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

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.
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