BSM 1st assumtion vs. 5th assumption

What is the difference between those BSM assumptions below?

1th assumption: The underlying asset price follows a geometric Brownian motion process. The return on the underlying asset follows a lognormal distribution.

5th assumption: The (continuously compounded) yield on the underlying asset is constant.

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.

Thank you for your reply. I will think more about 5th assumption, even though I think it means that the underlying asset’s cash flow is constant.

Hi Magician,

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.

See already my question here:

I am really confused.

+1 on the confusion… Could someone please clarify?

5th assumption: The (continuously compounded) yield on the underlying asset is constant. -> a constant (and continuous) dividend yield is known.

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.

actually I’m not sure

I checked John C. Hull and it says,

  • BSM assumes that % changes in stock price in a short period of time are normally distributed.

  • InST is normally distributed so that ST has a lognormal distribution.

  • continuously compounded rate of return per annum is normally distributed.

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.