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Illustrates an example of forward equation

Illustrates an example of forward equation?

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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

778_forward equation.png

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

2310_probability density function evolving by time.png

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

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