Hi, I am modelling stock returns. It seems there are two different ways in literature.1. Econometrics: we have CAPM framework2. Mathematics: we have a number of models, such as: jump-diffusion model, and pure jump model (Variance Gamma, CGMY)My question is: if I model one day return or one hour return (high frequency) whichframework I should use? Thanks a lot.

When modelling stock returns, there are two spectrums. One is whether you have true economic expectations right through to no arbitrage (ie risk neutral). The other is short term through to long term. For example, CAPM is long term, expectations (though can be calibrated). Alternatively, short term includes GARCH, jump-diffusion, etc, which again can be expectations or can be calibrated to be risk-neutral.So decide whether you want expecations or risk neutral. And make sure you have a short term framework.

Also look at whether you want to explain the returns of a portfolio. In that case one could use a CAPM model or a pure statistical model. The only difference would be that the CAPM model has a physical interpretation for the factors whereas the pure statistical model does not have such an interpretation. As gjlipman mentioned CAPM is favored for long term effects whereas pure statistical / mathematical models tend to be used for shorter term effects.

Thank you, gjlipman and HedgeQuant.I am trying to develop new trading strategy of stocks based on daily or hourly returns.Does that mean I have to use mathematical models instead?

Merton (1980) illustrates, in his paper "On Estimating the Expected Return on the Market", the estimates of variance and covariance from time series data is more accurate than the corresponding expected returns. If you consider the drift coefficient of the lognormal model within a 'Black Scholes' world, Rodgers (2001) illustrates, in his paper "The Relaxed Investor an Parameter Uncertainty", a 95% confidence interval of the drift coefficient would require over 1500 years of daily data. That is, a exogenous measure of market returns have large standard errors which subsequently requires a large amount of observations to have a statistically significant estimate. High frequency data may be able to provide this estimate, however one would doubt it would translate to 1500 high frequency pins.Coming back to your question, 'what model to use' is an important question in finance. Given we cannot measure returns exogenously, as described above, is there a generalised market model that will offer an accurate endogenous measure? If you believe Mehra and Prescott (1986), "The equity premium: A puzzle" then there is no consumption based economic model. I know of one model in development that can, perhaps the forum can offer advice of any other potential models?

hi topzhou,if your trading strategy is to capitalise on existing inefficiency(ies), please make sure that your model does not attribute inefficiency to stock-specific component )if you are to use apt type models). also, building on enPassants point, before giving preference to any model, make sure you have adequate data to callibrate your model on (for instance if you go for factor based models it is unlikely that you will be able to obtain daily/hourly data)best luck..