Black-Scholes Model for portfolios

Question

Given Black and Scholes model, consider the portfolio $a_t$ = 1/2, $b_t$ = $1/2$$S_t$ $exp(-rt)$.

1. Show that this portfolio replicates one share of stock.
2. Show if it is self-financing.
3. Find another portfolio which is self financing and replicates one share of stock.

My Attempt:

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.

Answer 1

To show whether it is self-financing, we need to show whether the equation \begin{align*} dV_t = a_t dS_t+b_t d\beta_t \end{align*} holds. Note that \begin{align*} V_t &= a_t S_t + b_t \beta_t\\ &=\frac{1}{2} S_t + \frac{1}{2} S_t e^{-rt} e^{rt}\\ &=S_t. \end{align*} Then \begin{align*} dV_t = dS_t. \end{align*} On the other hand, \begin{align*} a_t dS_t + b_t d\beta_t &=\frac{1}{2}dS_t + \frac{1}{2}S_t e^{-rt} \big(re^{rt}\big)dt\\ &=\frac{1}{2}dS_t + \frac{1}{2}rS_t dt\\ &\neq dS_t. \end{align*} Therefore, this is not a self-financing portfolio.

To find another self-financing portfolio that replicates one share of the stock, we can simply set $a_t=1$ and $b_t=0$.