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 Eqs. (6.11) and (6.12) into Eq. (6.14), we have
(6.15) 
This equation does not involve the stochastic component Δwt. Therefore, under the no arbitrage assumption, the portfolio Vt must be riskless during the small time interval Δt. In other words, the assumptions used imply that the portfolio must instantaneously earn the same rate of return as other short-term, risk-free securities. Otherwise there exists an arbitrage opportunity between the portfolio and the short-term, risk-free securities. Consequently, we have
where r is the risk-free interest rate. By Eqs. (6.13)–(6.16), we have
Therefore,
This is the Black–Scholes differential equation for derivative pricing. It can be solved to obtain the price of a derivative with Pt as the underlying variable. The solution so obtained depends on the boundary conditions of the derivative. For a European call option, the boundary condition is
where T is the expiration time and K is the strike price. For a European put option, the boundary condition becomes
Example 6.4
As a simple example, consider a forward contract on a stock that pays no dividend. In this case, the value of the contract is given by
where K is the delivery price, r is the risk-free interest rate, and T is the expiration time. For such a function, we have
Substituting these quantities into the left-hand side of Eq. (6.17) yields
which equals the right-hand side of Eq. (6.17). Thus, the Black–Scholes differential equation is indeed satisfied.