First, it’s the price that’s lognormally distributed, not the return.
This simply means that the underlying asset has no cash flows.
I believe that it’s possible that the price of the underlying could have a lognormal distribution even when the underlying has cash flows, but I’m not certain about that. In any case, this assumption makes the cash flow situation clear.
I thought that the cash flows were infinitesimal and continuous. An approximation (albeit unrealistic) unless you deal with portfolios of tens of dividend paying stocks who happen to pay dividends uniformly over the period.
I also thought that prices follows a lognormal distribution, but then more times in the Schweser (also on the daily questions) could find that returns are lognormally distributed as DavidYang said. I am confused too.
From Schweser notes:
One of the assumptions of the Black-Scholes-Merton model is th at the underlying asset returns are lognormally distributed. If the continuously compounded return (the natural log of the stock price) is distributed normally, then the returns are considered to be lognormally distributed.
If returns were lognormally distributed, then log(return) would be normally distributed. But that would mean that the return is never negative (nor zero) as you can take the log only of positive numbers.
Take my word for it: stocks can have negative returns.
If ST is lognormally distributed (ST>0), then is lnST/ST-1 normally distributed?
CFAI curriculum says, if ST follows GBM, which implies a lognormal distribution of the return, meaning that the logarithmic return, which is the continuously compounded return, is normally distributed.