Self-Financing Portfolios and Changes in Trading Positions
Summary
The document explains why a portfolio value written as holdings in a stock and a risk-free asset can be modeled using only the gains from those assets under a self-financing strategy. Applying the product rule to each holding reveals additional terms involving changes in the number of units held and, where relevant, covariation between holdings and asset prices. These terms do not disappear automatically; the self-financing condition requires their combined value to be zero.
The economic interpretation is that changes in positions must be funded by reallocating wealth within the portfolio. In the Black–Scholes example, the risk-free asset is nonrandom, so its covariation term with the number of units held is zero. The explanation gives the accounting condition for this setup, but does not develop details such as admissibility of trading strategies or the treatment of transaction costs. Those assumptions matter when applying the result beyond the idealized model.
Key ideas
- Applying the product rule to holdings and asset prices produces position-change and covariation terms.
- A self-financing strategy requires the net value of those position changes to be zero.
- Portfolio wealth changes through gains on held assets when the self-financing condition holds.
- In the Black–Scholes setup, a nonrandom risk-free asset has no covariation term with its holding.
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# Self-Financing Portfolio
# Self-Financing Portfolio
Why when we are using self-financing portfolios to replicate some external payoff we do not consider the quadratic variation of the portfolio weights? Say, in Black-Scholes world, when we are using $W_t = h_1(t)S_t + h_2(t)B_t $ where $S_t$ is the underlying stock which follows GBM and $B_t$ is the value of simple risk-free asset, we write the law of motion as follows?
$$ dW = h_1(t)dS_t + h_2(t)dB_t $$
Why we have no $dh $ or $(dS)(dh)$ term here?
## Answer by LocalVolatility (score 2, accepted)
https://quant.stackexchange.com/a/33118
As Quantuple implies in the comment, you do need to take all these terms into account. However, by the definition of a self-financing portfolio, they vanish. Using your notation, we generally have
\begin{equation} \mathrm{d}W_t = h_1(t) \mathrm{d}S_t + S_t \mathrm{d}h_1(t) + \mathrm{d} \langle h_1, S \rangle_t + h_2(t) \mathrm{d}B_t + B_t \mathrm{d}h_2(t) + \mathrm{d} \langle h_2, B \rangle_t. \end{equation}
We require that
\begin{equation} \mathrm{d}W_t = h_1(t) \mathrm{d}S_t + h_2 \mathrm{d}B_t. \end{equation}
The self-financing condition thus is
\begin{equation} S_t \mathrm{d}h_1(t) + \mathrm{d} \langle h_1, S \rangle_t + B_t \mathrm{d}h_2(t) + \mathrm{d} \langle h_2, B \rangle_t = 0. \end{equation}
Note that in a Black-Scholes world, you'd always have $\mathrm{d} \langle h_2, B \rangle_t = 0$ anyways as $B$ is non-random.
Go gain some intuition for what this means economically, it is useful to re-write the self-financing condition as
\begin{equation} \left( S_t + \mathrm{d}S_t \right) \mathrm{d}h_1(t) + \left( B_t + \mathrm{d}B_t \right) \mathrm{d}h_2(t) = 0. \end{equation}
I.e. we require that the value of the changes in the positions in $S$ and $B$ (at the new prices) cancel out.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.