Bayesian Predictive Distributions for Returns with Changing Volatility
Summary
The document considers fitting a probability density to historical stock returns and using it to estimate the chance of a future return falling within an interval. It raises a key limitation: pooling returns into one fitted distribution can obscure changes in volatility and other time variation. The accepted response proposes a Bayesian predictive distribution, which averages the return likelihood over uncertainty about its parameters given the observed data, rather than plugging in a single parameter estimate.
The answer argues that this approach accounts for parameter uncertainty and distinguishes prediction of future observations from estimating model parameters. However, its broader claims about the nonexistence of variance for returns and the universal superiority or admissibility of Bayesian methods are asserted rather than demonstrated in the excerpt, and should not be treated as settled general conclusions. No empirical comparison or specific volatility model is provided, so practical performance depends on the chosen likelihood, prior, and adequacy of the model.
Key ideas
- A single fitted return distribution may conceal volatility changes over time.
- A Bayesian predictive distribution integrates over parameter uncertainty conditional on observed data.
- Predicting future returns and estimating parameters are distinct statistical tasks.
- The predictive approach depends on the selected likelihood and prior, and the answer's sweeping methodological claims are not substantiated by evidence in the excerpt.
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Full text
# Predict probability of returns: How does changing volatility affect the return pdf?
# Predict probability of returns: How does changing volatility affect the return pdf?
I am trying to predict the future probability of stock returns based on the return distribution. Therefore I calculate the returns as $\frac{P(t)}{P(t-1)}$ for the whole daily data and fit a probability density function $f(x)$ to the data. Now the probability that the future return lies in the interval $[a,b]$ should be given as $\int_a^b f(x) dx$.
1.) What are some caveats about this approach?
I am particular concerned about how changing volatility could bias my results. As far as I understand it, my fitted pdf will take an average volatility value and yield the return probabilities based on that.
2.) How could I account for changing volatility? Pre selecting data on the most recent data?(but: fit will not be well with less data) Maybe usage of of volatility models?
3.)Are there different approaches to calculate the probability of future returns?
## Answer by Dave Harris (score 6, accepted)
https://quant.stackexchange.com/a/34750
I have written an entire paper on this approach at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2828744
As to your specifics
1) "Volatility" as defined by variance does not exist, which is why it is changing. The first moment is undefined so the second cannot exist. See the paper as to why. Your fitted pdf will treat the outcomes as having a single, joint scale parameter. You won't want to do fitting like this, though, because you won't be able to use a sufficient statistic to create it and so must lose information as to the true location. A Bayesian method will correct for this.
2) You do not need to, it is supposed to change. You will have to use a Bayesian method because there does not exist an admissible non-Bayesian method. The Bayesian likelihood function always is minimally sufficient and a statistic created using Bayesian methods is always admissible.
3) Yes, it is called the Bayesian Predictive distribution. It is defined as $$\Pr(\tilde{x}|\mathbf{x})=\int_{\theta\in\Theta}\Pr(\tilde{x}|\theta)\Pr(\theta|\mathbf{x})\mathrm{d}\theta,\forall\theta\in\Theta,$$ where $\Theta$ is the parameter space. What is important to note is that the prediction, $\tilde{x}$, only depends upon $\mathbf{x}$, the data. The integration process marginalizes the parameters, removing their effect. Your prediction does not depend upon a parameter estimate so $\hat{\theta}$ isn't used. This differs from Frequentist methods which depend upon a parameter estimate. The question you asked is "can I produce a distribution?" You did not ask "what are the parameter estimates?" Those are two different questions.
Do note that Bayesian methods are neither biased nor unbiased. They do not care about bias, it's not important in Bayesian methods. You should treat them as more accurate, but biased as both statements are generally true. This is because of the fact that you cannot stochastically dominate a Bayesian estimator.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.