Question:
Let $(S_t)_{t\ge 0}$ be a stock price process. Assume $u(.,.)$ satisfies the Black Scholes PDE with short rate $r=0$.
Assume that under a risk neutral measure P: $$ dS_t=\sigma_tS_tdW_t $$
Where $W_t$ is a P brownian motion. Assume $\phi_t$ is the stock holding and $\psi_t$ is the bond holding where:
$$ \phi_t=\frac {\partial(u(t,S_t))}{\partial S} $$ $$ \psi_t=u(t,S_t)-\phi_tS_t $$ Show that the trading strategy is self financing.
My Attempt
We have that $V_t=\phi_tS_t+\psi_tB_t=\phi_tS_t+\psi_t$
Further, $dV_t=\phi_tdS_t+S_td\phi_t+d\psi_t$
and the self financing condition is: $dV_t=\phi_tdS_t+\psi_tdB_t=\phi_tdS_t$
Meaning I just need to show that: $S_td\phi_t+d\psi_t=0$
When trying to evaluation the second term: $$ d\psi_t=d(u(t,S_t)-\phi_tS_t)=d(u(t,S_t))-d\phi_tS_t-\phi_tdS_t $$
Is it correct to assume that $d(u(t,S_t))=\phi_tdS_t$??
Thanks
You need to show that $$ S_t\,\mathrm d\phi_t+\mathrm d\psi_t=0, $$ i.e. $$ \mathrm d(u(t,S_t))-\phi_t\,\mathrm dS_t=0. $$ Now apply Itô's formula to compute $\mathrm d(u(t,S_t))$, and note that by the Black & Scholes PDE, $$ \frac{\partial u(t,S_t)}{\partial t} + \frac{1}{2}\sigma_t^2 S_t^2 \frac{\partial^2 u(t,S_t)}{\partial x^2} = 0. $$