# Self-financing portfolio

A self-financing portfolio is an important concept in financial mathematics.

A portfolio is self-financing if there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one.

## Mathematical definition

Let ${\displaystyle h_{i}(t)}$ denote the number of shares of stock number 'i' in the portfolio at time ${\displaystyle t}$, and ${\displaystyle S_{i}(t)}$ the price of stock number 'i' in a frictionless market with trading in continuous time. Let

${\displaystyle V(t)=\sum _{i=1}^{n}h_{i}(t)S_{i}(t).}$

Then the portfolio ${\displaystyle (h_{1}(t),\dots ,h_{n}(t))}$ is self-financing if

${\displaystyle dV(t)=\sum _{i=1}^{n}h_{i}(t)dS_{i}(t).}$[1]

### Discrete time

Assume we are given a discrete filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}$, and let ${\displaystyle K_{t}}$ be the solvency cone (with or without transaction costs) at time t for the market. Denote by ${\displaystyle L_{d}^{p}(K_{t})=\{X\in L_{d}^{p}({\mathcal {F}}_{T}):X\in K_{t}\;P-a.s.\}}$. Then a portfolio ${\displaystyle (H_{t})_{t=0}^{T}}$ (in physical units, i.e. the number of each stock) is self-financing (with trading on a finite set of times only) if

for all ${\displaystyle t\in \{0,1,\dots ,T\}}$ we have that ${\displaystyle H_{t}-H_{t-1}\in -K_{t}\;P-a.s.}$ with the convention that ${\displaystyle H_{-1}=0}$.[2]

If we are only concerned with the set that the portfolio can be at some future time then we can say that ${\displaystyle H_{\tau }\in -K_{0}-\sum _{k=1}^{\tau }L_{d}^{p}(K_{k})}$.

If there are transaction costs then only discrete trading should be considered, and in continuous time then the above calculations should be taken to the limit such that ${\displaystyle \Delta t\to 0}$.