Black-Scholes calculator

How does the Black-Scholes formula work?

Black-Scholes turns five inputs into one theoretical value for a European option, one that can be exercised only at expiry. With a dividend yield added, in the form Robert Merton published in 1973, the formulas are:

Call = S e^-qT N(d1) - K e^-rT N(d2)
Put = K e^-rT N(-d2) - S e^-qT N(-d1)

where d1 = [ln(S / K) + (r - q + σ² / 2) T] / (σ √T) and d2 = d1 - σ √T. S is the underlying price, K the strike, T the time to expiry in years, r the interest rate, q the dividend yield, σ the volatility and N() the standard normal cumulative distribution.

Take the default inputs: S = $100, K = $100, 365 days (T = 1), σ = 20%, r = 5% and q = 0. Then d1 = (0 + 0.05 + 0.02) / 0.20 = 0.35 and d2 = 0.15. N(0.35) = 0.6368 and N(0.15) = 0.5596. The discounted strike is 100 x e^-0.05 = $95.12.

• Call: 100 x 0.6368 - 95.12 x 0.5596 = 63.6831 - 53.2325 = $10.45.
• Put: 95.12 x 0.4404 - 100 x 0.3632 = 41.8905 - 36.3169 = $5.57.

N(d2) has a plain reading: under the model's own assumptions it is the chance that the call finishes in the money. N(d1) is the call's delta when there is no dividend yield. The working under the calculator prints every one of these steps with your numbers, to four decimals.

What are the option Greeks? Delta, gamma, theta, vega and rho

The Greeks measure how the option value responds when one input moves and the rest stay still. This page computes all five for both the call and the put, by differentiating the formula in closed form, so it also works as an option Greeks calculator. Each figure below uses the default inputs, and each is checked against a full re-pricing.

Delta: sensitivity to the underlying price

Call delta is e^-qT N(d1) = 0.6368; put delta is -e^-qT N(-d1) = -0.3632. If the underlying rises $1 to $101, delta predicts the call gains about $0.64. Re-pricing at $101 gives $11.10, a gain of $0.6461, and the put falls by $0.3539 to $5.22. The difference arises because delta itself changes as the price moves. Call delta minus put delta always equals e^-qT, which is 1 here.

Gamma: how fast delta changes

Gamma is e^-qT n(d1) / (S σ √T) = 0.01876, the same for the call and the put. A $1 rise should lift call delta from 0.6368 to about 0.6556; the exact delta at $101 is 0.6553. Gamma is highest for an option at the money close to expiry, which is why short-dated deltas swing hardest.

Theta: time decay per day

Theta here is the change in value as one calendar day passes, the annual figure divided by 365. Call theta is -0.0176 a day and put theta -0.0045 a day. Over seven days that suggests the call loses about $0.12; re-pricing with 358 days left gives $10.33, a fall of $0.1234. Put theta is smaller because the interest earned on the strike works in the put's favor. A deep in-the-money European put can even have positive theta.

Vega: sensitivity to volatility

Vega is S e^-qT n(d1) √T, shown per 1 volatility point: 0.3752 for both options. Raising volatility from 20% to 21% lifts the call to $10.83 and the put to $5.95, both up $0.3757. Vega is largest for long-dated options near the money.

Rho: sensitivity to the interest rate

Rho is shown per 1 interest-rate point. Call rho is K T e^-rT N(d2) / 100 = 0.5323, put rho -0.4189. At a 6% rate the call is worth $10.99 (up $0.5390) and the put $5.17 (down $0.4075). A higher rate lowers the present value of the strike, which helps the holder of a call and hurts the holder of a put. For the Greeks on crypto options, see crypto options Greeks explained.

Black-Scholes examples in dollars, euros and Bitcoin

The arithmetic is the same in every currency. These three runs show how differently the inputs can sit. Every value comes from the calculator's formulas, rounded to the cent.

The US stock. The $155 call is out of the money with only 45 days left, so the value is mostly time value: $4.05. The put is in the money by $5 and worth $8.38. Here d1 = -0.2405 and d2 = -0.3388.

The euro stock. A 3% dividend yield above a 2.5% interest rate puts the forward price at €81.87, below today's €82. That is why the call, €2 in the money, is worth €5.03 and not more. Put delta is -0.3990, and call delta minus put delta is e^-qT = 0.9902, not 1.

The Bitcoin option. European-style options on Bitcoin are listed on some crypto venues, and the model treats the coin like a stock with no yield. At 55% volatility the 30-day $65,000 call is worth $1,981.82. Vega is $63.15 per volatility point, so a move from 55% to 65% volatility adds $642.71 to the call. Gamma is only 0.0000388 per dollar because a one-dollar move is tiny next to a $60,000 price. With seven days left and the strike at $60,000, the call is worth $1,845.13 and its theta is -$133.29 a day.

How each input changes the call and put value

Holding the default inputs and changing one at a time shows the direction of each effect. Higher volatility raises both values, because the option holder gains from large moves and loses no more than the premium:

Time works in a similar direction for these at-the-money options, but not in a straight line. Value grows roughly with the square root of time, so the last weeks lose value fastest:

Dividend yield. A yield lowers the call and raises the put, because the holder of the option does not receive the payouts the underlying makes. At a 2% yield the default call falls to $9.23 and the put rises to $6.33; at 4% they are $8.10 and $7.15. This is the Merton form of Black-Scholes, and the same input handles a stock index, which pays dividends almost continuously. For an option on a currency, enter the foreign interest rate as the yield; that turns the formula into the version used for currency options.

Interest rate. A higher rate raises the call and lowers the put. Price and strike. Only their ratio and the discounting matter, so doubling both doubles both values.

Put-call parity: the check in the working

For European options on the same underlying, strike and expiry, the call and the put are tied together by call - put = S e^-qT - K e^-rT. The relationship follows from the payoffs alone, not from any view of volatility, so any correct Black-Scholes value must satisfy it.

With the default inputs, $10.4506 - $5.5735 = $4.8771, and $100 - $95.1229 = $4.8771. The working and the Inside the formula panel show this check for every run. If you price a put from a call in a spreadsheet, or check a put you calculated by hand, parity is the quickest test that nothing has gone wrong. It also explains why call delta minus put delta is e^-qT: differentiate both sides with respect to S.

What Black-Scholes assumes and what it leaves out

The model is exact only inside its own assumptions. The underlying follows a lognormal random walk with constant volatility; the interest rate and dividend yield are constant; trading is continuous and costless; there are no jumps. Real markets break each of these, so a model value is a reference point, not a forecast and not a statement about whether a quoted price is too high or too low. The calculator leaves out:

• American early exercise. An American-style option can be exercised before expiry, and that right can be worth something. With S = $60 and the other defaults, the European put is worth $35.18, which is $4.82 below its $40 intrinsic value. An American put can never be worth less than intrinsic, because the holder could exercise it now. A call on a stock paying no dividend is never worth exercising early, so for that case the European value also fits an American call.
• Discrete dividends. A real stock pays fixed amounts on set dates, not a smooth yield. A common adjustment is to subtract the present value of the dividends due before expiry from S and set the yield to zero.
• The volatility smile. Market prices imply different volatilities at different strikes, often higher for low strikes. One volatility for every strike, as the model assumes, cannot reproduce that. On crypto, see Bitcoin options skew explained.
• Costs and margin. Spreads, commission, financing and margin rules are outside the model.

Whether a contract is European-style or American-style is set by its specification, not by the model, so check that before using a European value for it.

Implied volatility: running Black-Scholes backwards

Every input except volatility can be read from the contract or the market. Volatility is the one you have to assume. Implied volatility reverses the problem: given an option's market price, find the volatility that makes the formula return that price. There is no closed form, so it is solved by trial, which is what the implied volatility calculator does.

You can do a rough version here. With the defaults, a call price of $14.23 corresponds to about 30% volatility and $18.02 to about 40%, so a market price between them implies a volatility in that range. Traders compare implied volatility with the volatility the underlying has actually shown, and quote options in volatility terms because that strips out the effect of price, strike and time. For the crypto version, read crypto implied volatility explained.

How to calculate Black-Scholes in Excel or Google Sheets

Put S, K, T in years, r, σ and q in cells B1 to B6, with rates and volatility as decimals (0.05, not 5). Then:

• d1 in B7: =(LN(B1/B2)+(B4-B6+B5^2/2)*B3)/(B5*SQRT(B3))
• d2 in B8: =B7-B5*SQRT(B3)
• Call: =B1*EXP(-B6*B3)*NORM.S.DIST(B7,TRUE)-B2*EXP(-B4*B3)*NORM.S.DIST(B8,TRUE)
• Put: =B2*EXP(-B4*B3)*NORM.S.DIST(-B8,TRUE)-B1*EXP(-B6*B3)*NORM.S.DIST(-B7,TRUE)

With the default inputs the sheet should return 0.35, 0.15, 10.4506 and 5.5735, the same as this page. Older spreadsheets use NORMSDIST(x) in place of NORM.S.DIST(x,TRUE). For delta use NORM.S.DIST(d1,TRUE) times EXP(-q*T); for vega and gamma you need the density, NORM.S.DIST(d1,FALSE). Enter days as a fraction of 365 to match this calculator, or of 252 if you count trading days, and say which you used.

Common Black-Scholes mistakes

• Entering percentages as whole numbers in a formula. 20% volatility is 0.20 in a spreadsheet. This calculator takes percent and converts for you.
• Mixing day counts. Volatility measured over 252 trading days and time measured over 365 calendar days give a slightly different T for the same option. This page uses 365 calendar days throughout.
• Reading theta per year as per day. The formula gives an annual theta. Divide by 365 for a calendar day, as this page does.
• Quoting vega per 100 points. Raw vega is per 1.00 of volatility, that is 100 points. This page divides by 100 to show the change per volatility point; rho is shown the same way.
• Using a European value for an American put deep in the money, where early exercise is worth something.
• Forgetting the contract multiplier. The value is per unit of the underlying. A contract covering 100 shares is worth 100 times the figure shown.

Glossary of option pricing terms

Call: the right to buy the underlying at the strike. Put: the right to sell it at the strike. European option: exercisable only at expiry. American option: exercisable at any time up to expiry. Strike: the agreed exercise price. Intrinsic value: what the option would pay if exercised now, max(S - K, 0) for a call. Time value: the option value above intrinsic value. Volatility: the annualized standard deviation of the underlying's returns. Risk-free rate: the interest rate the model discounts with. Forward price: S e^{(r - q)T}, the price at expiry that the model's pricing is centered on. d1 and d2: the standardized distances that feed N(). Moneyness: where the price sits against the strike: in, at or out of the money.

Frequently asked questions

Courses behind option pricing

Sources and review