When Time-Zero Conditional Expectation Equals Unconditional Expectation
Summary
This question examines when conditioning on the time-zero information set in a financial model gives the same expectation as an unconditional expectation. It highlights a potential tension: if the initial sigma algebra is trivial, every measurable random variable in it must be almost surely constant, while initial asset prices may be modeled as random variables.
The response suggests that equality can instead follow from independence: if a random variable is independent of the time-zero information, its conditional expectation given that information equals its unconditional expectation. This is a useful distinction between a trivial information set and independence from a nontrivial one. The answer is brief and does not resolve all modeling conventions for initial prices or filtrations; an explicit reference or fuller setup would be needed to determine which assumption applies in a particular finance text.
Key ideas
- A trivial sigma algebra contains only events of probability zero or one.
- A random variable measurable with respect to a trivial sigma algebra is almost surely constant.
- Conditional expectation equals unconditional expectation when the variable is independent of the conditioning information.
- Whether initial prices belong to a nontrivial time-zero information set depends on the model’s setup.
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# Unconditional Expectation vs. Conditional Expectation at time $0$
# Unconditional Expectation vs. Conditional Expectation at time $0$
In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional expectation. This is on a probability space equipped with the filtration generated by the standard Wiener Process. I know that this is true when $\mathcal F_0$ is the trivial sigma-algebra, but it seems like in the finance perspective, the $\mathcal F_0$ information also includes the time $0$ asset prices, which are random variables, and so $\mathcal F_0$ doesn't seem to be trivial in these cases.
I have seen that in some books, the stock/relevant prices are considered deterministic at time $0$, and therefore $\mathcal F_0$ is trivial. Is this a valid reasoning? I don't understand how the measurability of Random Variables is consistent then, since if one were to calculate the conditional expectation of a random variable, $Y$ that is $\mathcal F_0$- measurable (but not one of these deterministic time $0$ asset prices), then assuming $\mathcal F_0$ is trivial leads to $\mathbb E[Y|\mathcal F_0] = E[Y]$ and $\mathbb E[Y|\mathcal F_0] = Y$, which makes it seem like $Y$ is deterministic. And the notation is confusing since only constants are supposed to be measurable with respect to the trivial sigma algebra. So if $\mathcal F_0$ is considered trivial it seems like all random variables at time $0$ have to be deterministic.
I am not sure what I'm missing here. Thanks in advance!
## Answer by Tobsn (score 1)
https://quant.stackexchange.com/a/58007
An explicit reference could be helpful. It seems to me like an independence statement. For if $Y$ is independent of $\mathcal{F}_{0}$, then $\mathbb{E}[Y|\mathcal{F}_{0}]=\mathbb{E}[Y]$.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.