Given Black and Scholes model, consider the portfolio $a_t$ = 1/2, $b_t$ = $1/2$$S_t$ $exp(-rt)$.
- Show that this portfolio replicates one share of stock.
- Show if it is self-financing.
- Find another portfolio which is self financing and replicates one share of stock.
I'm fairly sure that for Q1, I need to show that this is a arbitrage free portfolio by showing $C_t$ = $V_t$, and not $C_t$ > $V_t$ or $C_t$ < $V_t$ with $V_t$ = $a_t$$S_t$+$b_t$$β_t$. However I'm not entirely sure how to find out $C_t$.
For Q2. I believe I need to show that $dV_t$ = $a_tdS_t+b_tdβ_t$ but am not sure how exactly to do that.
I have no idea how to attempt Q3.