Options, Futures, and Other Derivatives, 10th Edition

15.6 DERIVATION OF THE BLACK–SCHOLES–MERTON DIFFERENTIAL EQUATION

In this section, the notation is different from elsewhere in the book. We consider a derivative’s price at a general time t (not at time zero). If T is the maturity date, the time to maturity is $T-t$.

The stock price process we are assuming is the one we developed in Section 14.3:

$$dS=\mu Sdt+\sigma Sdz$$(15.8)

Suppose that f is the price of a call option or other derivative contingent on S. The variable f must be some function of S and t. Hence, from equation (14.14),

$$df=\left(\frac{\partial f}{\partial S}\mu S+\frac{\partial f}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}f}{\partial S^2}{\sigma }^2{S}^2\right)dt+\frac{\partial f}{\partial S}\sigma Sdz$$(15.9)

The discrete versions of equations (15.8) and (15.9) are

$$\text{ Δ }S=\mu S\text{ Δ }t+\sigma S\text{ Δ }z$$(15.10)

and

$$\text{ Δ }f=\left(\frac{\partial f}{\partial S}\mu S+\frac{\partial f}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}f}{\partial S^2}{\sigma }^2{S}^2\right)\text{ Δ }t+\frac{\partial f}{\partial S}\sigma Sdz$$(15.11)

where Δf and Δ ...