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 $h_i(t)$ denote the number of stock number 'i' in the portfolio at time $t$, and $S_i(t)$ the price of stock number 'i' in a frictionless market with trading in continuous time. Let

$V(t) = \sum_{i=1}^{n} h_i(t) S_i(t).$

Then the portfolio $(h_1(t), \dots, h_n(t))$ is self-financing if

$dV(t) = \sum_{i=1}^{n} h_i(t) dS_{i}(t).$^[1]

Discrete time

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

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

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.