I want to estimate an empirical pricing kernel for an index. Hence, I need to estimate a physical and risk neutral density. For estimating the physical density, only the index data in an observed time interval is needed. Moreover, I know for estimating the risk neutral density I can use option prices with different strike prices and time to maturity. However, I think the observation date for the option prices should somehow correspondent to the time interval used for estimating the physical density.
Therefore, my question is how the observation date for the option prices has to correspond to the used time interval.
Note: The empirical pricing kernel equal is also referred to as the stochastic discount factor.