6.5 Derivation of Black–Scholes Differential Equation
In this section, we use Ito's lemma and assume no arbitrage to derive the Black–Scholes differential equation for the price of a derivative contingent to a stock valued at Pt. Assume that the price Pt follows the geometric Brownian motion in Eq. (6.8) and Gt = G(Pt, t) is the price of a derivative (e.g., a call option) contingent on Pt. By Ito's lemma,
The discretized versions of the process and previous result are
where ΔPt and ΔGt are changes in Pt and Gt in a small time interval Δt. Because 
for both Eqs. (6.11) and (6.12), one can construct a portfolio of the stock and the derivative that does not involve the Wiener process. The appropriate portfolio is short on derivative and long ∂Gt/∂Pt shares of the stock. Denote the value of the portfolio by Vt. By construction,
The change in Vt is then
Substituting ...
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