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 Tt.

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

dS=μSdt+σSdz(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=(fSμS+ft+122fS2σ2S2)dt+fSσSdz(15.9)

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

 Δ S=μS Δ t+σS Δ z(15.10)

and

 Δ f=(fSμS+ft+122fS2σ2S2) Δ t+fSσSdz(15.11)

where Δf and Δ ...

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