Black-Scholes-Merton and the Greeks
The closed-form price of a European option with a dividend yield, and its sensitivities to each input.
What it does
The Black-Scholes-Merton formula gives the price of a European option, one that can only be exercised at expiry, when the underlying price follows a lognormal random walk with constant volatility. Merton's extension adds a continuous dividend yield, which also covers stock indices and, with the foreign rate as the yield, currencies.
The Greeks are the formula's partial derivatives: how much the price changes when one input moves and the others stay fixed. They are computed analytically.
Why it is used
It is the benchmark every other method on the page is checked against, and the convention markets use to quote options: a price is usually stated as the volatility that makes this formula reproduce it (its implied volatility). Where a closed form exists, it is exact under its assumptions and instant to compute.
Inputs
- Spot price S and strike K, in the same currency.
- Time to expiry T in years (calendar time).
- Risk-free rate r and dividend yield q, both continuously compounded.
- Volatility σ: the annualized standard deviation of log returns.
- Call or put.
Formulas
Assumptions
- The underlying follows geometric Brownian motion: log returns are normal, independent, with constant volatility.
- Interest rates and the dividend yield are constant and paid continuously.
- Trading is continuous and frictionless (no costs, no taxes, short selling allowed), and there is no arbitrage.
- European exercise only. American options are priced with the binomial tree instead.
- At expiry (T = 0) or with zero volatility the formula's limit is used: the option is worth its discounted forward intrinsic value, max(S·e^(−qT) − K·e^(−rT), 0) for a call.
How to read the results
The price is in the same units as the spot price. On the page, vega is shown per one percentage point of volatility (the formula's vega divided by 100), rho per one percentage point of rates, and theta per calendar day (the annual figure divided by 365).
The "chance of finishing in the money", N(d2) for a call, is a probability under the risk-neutral measure, where the underlying is assumed to grow at the risk-free rate. It is a pricing device and not a forecast.
Limitations
- Real markets price different strikes and expiries at different volatilities (the volatility smile and term structure). One constant σ cannot match them all.
- Prices can jump (earnings, news), and returns have fatter tails than the normal distribution, so far out-of-the-money options tend to be worth more in markets than the formula says.
- Discrete cash dividends are approximated by a continuous yield.
- Greeks describe small moves with everything else fixed. In a large move, gamma and changing volatility dominate.
Where it can fail
- Very short-dated, far out-of-the-money options have prices that round to zero; their Greeks are then zero too.
- The model's σ is an input. Garbage volatility gives a precise-looking wrong price.
Validation on current data
Worked examples from Hull's textbook, recomputed by the running engine. Hull rounds to the digits shown; the engine matches each one. The test suite also checks put-call parity and every Greek against finite differences on a grid of inputs.
References
- Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3).
- Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1).
- Hull, J. C. Options, Futures, and Other Derivatives. Pearson (chapters on the Black-Scholes-Merton model, index options and the Greek letters).