Chapter 2: FAQs 121
Figure 2-6:The probability density function for the lognormal ran-
dom walk evolving through time.
The backward equation Also known as thebackward Kol-
mogorov equationthis is
∂p
∂t+^12 B(y,t)^2∂^2 p
∂y^2+A(y,t)∂p
∂y= 0.This must be solved backwards intwith specified final
data.
For example, if we wish to calculate the expected value
of some functionF(S)attimeTwe must solve this
equation for the functionp(S,t)with
p(S,T)=F(S).Option prices If we have the lognormal random walk for
S, as above, and we transform the dependent variable
using a discount factor according to
p(S,t)=er(T−t)V(S,t),then the backward equation forpbecomes an equation
forVwhich is identical to the Black–Scholes partial
differential equation. Identical but for one subtlety, the