# Estimating and forecasting stock and option prices with GARCH models

I am new in the field of time series. I wonder why there is not enough literature about GARCH models used to predict stock or option prices? In other words, is it reasonable to use a general regression model with autocorrelated error and conditional heteroscedasticity for the error variance

\begin{align} Y_{t} = X_{t}^{'}\beta_{t} + \epsilon_{t} \end{align}

where

\begin{align} &\epsilon_{t} = \varphi_{1}\epsilon_{t-1}+ \cdots + \varphi_{p}\epsilon_{t-p} + n_{t} \\ &n_{t} = \sigma_{t}e_{t} \\ & \sigma_{t}^{2} = \theta_{0} + \phi_{1}\sigma_{t-1}^{2} + \cdots + \phi_{r}\sigma_{t-r}^{2}+\theta_{1}n_{t-1}^{2} + \cdots + \theta_{s}n_{t-s}^{2} \end{align} and the $$e_{t}$$ are i.i.d. $$\mathcal{N}(0,1)$$ and independent of past realization of $$n_{t-i}$$? Is there some technical issue when this model is fit to data of stock and option prices?

I know that for options we can use the Black-Scholes-Merton model. However, these models have stationary and independent increments, but this assumption does not happen in the real world. As an example, when we are in a bullish market it is more likely to occur an uptrend rather than a downtrend. As a result, it is reasonable to think about a model of autocorrelated errors.