ but it still doesn't give me a self-financed portfolio. What am I doing wrong?

Answer 1

I found the mistake I was making. Its the same mistake I made a variant of this problem about two years ago and Quantuple answered it. I am giving the correct answer here for completeness.

From the Martingale representation theorem we have

$$ d(\frac{C}{S}) = \alpha d (\frac{B}{S}) $$

Now the LHS is

$$ LHS = \frac{dC}{S} - \frac{CdS}{S^2} + \frac{CdS^2}{S^3} - \color{red}{\frac{dCdS}{S^2}} $$

I was missing the red term in my derivation. We are keeping terms only upto order $dt$ and the piece that contributes from the $dC$ in the last term will be $\mathcal O(dW)$.

The RHS is

$$ \alpha ( \frac{dB}{S} - \frac{BdS}{S^2} + \frac{BdS^2}{S^3}) $$

and there is no equivalent term for the red one because $dB$ has not stochastic piece so such a term would not appear at order $dt$.

From these one gets

$$ dC= \alpha dB + (C-\alpha B) \frac{dS}{S} $$

making $C$ a self-financed strategy.