Black-Scholes-Merton model

10 minute read · By Krishna Avadhani, Vestora Academy · Updated 18 October 2026

In 1973 Fischer Black, Myron Scholes and Robert Merton published a formula to price European options. Scholes and Merton received the 1997 Nobel memorial prize in economics for it (Black had died in 1995).

Inputs: share price S, strike K, time to expiry T (in years), risk-free rate r, volatility σ (and dividend yield q in Merton's extension). The only input not directly observable is volatility, which is why traders quote options in implied volatility.

Key assumptions: prices follow geometric Brownian motion with constant volatility, no transaction costs, continuous trading, and European exercise. Real markets break these assumptions (jumps, fat tails, volatility skew), so practitioners treat Black-Scholes as a common language and adjust volatility by strike and expiry.

Formula

C = S e^(-qT) N(d1) - K e^(-rT) N(d2)
P = K e^(-rT) N(-d2) - S e^(-qT) N(-d1)
d1 = [ln(S/K) + (r - q + σ²/2)T] / (σ√T)
d2 = d1 - σ√T

Key takeaways

Five inputs: S, K, T, r, σ (plus q).
Volatility is the only unobservable input.
Real-world departures show up as the volatility smile.

Check your understanding

1. Which Black-Scholes input cannot be observed directly in the market?

Share price
Strike
Volatility
Time to expiry

Show the answer

Volatility. Volatility must be estimated or implied.

2. Which assumption is most clearly violated in real markets?

Strikes are fixed
Constant volatility and no jumps
Time passes
Interest rates exist

Show the answer

Constant volatility and no jumps. Prices jump and volatility changes, causing the smile.

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General education only, not financial advice. Examples are illustrations. Rules, tax settings and products change; check official sources.