Why Self-Financing Strategies Must Be Predictable
Summary
The document explains why a self-financing trading strategy is modeled as predictable, or previsible, while asset prices and portfolio value remain adapted. The key timing idea is that holdings for the next interval must be chosen using information available before the next price movement. Predictability makes that trading decision explicit in advance.
The discussion also highlights why the distinction becomes important for jump processes: an adapted strategy alone may not specify whether a holding changes at the instant of a jump. Predictability resolves this ambiguity by requiring holdings to be determined from prior information. The text offers intuition rather than a formal derivation, and briefly notes that for continuous processes the distinction may be less apparent.
Key ideas
- A self-financing strategy specifies holdings before the subsequent price movement.
- Predictable holdings are determined using information available before the time they apply.
- Asset prices and the resulting portfolio value can remain adapted without being predictable.
- For jump processes, predictability clarifies whether holdings change at the jump time.
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Full text
# Why should a self-financing strategy be previsible?
# Why should a self-financing strategy be previsible?
There is an amazing answer on mathematics stackexchange which defines what a self-financing strategy is—both in the discrete and continuous sense. Please check out this short answer to better understand my question.
I have a short follow-up question: Baxter and Rennie—while defining a self-financing strategy—requires the portfolio process to be previsible. The way the answer in the link arrives at the definition of self-financing, which is also the way I derive it, doesn't seem to require previsiblilty.
We are holding $(\Delta_t, E_t)$ over $t$ to $t+1$, which will be know to us at time $t$, even if the process is just adapted; previsiblilty is not required to know $(\Delta_t, E_t)$ at time $t$.
Extra for those who are interested: As always, things get muddier in the continuous time setting.
I think I have an intuition for what previsibility means when dealing with continuous processes: if a process $\phi$ is left-continuous, then we can know it's value $\phi(t)$ at a particular time $t$ with arbitrary precision by pushing the inputs close enough to $t$ from below, without actually having to reach $t$; this makes the value of $\phi(t)$ predictable with information upto but not including time $t$.
But it isn't clear to me why this previsibility is required while arriving at a sensible definition of a self-financing strategy —as the answer in the link succeeds to do.
## Answer by Kevin (score 5)
https://quant.stackexchange.com/a/47316
A self-financing strategy needs to be previsible (aka predictable) since at time $t$, you need to decide (with the information from $\mathcal{F}_t$) how much you want to be invested in the different assets at time $t+1$. So, you need to decide in advance which makes the trading strategy predictable.
Of course, the asset prices (and hence the value process of your strategy) remain adapted and are not previsible.
## Answer by Magic is in the chain (score 2)
https://quant.stackexchange.com/a/47323
When you think in continuous time, for continuous processes, the distinction does not matter much. But now consider a jump process. You want the strategy to be predictable because adapted won’t do- did you change your holding at the time of the jump? Predictable removes any ambiguity.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.