This is the risk-neutral probability of expiring ITM. DV - Ct - 12 2 CS 2 q 2 S 2 dt. The price of the underlying asset follows a geometric Brownian motion.
It was originally derived by Fischer Black and Myron Scholes by using arbitrage to create a partial differential equation which turned out to be the well-known heat equation from physics.
Note that the title of this chapter doesnt explicitly refer to the Black-Scholes equation or the Black-Scholes formula. The Black-Scholes formula is a solution to the Black-Scholes PDE given the boundary conditions below eq. The first eight ways of deriving the Black-Scholes equationformula are taken from the excellent paper by Jesper Andreason Bjarke Jensen and Rolf Poulsen 1998. Suppose that stock price S follows a geometric Brownian motion dS µSdtsSdw other assumptions in a moment We derive a partial differential equation for the price of a derivative Two ways of derivations.