210 Frequently Asked Questions In Quantitative Finance
You would expect equity prices to follow a random
walk around an exponentially growing average. So take
the logarithm of the stock price and you might expect
that to be normal about some mean. That is the non-
mathematical explanation for the appearance of the
lognormal distribution.
More mathematically we could argue for lognormality
via theCentral Limit Theorem.UsingRito represent
the random return on a stock price from dayi−1to
dayiwe have
S 1 =S 0 (1+R 1 ),
the stock price grows by the return from day zero, its
starting value, to day 1. After the second day we also
have
S 2 =S 1 (1+R 2 )=S 0 (1+R 1 )(1+R 2 ).
Afterndays we have
Sn=S 0
∏n
i= 1
(1+Ri),
Figure 2-12:The probability density function for the lognormal ran-
dom walk evolving through time.
