Black-Scholes and Greeks Calculator

The six inputs and what each one does

Black-Scholes turns spot price, strike, time, the risk free rate, volatility and dividend yield into a single fair value. Five of the six can be looked up. Volatility cannot, because it describes the future, and that is the whole reason the options market has something to argue about. Everything the Greeks measure is a partial derivative of the price with respect to one of those inputs while the others sit still.

d1 = [ln(S/K) + (r - q + 0.5 x sigma^2) x T] / (sigma x sqrt(T))

Call = S x e^(-qT) x N(d1) - K x e^(-rT) x N(d2), where d2 = d1 - sigma x sqrt(T)

The worked example behind the defaults

The calculator loads at the money: spot $100, strike $100, 365 days, a 5% rate, 20% volatility and no dividend. With those numbers d1 works out to 0.3500 and d2 to 0.1500, and the model returns $10.45 for the call and $5.57 for the put. Both are pure extrinsic value, because neither has any intrinsic value with spot sitting exactly on the strike.

The difference between the two prices is not an accident. Put-call parity says C - P = S x e^(-qT) - K x e^(-rT), which here is 100 - 100 x e^(-0.05) = 4.877. Subtract 5.57 from 10.45 and you get 4.88. The call costs more because owning it defers paying for the stock, and a year of deferral is worth the interest on $100.

The Greeks on that call read: delta 0.6368, gamma 0.0188, theta minus $0.0176 a day, vega $0.3752 per volatility point and rho $0.5323 per percentage point of rates. A one dollar rise in the stock adds about 64 cents to the call. A one point rise in implied volatility, from 20% to 21%, adds about 38 cents. A day passing costs under two cents at this distance from expiry, which is a long way from the daily bleed a weekly option suffers.

Reading implied volatility

The solver at the bottom of the form runs the model backwards. Leave the inputs alone and enter a call price of $12.00, which is above the $10.45 the model produces at 20% volatility. The solver finds that a market paying $12.00 is pricing 24.11% volatility, about four points higher than the assumption you fed in. That gap is the difference between what you think will happen and what the option chain is charging for.

The method is bisection. Option price rises steadily as volatility rises, so the solver can halve the search range on every pass and be confident it never loses the answer. It needs a market price above the option's intrinsic value to work, since no positive volatility can produce a price below that floor. Implied volatility explained covers IV rank, term structure, and why implied volatility collapses the morning after earnings.

Where the model is wrong on purpose

Black-Scholes assumes constant volatility, continuous trading, no transaction costs and a lognormal distribution of returns. Real markets break all four. The clearest evidence is the volatility smile: strikes far below the money consistently trade at higher implied volatility than at the money strikes, because crash risk is real and the lognormal assumption underprices it. Traders keep using the model anyway, as a common language for quoting prices in volatility terms rather than as a description of reality. Volatility and the VIX follows that thread into the index that measures it.

Once you know what a single option is worth, combine several with the options strategy builder, or work out the income on shares you already own with the covered call calculator. For the theory behind each Greek in turn, read option Greeks explained, and for the contract mechanics start with options trading for beginners. Vertical spreads explained shows how two of these prices become one defined risk trade.