I am trying calculate expected return and risk (stdev) based on historical data using Mean Variance Analysis framework. Let's say the portfolio has 10 stocks, 9 of them have more than 10 years history but one of them only have 1 month historical data. If I want every stock has the same length of historical data for correlation matrix, shall i just use 1 month of data to estimate? or fill the one with short historical data with 0?
There are two common solutions.
You exclude from your universe stocks whose history is too short (i.e. only recently went public). Sometimes it breaks your heart to do that, because you really like the stock. (If the time series is long enough, but still has small gaps, there are ways to "fix" the covariance matrix.)
You use models to predict the covariance. The most famous example of this approach is MSCI Barra predicted beta (see, for example, https://doi.org/10.3905/jpm.2014.41.1.057 for its description). However they don't try to predict the covarance to all other stocks. Rather, they use multifactor model, and predict, based on fundamentals, the correlation to the few factors that they assume to drive the market. They do it right at IPO time with no history.