Estimating Conditional Stock-Move Probabilities from Historical Data
Summary
The document explains how to estimate the chance that stock B moves up when stock A moves up. If the joint distribution of their prices or returns is known, the conditional probability follows from the joint probability divided by the probability of A’s move. For historical data, one answer proposes an empirical estimator: count observations where both stocks move, then divide by observations where A moves. This is a direct frequency estimate of the conditional event.
A further approach models the return distributions with marginal GARCH processes and connects them using a copula, which can represent dependence between the stocks. The discussion does not provide a worked comparison, sample size guidance, uncertainty estimates, or evidence that any particular model predicts future co-movement. The estimate depends on how “moves” is defined and on whether historical relationships remain representative. Correlation alone is not enough to determine the conditional probability.
Key ideas
- Conditional probability is the joint probability of both moves divided by the probability of the conditioning move.
- Historical observations can estimate the conditional probability by counting joint moves relative to moves in stock A.
- Copulas can model dependence between return distributions, with GARCH processes as a possible choice for the marginals.
- The definition of a move and the stability of historical dependence affect the estimate.
Tags
Full text
# How do I estimate the joint probability of stock B moving, if stock A moves?
# How do I estimate the joint probability of stock B moving, if stock A moves?
I have two stocks, A and B, that are correlated in some way.
If I know (hypothetically) that stock A has a 60% chance of rising tomorrow, and I know the joint probability between stocks A and B, how do I calculate the probability of stock B moving tomorrow?
For bonus upvotes - do you know of any standard libraries that can calculate the joint probability of stocks A and B, given a time series of historical data?
Update:
The phrase "conditional probability" is also applicable.
- See Wikipedia on Conditional Probability.
- See Tutorial video on Conditional Probability from Khan Academy.
## Answer by user3296 (score 7, accepted)
https://quant.stackexchange.com/a/1423
So you want to calculate $\mathbb{P}[B_1 > B_0 + \varepsilon \;|\; A_1 > A_0 + \varepsilon]$? If you truly have the joint distribution of $A_1$ and $B_1$ and the current prices $A_0$ and $B_0$, this just becomes a simple exercise in integration, by the definition of probability density. Are you asking how to find a conditional probability in general, or is your question about something else?
## Answer by Beer4All (score 9)
https://quant.stackexchange.com/a/1363
Why not using the so simple Monte-Carlo estimator
$ \hat{p}_N =\frac{ \sum_{i=1}^N 1_{|A_{i+1}-A_i|>0 \cap |B_{i+1}-B_i|>0}} {\sum_{i=1}^N 1_{|A_{i+1}-A_i|>0 }}$
where $1_{|A_{i+1}-A_i|>0}$ is $1$ if stock $A$ has moved at time $i+1$
## Answer by babelproofreader (score 5)
https://quant.stackexchange.com/a/1294
> ...do you know of any standard libraries that can calculate the joint probability of stocks A and B, given a time series of historical data?
Using R and the LSPM package with the code posted here might be what you are looking for.
## Answer by Jase (score 3)
https://quant.stackexchange.com/a/8710
You can use copulas. The probability that B rises given A rises is $P(- R_B < 0 | - R_A < 0) = \frac{P(-R_B < 0, - R_A < 0)}{P(-R_A < 0)} = \frac{C(F_{-B}(0),F_{-A}(0))}{F_{-A}(0)}$.
You can specify the marginals as a GARCH process and use either non parametric or parametric copulas to get your final conditional probability.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.