Black-Scholes-Merton formula

This formula was developped independently by Black and Scholes, and by Merton in 1980/81. They received the Nobel prize in 1997 for this “discovery” of such an important formula to prize options as financial instruments.

The formula depends on:

S: the spot price of the underlying asset
K: the strike price to buy/sell at maturity date
T: the time to maturity (in years)
r: the interest rate (now till maturity)
q: the dividend yield
σ: the volatility of the underlying (or implied volatility)

We first calculate two terms that are subsequently used in simplified formulas:

$d_1=\frac{log(S/K) + (r -q +\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}$ $d_2=\frac{log(S/K) + (r -q -\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}$

Value $V=Se^{-qT}N(d_1) - Ke^{-rT}N(d_2)$

Delta: $\frac{\partial V}{\partial S} = e^{-qT}N(d_1)$

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