The lognormal random variable is particularly useful when security returns or interest are compounded over time. First, consider the non-random model based on compound interest. Suppose that a security with value *Y*(*t*) at time *t* pays interest at a rate *r* compounded continuously. This means that the instantaneous rate of change of *Y*(*t*) is equal to *rY*(*t*):

$\frac{\mathrm{d}Y}{\mathrm{d}t}=\mathrm{rY}$

We know from integral calculus that the solution for *Y*(*t*) is:

$Y(t)={Y}_{0}{e}^{\mathrm{rt}}$

where *Y*_{0} is the value of *Y* at time 0. Observe that ln(*Y*(*t*)/*Y*_{0})=*rt*. Of course, ...

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