Self-financing portfolio


In financial mathematics, a self-financing portfolio is a portfolio having the feature that, 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 shares of stock number 'i' in the portfolio at time, and the price of stock number 'i' in a frictionless market with trading in continuous time. Let
Then the portfolio is self-financing if

Discrete time

Assume we are given a discrete filtered probability space, and let be the solvency cone at time t for the market. Denote by. Then a portfolio is self-financing if
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 then only discrete trading should be considered, and in continuous time then the above calculations should be taken to the limit such that.