Deriving a Self-Financing Portfolio from a Time-Varying Stock Holding
Summary
The document derives the cash account needed to keep a portfolio self-financing when its stock holding changes over time. Portfolio value is written as the shares held times the stock price plus the balance in an interest-bearing account. Applying the stochastic product rule reveals terms from changes in both holdings and asset prices; when the holdings are deterministic and the cash account has no diffusion term, the relevant cross terms vanish.
For a stock position that declines as one minus time, the change in the stock holding creates a cash flow of minus the stock price times the time increment. The cash balance must offset that flow, giving a differential condition in which the change in the cash account, scaled by its value, equals the stock price times time. Integrating yields the cash position from its initial balance and the stock-price path. An explicit result requires a model for the stock price. The derivation assumes the stated trading dynamics and account structure.
Key ideas
- A self-financing portfolio's value change comes from asset price changes, not external cash injections.
- Applying the product rule to changing holdings introduces terms for changes in positions.
- With deterministic holdings, the relevant stock and cash cross terms vanish under the stated setup.
- A declining stock allocation requires an offsetting cash-account adjustment to preserve self-financing.
- An explicit cash balance depends on the stock-price model and the initial cash position.
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Full text
# How can I make this portfolio self-financing?
# How can I make this portfolio self-financing?
$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$
$b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate
So the value of the portfolio:
$$V_t = a_t S_t + b_t \beta_t$$
Is self-financing if
$$dV_t = a_t dS_t + b_t d \beta_t$$
If $a_t = 1-t$, how can I choose $b_t$ such that my portfolio is self-financing?
$$V_t = (1-t)S_t + b_t \beta_t$$
How do I formulate $dV_t$ now? Don't I require more information, in particular, what is $S_t$?
Is there a need to use the stochastic product rule?
## Answer by bcf (score 2, accepted)
https://quant.stackexchange.com/a/17881
We have $$ V_t = a_t S_t + b_t \beta_t. $$
By Ito's product rule, \begin{align*} dV_t & = d(a_t S_t) + d(b_t \beta_t) \\ & = a_t dS_t + S_t da_t + da_t dS_t + b_t d\beta_t + \beta_t db_t + db_td\beta_t. \end{align*}
Since $da_t$ and $db_t$ have no $dW_t$ term, the cross terms are both zero and we have \begin{align*} dV_t & = a_t dS_t + S_t da_t + b_t d\beta_t + \beta_t db_t. \end{align*}
Now just plug in your value for $da_t$ and solve one equation in the unknown $b_t$: \begin{align*} dV_t & = a_tdS_t - S_t dt + b_t d\beta_t + \beta_t db_t \triangleq a_t dS_t + b_t d\beta_t \\ & \iff \beta_t db_t = S_t dt \\ & \iff b_t = b_0 + \int_0^t S_u/\beta_u du. \end{align*}
Now you may be able to solve for $b_t$ explicitly depending on your model for $S_t$.
## Answer by torbonde (score 1)
https://quant.stackexchange.com/a/17873
In the Black-Scholes model, you would have $d S_t = \mu\, d t + \sigma\, d W_t$ where $W$ is a Brownian motion. So if $V_t = a_t S_t + b_t \beta_t$, then $$ dV_t = a_t\, d S_t + S_t\, d a_t + da_t\,dS_t + b_t\,d\beta_t + \beta_t\,d b_t + db_t\, d\beta_t $$ by the product rule. In your case, when $a_t = 1-t$ you will have $$ dV_t = (1-t) \, dS_t - S_t\, dt + b_t\,d\beta_t + \beta_t\,db_t $$ since $da$ and $d\beta$ have no $dW$-term. Hence, you will need to pick $b$ such that $\beta\,db_t = S_t\,dt$, for the portfolio to be self-financing.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.