Self-financing portfolio

A self-financing portfolio is 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 ${\displaystyle h_{i}(t)}$ denote the number of shares of stock number 'i' in the portfolio at time $t$, and ${\displaystyle S_{i}(t)}$ the price of stock number 'i' in a frictionless market with trading in continuous time. Let

${\displaystyle V(t)=\sum _{i=1}^{n}h_{i}(t)S_{i}(t).}$

Then the portfolio ${\displaystyle (h_{1}(t),\dots ,h_{n}(t))}$ is self-financing if

${\displaystyle dV(t)=\sum _{i=1}^{n}h_{i}(t)dS_{i}(t).}$^[1]

Discrete time

Assume we are given a discrete filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}$, and let $K_{t}$ be the solvency cone (with or without transaction costs) at time t for the market. Denote by ${\displaystyle L_{d}^{p}(K_{t})=\{X\in L_{d}^{p}({\mathcal {F}}_{T}):X\in K_{t}\;P-a.s.\}}$. Then a portfolio ${\displaystyle (H_{t})_{t=0}^{T}}$ (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 ${\displaystyle t\in \{0,1,\dots ,T\}}$ we have that ${\displaystyle H_{t}-H_{t-1}\in -K_{t}\;P-a.s.}$ with the convention that ${\displaystyle H_{-1}=0}$.^[2]

If we are only concerned with the set that the portfolio can be at some future time then we can say that ${\displaystyle H_{\tau }\in -K_{0}-\sum _{k=1}^{\tau }L_{d}^{p}(K_{k})}$.

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

References

1. Björk, Tomas (2009). Arbitrage theory in continuous time (3rd ed.). Oxford University Press. p. 87. ISBN 978-0-19-877518-8.
2. Hamel, Andreas; Heyde, Frank; Rudloff, Birgit (November 30, 2010). "Set-valued risk measures for conical market models" (pdf). Retrieved February 2, 2011.