Self-financing portfolio

Template:Refimprove In financial mathematics, a self-financing portfolio is a portfolio having the feature that, 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.Template:Cn This concept is used to define for example admissible strategies and replicating portfolios, the latter being fundamental for arbitrage-free derivative pricing.

Mathematical definition

Discrete time

Assume we are given a discrete filtered probability space $(\mathrm{\Omega },ℱ,\{{ℱ}_t{\}}_{t=0}^T,P)$, and let $K_t$ be the solvency cone (with or without transaction costs) at time t for the market. Denote by $L_d^p(K_t)=\{X\in L_d^p({ℱ}_T):X\in K_{t}P-a.s.\}$. 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 $\mathrm{\Delta }t\to 0$.

Continuous time

Let $S=(S_t{)}_{t\ge 0}$ be a d-dimensional semimartingale frictionless market and $h=(h_t{)}_{t\ge 0}$ a d-dimensional predictable stochastic process such that the stochastic integrals $h^i\cdot S^i$ exist $\mathrm{\forall }i=1,\dots ,d$. The process $h_t^i$ denote the number of shares of stock number $i$ in the portfolio at time $t$, and $S_t^i$ the price of stock number $i$. Denote the value process of the trading strategy $h$ by

$V_t=\sum _{i=1}^n{h}_t^i{S}_t^i.$

Then the portfolio/the trading strategy $h={((h_t^1,\dots ,h_t^d))}_t$ is called self-financing if

$V_t=\sum _{i=1}^n\{h_0^i{S}_0^i+\underset{0}{\overset{t}{\int }}{h}_u^i\mathrm{d}{S}_u^i\}=h_0\cdot S_0+\underset{0}{\overset{t}{\int }}{h}_u\cdot \mathrm{d}{S}_u$.

The term $h_0\cdot S_0$ corresponds to the initial wealth of the portfolio, while $\underset{0}{\overset{t}{\int }}{h}_u\cdot \mathrm{d}{S}_u$ is the cumulated gain or loss from trading up to time $t$. This means in particular that there have been no infusion of money in or withdrawal of money from the portfolio.

See also

• Replicating portfolio
• Self-financing