Chapter 2: FAQs 211
the stock price is the initial value multiplied bynfac-
tors, the factors being one plus the random returns.
Taking logarithms of this we get
ln(Sn)=ln(S 0 )+
∑n
i= 1
ln(1+Ri),
the logarithm of a product being the sum of the loga-
rithms.
Now thinkCentral Limit Theorem. If eachRiis ran-
dom, then so is ln(1+Ri). So the expression for ln(Sn)
is just the sum of a large number of random num-
bers. As long as theRiare independent and identically
distributed and the mean and standard deviation of
ln(1+Ri) are finite then we can apply the CLT and con-
clude that ln(Sn) must be normally distributed. Thus
Snis normally distributed. Since herenis number of
‘days’ (or any fixed time period) the mean of ln(Sn)is
going to be linear inn, i.e., will grow linearly with time,
and the standard deviation will be proportional to the
square root ofn, i.e., will grow like the square root of
time.
References and Further Reading
Wilmott, P 2006Paul Wilmott On Quantitative Finance, second
edition. John Wiley & Sons
