Black-Scholes, intuitively
In 1973 Fischer Black and Myron Scholes, building on work with Robert Merton, published a formula for pricing options. It transformed derivatives markets and earned Scholes and Merton the 1997 Nobel Prize (Black had died in 1995).
The big idea
An option can be replicated by holding a changing amount of the underlying shares and borrowing or lending cash. If you can replicate it, it must cost the same as the replication, otherwise there's free money. So the price doesn't depend on anyone's view of where the stock is going. It depends on volatility.
The formula
Call = S × N(d1) − K × e^(−rT) × N(d2)
In words: the expected value of receiving the stock if the option finishes in the money, minus the present value of paying the strike, each weighted by a probability. N(d1) is the option's delta. The inputs are:
- S: share price
- K: strike
- T: time to expiry
- r: interest rate
- σ: volatility
Everything except volatility is known. That's why options are quoted and traded in terms of implied volatility: the σ that makes the formula match the market price.
The assumptions
- Volatility is constant over the option's life.
- Prices move continuously, with no jumps or gaps.
- Returns are normally distributed (prices are lognormal).
- You can hedge continuously with no trading costs.
- Interest rates are constant and borrowing is unlimited.
Where it breaks
Real markets have jumps, fat tails and changing volatility. Crashes happen far more often than a normal distribution predicts. After the 1987 crash, implied volatility stopped being the same across strikes: downside puts became permanently more expensive (lesson 4.2). If Black-Scholes were right, implied volatility would be flat.
Why it's still used
Black-Scholes is less a forecast than a common language. Traders quote in implied volatility, risk-manage with its Greeks and then adjust for its known flaws using volatility surfaces and more advanced models. As the saying goes, it is the wrong number you put into the wrong formula to get the right price.
Key takeaway
Black-Scholes showed options can be priced by replication rather than prediction, and turned volatility into the thing that is actually traded.
Educational content only, not financial advice. Investing involves risk and you can lose money.