So I was just reading through a proof of the Black-Scholes-Merton model for options pricing, and a part of the proof says that the model considers a non-dividend paying stock and assumes that the return on the stock in a short period of time is normally distributed.
We define:
to be the expected return on the stock and
to be the volatility of the stock price.
Then from this we can deduce that the mean of the return on the stock in a short period of time
is
and the standard deviation of this return in this period of time is
. Now the assumption underlying the Black-Scholes-Merton model is that
)
where
is the change in the stock price
in time
and
)
is a normal distribution with mean m and variance n.
However then the proof says that the assumption in
implies the stock price at any future time has a
lognormal distribution.
My question is: How does
imply that the stock price has a lognormal distribution? My understanding of what the equation states is that the return on the stock (
) in a short time period (
) is given by a normal distribution, how does this imply then that the stock price
itself is given by a lognormal distribution (during the short period of time)?
I know that if Y is log-normally distributed random variable, then X = log(Y) is normally distributed, so if we take
how does this then imply Y is S?
Thanks!
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