Black-Scholes-Merton Derivative Pricing Formula

thm. BSM Derivative Pricing. Generalized pricing of any derivative. Using different constraints, one can price any derivative.

$$\frac{\partial f}{\partial S}+\frac{{\sigma }^2{S}^2}{2}\frac{{\partial }^{2}f}{\partial S^2}+(r-q)S\frac{\partial f}{\partial S}-rf=0$$

where…

• $f[S(t),t]$ is the price of the derivative at time $t$
• $S(t)$ is a Geometric Brownian Motion s.t. $dS=(m-q)S\cdot dt+\sigma S\cdot dB(t)$ with $B(t)$ a standard brownian motion
  • $\sigma$ is the scale of process $S$
  • $\mu$ is the drift of process $S$
  • $m$ is the rate of return of process $S$
  • $q$ is the dividend rate (any dividend is immediately reinvested)
• $r$ is the Discount Rate