Self-Financing Portfolio Dynamics Do Not Depend on the Probability Measure

Summary

A self-financing portfolio changes value only through changes in the assets it holds, with no external deposits or withdrawals. For holdings phi_i in assets priced S_i, its infinitesimal value change is the sum of each holding multiplied by that asset’s price change. This accounting condition defines self-financing behavior.

The answer explains that the condition applies whether asset dynamics are expressed under the physical probability measure P or a martingale measure Q. Changing the probability measure changes how price movements are described probabilistically, but it does not add cash flows to the portfolio or alter the self-financing identity. The discussion is brief and assumes the usual continuous-time portfolio setup; it does not address complications such as transaction costs, trading constraints, or portfolio rebalancing conventions.

Key ideas

• A self-financing portfolio has no external cash flows after its initial funding.
• Its value change equals the sum of asset price changes weighted by current holdings.
• The self-financing accounting identity holds under both physical and risk-neutral measures.
• A probability measure changes the modeled dynamics, not the portfolio’s funding condition.

Tags

# Self-financing portfolio under $Q$-dynamics


# Self-financing portfolio under $Q$-dynamics












I know what given stocks $S_1, ..., S_N$ with SDE's, a portfolio must have a particular value dynamics shape (which depends on the dynamics of $S_1,...,S_N$), if that portfolio is to be self-financing.

However, does this have to hold under every probability measure? Usually we are given dynamics of stocks in P-world, but we also later study the martingale measure $Q$, and we know that the stocks have some known $Q$-dynamics as well.

So, given a portfolio, do the same restrictions on its value dynamics apply in $Q$-world?

## Answer by Quantuple (score 3)

https://quant.stackexchange.com/a/32795

Infinitesimally, a self-financing portfolio is a portfolio

$$ V_t = \sum_{i=1}^N \phi_i(t) S_{t,i} $$

whose value changes only because the values of the assets in which it invests change (no in/out exogenous cash flows, hence its name), i.e.

$$ dV_t = \sum_{i=1}^N \phi_i(t) dS_{t,i} $$

Whether these price changes $dS_{t,i}$ are described under this or that probability measures does not matter.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.