Testing Whether a Portfolio Replicates Stock and Is Self-Financing
Summary
The document examines a portfolio made from a fraction of the stock and a cash account, with holdings chosen so its value equals one share at each instant. It distinguishes matching the stock’s value from being self-financing: a portfolio can have the same marked value without its holdings evolving through trading gains alone.
The answer checks the self-financing condition by comparing the change in portfolio value with the gains from the current stock and cash positions. Although the specified portfolio’s value simplifies to the stock price, its gain process includes only half of the stock movement plus interest on the cash position, so it does not satisfy the condition. As a simple alternative, holding one share and no cash replicates one share and is self-financing. The example is a basic continuous-time finance exercise; it does not address transaction costs, constraints, or more complex replication problems.
Key ideas
- A portfolio can match an asset’s value without being self-financing.
- The self-financing condition equates portfolio value changes to gains from existing holdings.
- The proposed stock-and-cash holdings have the same value as one share but fail that condition.
- Holding one share and no cash is a self-financing replication of one share.
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Full text
# Black-Scholes Model for portfolios
# Black-Scholes Model for portfolios
Given Black and Scholes model, consider the portfolio $a_t$ = 1/2, $b_t$ = $1/2$$S_t$ $exp(-rt)$.
- Show that this portfolio replicates one share of stock.
- Show if it is self-financing.
- Find another portfolio which is self financing and replicates one share of stock.
My Attempt:
I'm fairly sure that for Q1, I need to show that this is a arbitrage free portfolio by showing $C_t$ = $V_t$, and not $C_t$ > $V_t$ or $C_t$ < $V_t$ with $V_t$ = $a_t$$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 by Gordon (score 2, accepted)
https://quant.stackexchange.com/a/26169
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 $b_t=0$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.