Misplaced Pages

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Self-financing portfolio" – news · newspapers · books · scholar · JSTOR (March 2024) (Learn how and when to remove this message)

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. 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 ${\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}$, and let ${\displaystyle K_{t}}$ be the solvency cone (with or without transaction costs) at time t for the market. Denote by ${\displaystyle L_{d}^{p}(K_{t})=\{X\in L_{d}^{p}({\mathcal {F}}_{T}):X\in K_{t}\;P-a.s.\}}$. Then a portfolio ${\displaystyle (H_{t})_{t=0}^{T}}$ (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 ${\displaystyle t\in \{0,1,\dots ,T\}}$ we have that ${\displaystyle H_{t}-H_{t-1}\in -K_{t}\;P-a.s.}$ with the convention that ${\displaystyle H_{-1}=0}$.

If we are only concerned with the set that the portfolio can be at some future time then we can say that ${\displaystyle H_{\tau }\in -K_{0}-\sum _{k=1}^{\tau }L_{d}^{p}(K_{k})}$.

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

Continuous time

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

${\displaystyle V_{t}=\sum _{i=1}^{n}h_{t}^{i}S_{t}^{i}.}$

Then the portfolio/the trading strategy ${\displaystyle h=\left((h_{t}^{1},\dots ,h_{t}^{d})\right)_{t}}$ is called self-financing if

${\displaystyle V_{t}=\sum _{i=1}^{n}\left\{h_{0}^{i}S_{0}^{i}+\int _{0}^{t}h_{u}^{i}\mathrm {d} S_{u}^{i}\right\}=h_{0}\cdot S_{0}+\int _{0}^{t}h_{u}\cdot \mathrm {d} S_{u}}$.

The term ${\displaystyle h_{0}\cdot S_{0}}$ corresponds to the initial wealth of the portfolio, while ${\displaystyle \int _{0}^{t}h_{u}\cdot \mathrm {d} S_{u}}$ is the cumulated gain or loss from trading up to time ${\displaystyle t}$. 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

References

1. Hamel, Andreas; Heyde, Frank; Rudloff, Birgit (November 30, 2010). "Set-valued risk measures for conical market models". arXiv:1011.5986v1 .
2. Björk, Tomas (2009). Arbitrage theory in continuous time (3rd ed.). Oxford University Press. p. 87. ISBN 978-0-19-877518-8.

• Portfolio theories