# 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 $\int_a^b f(x) dx$. 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?

• You can imply the future distribution of a stock from the second derivative of the u discounted option prices. Try comparing this to the distributions you're generating (which won't account for the market's expectation of the outcomes of any upcoming events...).
– will
Jun 17, 2017 at 16:01

I have written an entire paper on this approach at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2828744

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.