Self-financing portfolio
title: "Self-financing portfolio" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["portfolio-theories"] topic_path: "general/portfolio-theories" source: "https://en.wikipedia.org/wiki/Self-financing_portfolio" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 (\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 \Delta t \to 0.
Continuous time
Let S=(S_t){t\geq 0} be a d-dimensional semimartingale frictionless market and h=(h_t){t\geq 0} a d-dimensional predictable stochastic process such that the stochastic integrals h^i\cdot S^i exist \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=\left( (h_t^1, \dots, h_t^d)\right)_t is called self-financing if
: 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 h_0 \cdot S_0 corresponds to the initial wealth of the portfolio, while \int_0^t 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.
References
References
- (November 30, 2010). "Set-valued risk measures for conical market models".
- Björk, Tomas. (2009). "Arbitrage theory in continuous time". Oxford University Press.
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