Using first order expansion we have

                          

                                     

                                     

Let us now first divide both sides of the equations with  and take limits as . Furthermore we also divide and simultaneously multiply the first term and the second term on the right hand side of equality by  and  respectively. This results in the following form which is as follows

This is the fundamental partial differential equation (PDE) for pricing derivatives under the underlying assumption that logarithmic price of the underlying financial asset has Brownian motion.

In case one is interested to understand how the theoretical relationship between  and  varies then Figure gives a fair idea how that behaves.

Figure 10.12: Illustration for PDE for pricing derivatives

Recall from Markov Chain Theory that . Now consider a simple random walk with equal probability of a step up and step down i.e.,