Jump to content

Self-financing portfolio

From Wikipedia, the free encyclopedia

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.

Mathematical definition

[edit]

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 .[1]

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 .

Continuous time

[edit]

Let 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 , and 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

.[2]

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.

See also

[edit]

References

[edit]
  1. ^ Hamel, Andreas; Heyde, Frank; Rudloff, Birgit (November 30, 2010). "Set-valued risk measures for conical market models". arXiv:1011.5986v1 [q-fin.RM].
  2. ^ Björk, Tomas (2009). Arbitrage theory in continuous time (3rd ed.). Oxford University Press. p. 87. ISBN 978-0-19-877518-8.