Self financing portfolio

Mathematical definition

Let h i ( t ) denote the number of shares 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 ) = ∑ i = 1 n h i ( t ) S i ( t ) .

Then the portfolio ( h 1 ( t ) , … , h n ( t ) ) is self-financing if

d V ( t ) = ∑ i = 1 n h i ( t ) d S i ( t ) .

Discrete time

Assume we are given a discrete filtered probability space ( Ω , F , { 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 ∈ L d p ( F T ) : X ∈ 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

If we are only concerned with the set that the portfolio can be at some future time then we can say that H τ ∈ − K 0 − ∑ k = 1 τ 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 Δ t → 0 .