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.
[citation needed] This concept is used to define for example admissible strategies and replicating portfolios, the latter being fundamental for arbitrage-free derivative pricing.
Assume we are given a discrete filtered probability space
(
, {
t
}
,
be the solvency cone (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 If we are only concerned with the set that the portfolio can be at some future time then we can say that
τ
τ
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
be a d-dimensional semimartingale frictionless market and
a d-dimensional predictable stochastic process such that the stochastic integrals
exist
The process
denote the number of shares of stock number
in the portfolio at time
the price of stock number
Denote the value process of the trading strategy
by Then the portfolio/the trading strategy
is called self-financing if The term
corresponds to the initial wealth of the portfolio, while
is the cumulated gain or loss from trading up to time
This means in particular that there have been no infusion of money in or withdrawal of money from the portfolio.