en>Headbomb: Various citation cleanup (identifiers mostly), replaced: |id={{MR|1129778}} → |mr=1129778 using AWB

2011-08-31T04:46:19Z

Various citation cleanup (identifiers mostly), replaced: |id={{MR|1129778}} → |mr=1129778 using AWB

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'''Self-financing portfolio''', 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 ==
Let <math> h_i(t) </math> denote the number of stock number 'i' in the portfolio at time <math> t </math>, and <math> S_i(t) </math> the price of stock number 'i' in a [[frictionless market]] with trading in continuous time. Let

<math>
V(t) = \sum_{i=1}^{n} h_i(t) S_i(t).
</math>

Then the portfolio <math> (h_1(t), \dots, h_n(t)) </math> is self-financing if

<math>
dV(t) = \sum_{i=1}^{n} h_i(t) dS_{i}(t).
</math><ref>{{cite book|first=Tomas|last=Björk|title=Arbitrage theory in continuous time|edition=3rd|page=87|publisher=Oxford University Press|year=2009|isbn=978-0-19-877518-8}}</ref>

=== Discrete time ===
Assume we are given a discrete [[filtered probability space]] <math>(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t=0}^T,P)</math>, and let <math>K_t</math> be the [[solvency cone]] (with or without [[transaction costs]]) at time ''t'' for the market. Denote by <math>L_d^p(K_t) = \{X \in L_d^p(\mathcal{F}_T): X \in K_t \; P-a.s.\}</math>. Then a portfolio <math>(H_t)_{t=0}^T</math> (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 <math>t \in \{0,1,\dots,T\}</math> we have that <math>H_t - H_{t-1} \in -K_t \; P-a.s.</math> with the convention that <math>H_{-1} = 0</math>.<ref>{{cite journal|last=Hamel|first=Andreas|last2=Heyde|first2=Frank|last3=Rudloff|first3=Birgit|date=November 30, 2010|title=Set-valued risk measures for conical market models|url=http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.5986v1.pdf|accessdate=February 2, 2011|format=pdf}}</ref>

If we are only concerned with the set that the portfolio can be at some future time then we can say that <math>H_{\tau} \in -K_0 - \sum_{k=1}^{\tau} L_d^p(K_k)</math>.

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 <math>\Delta t \to 0</math>.

==See also==
* [[Replicating portfolio]]

==References==
{{Reflist}}

[[Category:Mathematical finance]]

Hardy hierarchy - Revision history

en>Headbomb: Various citation cleanup (identifiers mostly), replaced: |id={{MR|1129778}} → |mr=1129778 using AWB