What Black-Scholes gives you
The Black-Scholes model prices a European option from five inputs: the share price, the strike, the time to expiry, the interest rate and the volatility, with dividends added as a continuous yield. Four of those are observable. Volatility is not, which is why the model is used as often backwards, to read the implied volatility out of a market price, as forwards.
call = S·e−qT·N(d1) − K·e−rT·N(d2)
d1 = [ln(S/K) + (r − q + σ²/2)T] ÷ σ√T · d2 = d1 − σ√T
A worked example
With the figures loaded above, an at-the-money option on a $100 share with 30 days left, 25% volatility and a 4% rate:
- Call is worth $3.02 and the put $2.69, or $302 and $269 per contract.
- Delta is 0.533 for the call and −0.467 for the put: a $1 rise in the shares adds about $53 to one call contract.
- Gamma is 0.0555: delta changes by about 0.055 for each $1 move.
- Theta is −$0.053 a day for the call, about −$5.30 per contract per day, and −$4.21 for the put.
- Vega is 0.114: each 1-point change in implied volatility moves either option by about $11.40 per contract.
Enter a market price of $3.00 for the call in the solver and the implied volatility comes out at 24.81%. That is how option chains show IV: the volatility that makes the model match the price.
Per share, per contract, per day
Many calculators quote Greeks per share and theta per year, which is correct and hard to use. This one shows theta per calendar day and vega per volatility point, each per share and per 100-share contract, because that is how a position actually moves on your screen.
Where the model stops working
- Early exercise. Listed US equity options are American-style. Black-Scholes ignores early exercise, which matters most for deep in-the-money puts and for calls just before a dividend.
- Volatility is not constant. Different strikes trade at different implied volatilities (the skew), and volatility changes over time. One number cannot describe the chain.
- Fat tails. Real returns produce big moves more often than the model’s lognormal curve allows.
- Discrete dividends are approximated by a yield. A large one-off dividend before expiry is better handled explicitly.
For how delta, theta and breakeven interact in a real trade, read where an option actually breaks even; for what happens to vega around earnings, the IV crush guide.