Stock Market · Options Theory

Black-Scholes

In 1973, Fischer Black and Myron Scholes published a mathematical model for pricing options. It transformed financial markets, and earned Scholes the Nobel Prize in Economics in 1997 (Black had died).

What Black-Scholes does

Given five inputs, Black-Scholes calculates the theoretically fair price for a European-style option:

1. Current stock price (S)

2. Strike price (K)

3. Time to expiry (T) in years

4. Risk-free interest rate (r)

5. Volatility (σ, sigma) of the underlying

The formula (simplified)

Call price = S × N(d1) − K × e^(−rT) × N(d2)

Where N() = cumulative normal distribution function, d1 and d2 are calculated from the inputs.

> You don't need to memorise the formula. You need to understand that volatility (σ) is the ONLY unobservable input, and it drives most of the model's output.

Black-Scholes and implied volatility

Since option prices are observable in the market, traders reverse-engineer the formula: plug in the market price and solve for σ. This gives you Implied Volatility (IV). The market's consensus estimate of future volatility embedded in the option price.

IVthe market's volatility expectation baked into the option price

Limitations of Black-Scholes

• Assumes constant volatility (reality: IV changes constantly)
• Assumes normally distributed returns (reality: fat tails. Extreme events happen more than the model predicts)
• Doesn't account for jumps (overnight gaps, crashes)

Despite limitations, Black-Scholes remains the market standard for quoting and understanding option pricing.

Takeaway. Black-Scholes prices options from 5 inputs. Volatility is the key variable. When reversed, it gives Implied Volatility. The market's expectation of future moves baked into option prices.