Self-financing portfolio

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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[edit]

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[edit]

Assume we are given a discrete filtered probability space , and let 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

for all we have that with the convention that .[2]

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 .

See also[edit]


  1. ^ Björk, Tomas (2009). Arbitrage theory in continuous time (3rd ed.). Oxford University Press. p. 87. ISBN 978-0-19-877518-8. 
  2. ^ Hamel, Andreas; Heyde, Frank; Rudloff, Birgit (November 30, 2010). "Set-valued risk measures for conical market models" (pdf). Retrieved February 2, 2011.