Black-Scholes model
The Black-Scholes model prices European options by assuming the underlying follows a lognormal random walk with constant volatility and interest rate, and that the option can be replicated by continuous delta hedging. Its closed-form price and greeks remain the common language of options markets, even though real markets violate several of its assumptions.
Senzoukria · Glossary · Updated September 2026
At a glance
- Call price
- C = S·e^(−qτ)·N(d1) − K·e^(−rτ)·N(d2)
- Put price
- P = K·e^(−rτ)·N(−d2) − S·e^(−qτ)·N(−d1)
- Exercise style
- European (exercise at expiry only)
The formula
With spot S, strike K, time to expiry τ in years, risk-free rate r, continuous dividend yield q and volatility σ, the call value is S·e^(−qτ)·N(d1) − K·e^(−rτ)·N(d2) and the put value K·e^(−rτ)·N(−d2) − S·e^(−qτ)·N(−d1), where d1 = [ln(S/K) + (r − q + σ²/2)·τ]/(σ√τ), d2 = d1 − σ√τ and N is the standard normal distribution function. Every standard greek is a derivative of these expressions.
Example: spot 100, strike 100, 30 days, σ = 20%, r = q = 0. Then d1 = 0.0287, d2 = −0.0287, and both the call and the put are worth 2.29. With a 4% rate the call is worth more than the put, as put-call parity requires.
Assumptions and where markets depart from them
| Assumption | What markets show |
|---|---|
| Constant volatility across strikes and expiries | Implied volatility varies by strike (smile, skew) and by expiry (term structure) |
| Continuous lognormal prices, no jumps | Index returns have fat tails and occasional gaps, especially at the open and on news |
| Continuous, costless hedging | Hedging is discrete and pays spreads and fees |
| Known constant rate and dividend yield | Both change and must be estimated for each expiry |
| European exercise | Many equity and ETF options, and some options on futures, are American-style |
Why it is still used
Practitioners use Black-Scholes as a translation device rather than as a belief about markets. Quoting an option by the volatility that makes the formula return its price, the implied volatility, lets contracts with different strikes and maturities be compared on one scale. The smile is then simply a record of how far the market departs from the model's constant-volatility assumption. Greeks computed from the formula at each strike's own implied volatility are the standard input for hedging and for exposure aggregates such as gamma exposure.
In Senzoukria
The GEX module uses greeks published by the options provider when there are any, and names the provider as the model origin because providers do not publish their model. When the app computes greeks itself, it uses Black-Scholes and labels the model Black-Scholes (European) in the Calculation assumptions panel: on the Databento path it solves implied volatility from the quote midpoint and derives the greeks, and on the ThetaData path it computes gamma from the provider's implied volatility. Black-Scholes gamma also rescales exposures in the Gamma profile by price, and Black-Scholes delta locates the 25-delta points when the app computes the skew itself. For American-style ETF options this is a European approximation.
Related
- d1 and d2 in Black-Scholes
- Implied volatility calculation
- Black-76 model
- Option greeks
- GEX assumptions documentation
In the same section
- Exercise style
- Binomial model
- Futures block trade
- Block trade
- Book depth
- Big trades
- Bootstrap
- Bid-ask spread
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Frequently asked questions
- Is Black-Scholes wrong because of the volatility smile?
- It is incomplete rather than wrong. The smile shows that no single volatility prices all strikes, so markets use the formula with a different implied volatility for each strike and expiry. The model remains an accurate mapping between price and implied volatility for European options.
- Can Black-Scholes price SPY options?
- Approximately. SPY options are American-style and SPY pays dividends, so early exercise can have value, notably for deep in-the-money calls before an ex-dividend date and for deep in-the-money puts. Binomial or other American models handle this; the difference is small for most near-the-money, short-dated contracts.