Illustrates an example of forward equation

Question

Illustrates an example of forward equation?

Answer 1

Example: A significant example is the distribution of equity prices within the future. When we have the random walk

dS = µSdt + σSdX

then the forward equation turns into

∂p/∂t' = ½ ∂²/∂S²(σ² S'²p) - ∂/∂S'(µS'p).

A particular solution of this representing a variable which begins with certainty along with value S at time t is as

It is plotted as a function of both S' and t' demonstrated in figure.

Figure: The probability density function evolving by time for the lognormal random walk.