Risk Neutral Pricing and Financial Mathematics: A Primer

Figure 2.4 The lognormal density curve.

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 Y₀ is the value of Y at time 0. Observe that ln(Y(t)/Y₀)=rt. Of course, ...