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What is the general usage of the term daily log returns $Y_t$ of an asset? (1) or (2)?
$$(1) \text{ } Y_t = log (\frac{p_t}{p_{t-1}})$$ OR $$(2) \text{ } Y_t = log (\frac{p_t-p_{t-1}}{p_{t-1}})$$ for $p_t$ being the close price of day $t$.

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  • $\begingroup$ I am voting to close this question as off topic for being too basic. Hint: what would happen to your second expression when $p_t < p_{t - 1}$? $\endgroup$
    – LocalVolatility
    Nov 1, 2017 at 7:53

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The first one is a correct way. The second one is simply taking log to the relative returns which can be negative and natural log is not defined for negative values...

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    $\begingroup$ small correction: The first one is the correct way if and only if there are no distributions (eg. dividends etc...) at time $t$. The return from $t-1$ to $t$ is $R_t = \frac{P_t + D_t}{P_{t-1}}$. The log return is $ r_t = \log R_t = \log \left( P_t + D_t \right) - \log P_{t-1}$. $\endgroup$
    – Matthew Gunn
    Nov 1, 2017 at 14:59