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.[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

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Discrete time

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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

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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

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References

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  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.