Self-financing portfolio
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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 h_i(t) denote the number of shares of stock number 'i' in the portfolio at time t , and S_i(t) the price of stock number 'i' in a
frictionless market Frictionless can refer to: * Frictionless market * Frictionless continuant * Frictionless sharing * Frictionless plane The frictionless plane is a concept from the writings of Galileo Galilei. In his 1638 '' The Two New Sciences'', Galileo prese ...
with trading in continuous time. Let : V(t) = \sum_^ h_i(t) S_i(t). Then the portfolio (h_1(t), \dots, h_n(t)) is self-financing if : dV(t) = \sum_^ h_i(t) dS_(t).


Discrete time

Assume we are given a discrete
filtered probability space Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
(\Omega,\mathcal,\_^T,P), and let K_t be the
solvency cone The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone of portfolios that can ...
(with or without
transaction costs In economics and related disciplines, a transaction cost is a cost in making any economic trade when participating in a market. Oliver E. Williamson defines transaction costs as the costs of running an economic system of companies, and unlike pro ...
) at time ''t'' for the market. Denote by L_d^p(K_t) = \. Then a portfolio (H_t)_^T (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 t \in \ we have that H_t - H_ \in -K_t \; P-a.s. with the convention that H_ = 0. If we are only concerned with the set that the portfolio can be at some future time then we can say that H_ \in -K_0 - \sum_^ L_d^p(K_k). 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 \Delta t \to 0.


See also

*
Replicating portfolio In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties (especially cash flows). This is meant in two distinct senses: static replication, where the portfolio has ...


References

{{Reflist Mathematical finance