In
financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, a self-financing portfolio is a
portfolio
Portfolio may refer to:
Objects
* Portfolio (briefcase), a type of briefcase
Collections
* Portfolio (finance), a collection of assets held by an institution or a private individual
* Artist's portfolio, a sample of an artist's work or a ...
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
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
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 present ...
with trading in continuous time. Let
:
Then the portfolio
is self-financing if
:
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 filter ...
, and let
be the
solvency cone
The solvency cone is a concept used in financial mathematics which models the possible trades in the Market (economics), financial market. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone of po ...
(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 produ ...
) 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
.
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
*
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