Black-Scholes

## Reconciliation of Black-Scholes Variants

##### Apr 17, 2018

This post is just to show that the different variants of Black-Scholes formula are in fact the same. $S$: Underlying share price $t$: Time to maturity $\sigma$: Standard deviation of underlying share price $K$: Exercise price $r_f$: Risk-free rate Variant 1 This is the one shown in our formula sheet, and is also the traditional presentation of Black-Scholes model. $$$$C=SN(d_1)-N(d_2)Ke^{-r_f t}$$$$ $$$$d_1=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}}$$$$ ...