Self-financing portfolio, 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 denote the number of stock number 'i' in the portfolio at time , and the price of stock number 'i' in a frictionless market

Frictionless market

A Frictionless market is a financial market without transaction costs. Friction is a type of market incompleteness. Every complete market is frictionless, but the converse does not hold. In a frictionless market the solvency cone is the halfspace normal to the unique price vector. The...

 with trading in continuous time. Let



Then the portfolio is self-financing if

Alternative definition

Assume we are given a discrete filtered probability space , and let be the solvency cone

Solvency cone

The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs...

 (with or without transaction costs) at time t for the market. Denote by . Then a portfolio (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 we have that with the convention that .

If we are only concerned with the set that the portfolio can be at some future time then we can say that .

If there are transaction costs than only discrete trading should be considered, and in continuous time then the above calculations should be taken to the limit such that .