Hardy hierarchy

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 $h_i(t)$ 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

$dV(t)=\sum _{i=1}^n{h}_i(t)d{S}_i(t).$

Discrete time

Assume we are given a discrete filtered probability space $(\mathrm{\Omega },\mathcal{ℱ},\{{\mathcal{ℱ}}_t{\}}_{t=0}^T,P)$, 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 $H_{-1}=0$.

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 $\mathrm{\Delta }t\to 0$.

See also

• Replicating portfolio

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.