11.14 3 of 3: The derivation of the Black-Scholes Differential Equation The return on the portfolio must be the risk-free rate. Hence △I=r∏△t We substitute for Af and As in these equations to get the Black-Scholes differential equation d+.02o28f=rf +rS=+ Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University11.14 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 3 of 3: The Derivation of the Black-Scholes Differential Equation 2 2 2 The return on the portfolio must be the risk-free rate. Hence We substitute for and in these equations to get the Black-Scholes differential equation: 1 2 f f S f f rS S t S r t S        + = +  D D D  D 2 = rf