Hi All, I have been reading for days trying to get to the bottom of this question, however I don’t have a solid enough math background to reach the answer. I want to simulate the performance of an asset and I am assuming it is normally distributed and it will behave in a similar manner in the future.

1-) Take the daily changes in an assets price

2-) Apply natural log on those price (LN)

3-) Calculate the Average and Sd of (2)

Multiple the average by 250 to arrive at the annual expected return and Sd by Sqrt(250) to arrive at annual volatility

If I am running a simulation with annual periods, how do I correctly use the data from step 4. If I draw value from a distribution with the return and volatility from step 4. Should I apply exponential to that randomly generated before using it in the simulation?

I would say just use the log return data calculated from step 4 - stock (log) returns are normally distributed meaning the ending price will be log-normally distributed.

Or you could calculate the arithmetic returns and then apply the exponential:

So let us say step 4 gave a mean of 4% and Sd of 5%

If we draw a value from a normal distribution with a mean of 4% and Sd of 5%, say that value came out to be 4.5%, should I use 4.5% in my simulation as the return for that year as in 100 *1.044 or should I used 100* exp(0.045) since the rate calculated in step 4 is actually a continuously compounded rate?