# Black-Scholes Model for portfolios

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.

• You have a typo in the expression for $b_t$, the xponent is $-r t$. In Q1 you need to show that $S_t = V_t$, which is obvious once you plug in $a_t,b_t, \beta_t$ into $V_t=a_t S_t+b_t \beta_t$ – Alex C May 22 '16 at 6:28
• Thanks for spotting the mistake. When I plug those values in, I should be getting $V_t = 1/2S_t + 1/2 S_t e^0$ = $S_t$, correct? For Q2, am I heading in the right direction? – user153603 May 22 '16 at 6:41
• That makes a lot of sense, thanks very much :) Given the way it's worded, I'm assuming I can just write down another random portfolio which is SF and is arbitrage free for Q3. – user153603 May 22 '16 at 12:01

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