Black–Scholes and binomial option calculator with Greeks

10.4506call5.5735put (European, Black–Scholes–Merton)

A 100-step binomial tree gives 10.4306 and 5.5536; with American exercise 10.4306 and 6.0824. Early exercise is worth 0.0000 on the call and 0.5288 on the put.

Put–call parity: C − P = 4.8771, and S e^−qT − K e^−rT = 4.8771.

Prices per unit of the underlying
Option	Black–Scholes–Merton	Tree, European	Tree, American
Call	10.4506	10.4306	10.4306
Put	5.5735	5.5536	6.0824

Greeks of the European options (Black–Scholes–Merton)
Greek	Call	Put
Delta	0.6368	−0.3632
Gamma	0.018762	0.018762
Vega, per 1 percentage point of volatility	0.3752	0.3752
Theta, per calendar day	−0.0176	−0.0045
Rho, per 1 percentage point of the rate	0.5323	−0.4189

In the tree, the hedge after the first step is 0.6365 units of the underlying per call and −0.3635 per put; Black–Scholes–Merton deltas are 0.6368 and −0.3632.

Put–call parity check
S e^−qT − K e^−rT	4.8771
C − P, Black–Scholes–Merton	4.8771 (difference 0.0000)
C − P, tree, European	4.8771 (difference 0.0000)
C − P, tree, American	4.3483 – within 0.0000 to 4.8771

European prices must satisfy parity exactly. American prices need not; without a dividend yield, their C − P lies between S − K and S − K e^−rT.

Tree diagram shows theCallPut

Top line: price of the underlying; bottom line: call value. Values come from the full 100-step tree.

Tree as a table

European call: the first 4 of 100 steps
Step, up moves	Underlying	Option value
0, 0	100.0000	10.4306
1, 1	102.0201	11.6898
1, 0	98.0199	9.1436
2, 2	104.0811	13.0484
2, 1	100.0000	10.3017
2, 0	96.0789	7.9595
3, 3	106.1837	14.5074
3, 2	102.0201	11.5583
3, 1	98.0199	9.0173
3, 0	94.1765	6.8776
4, 4	108.3287	16.0671
4, 3	104.0811	12.9150
4, 2	100.0000	10.1719
4, 1	96.0789	7.8367
4, 0	92.3116	5.8962

How it is calculated

d₁[ln(100 ÷ 100) + (0.05 − 0 + 0.2² ÷ 2) × 1] ÷ (0.2 × √1) = 0.350000
d₂d₁ − σ√T = 0.350000 − 0.200000 = 0.150000
N(d₁) and N(d₂)Standard normal distribution function at d₁ and d₂ = 0.636831 and 0.559618
Discount factorse^−qT = e⁰ and e^−rT = e^−0.05 = 1.000000 and 0.951229
CallS e^−qT N(d₁) − K e^−rT N(d₂) = 100 × 1.000000 × 0.636831 − 100 × 0.951229 × 0.559618 = 10.4506
PutK e^−rT N(−d₂) − S e^−qT N(−d₁) = 100 × 0.951229 × 0.440382 − 100 × 1.000000 × 0.363169 = 5.5735
Tree with 100 stepsΔt = 1 ÷ 100 = 0.01; u = e^σ√Δt, d = 1 ÷ u = u 1.020201, d 0.980199
Risk-neutral probabilityp = (exp((r − q)Δt) − d) ÷ (u − d) = (1.000500 − 0.980199) ÷ (1.020201 − 0.980199) = 0.507502
Put–call parityC − P = 10.4506 − 5.5735; S e^−qT − K e^−rT = 100 × 1.000000 − 100 × 0.951229 = 4.8771 and 4.8771

Limits: the model assumes a constant volatility and rate, a continuous dividend yield and lognormal prices. Real options with different strikes trade at different implied volatilities, and discrete dividends need other methods.

Key facts

Call = S e^(−qT) N(d₁) − K e^(−rT) N(d₂) and put = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁), with d₁ = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T) and d₂ = d₁ − σ√T.
European prices satisfy put–call parity, C − P = S e^(−qT) − K e^(−rT): 4.8771 for the one-year at-the-money option at a 5% rate.
For that call, delta is 0.6368, gamma 0.018762, vega 0.3752 per percentage point of volatility, theta −0.0176 per calendar day and rho 0.5323 per percentage point of the rate.
The Cox–Ross–Rubinstein tree uses u = e^(σ√Δt), d = 1/u and p = (e^((r − q)Δt) − d) ÷ (u − d); with 1,000 steps the call is within 0.0020 of the formula price.
An American put can be worth more than a European one even without dividends: 6.0824 against 5.5536 in the example (100 steps). An American call on an asset without dividends is worth the same as the European call while the rate is positive.

How is the result calculated?

Black–Scholes–Merton. For a European option on an asset with a continuous dividend yield $q$ , the call and put prices are (Back, Liu and Loewenstein, Black–Scholes formula (secondary source)):

c = S e^{-qT} N(d_1) - K e^{-rT} N(d_2)\qquad p = K e^{-rT} N(-d_2) - S e^{-qT} N(-d_1)

d_1 = \frac{\ln(S/K) + (r - q + \tfrac12\sigma^2)\,T}{\sigma\sqrt{T}}\qquad d_2 = d_1 - \sigma\sqrt{T}

$S$ is the price of the underlying, $K$ the strike, $T$ the time to expiry in years, $r$ the continuously compounded risk-free rate and $\sigma$ the volatility a year; rates, yield and volatility go into the formula as decimals (20% = 0.20). $N(\cdot)$ is the standard normal distribution function, which the calculator works out to about 15 decimal places instead of reading it from a table.

Greeks. With $\varphi$ the standard normal density, the call’s sensitivities are (Back, Liu and Loewenstein, Black–Scholes formula (secondary source)):

\Delta_c = e^{-qT}N(d_1)\qquad \Gamma = \frac{e^{-qT}\varphi(d_1)}{S\sigma\sqrt{T}}\qquad \mathcal{V} = S e^{-qT}\varphi(d_1)\sqrt{T}

\Theta_c = -\frac{S e^{-qT}\varphi(d_1)\,\sigma}{2\sqrt{T}} + qSe^{-qT}N(d_1) - rKe^{-rT}N(d_2)\qquad \rho_c = TKe^{-rT}N(d_2)

Gamma and vega are the same for the put; its delta is $-e^{-qT}N(-d_1)$ . The put’s theta and rho follow from put–call parity: $\Theta_p = \Theta_c - qSe^{-qT} + rKe^{-rT}$ and $\rho_p = \rho_c - TKe^{-rT}$ . Theta here is the change in value as time passes, so a negative theta means the option loses value. The formulas give vega and rho per 1.00 of volatility and rate and theta per year. The calculator shows vega and rho per percentage point and theta per day, as the options guides of the Australian Securities Exchange and the Options Industry Council do (ASX, option Greeks (secondary source)) (OIC, theta (secondary source)), dividing by 100 and by 365 calendar days.

Binomial tree. The Cox–Ross–Rubinstein tree splits the time to expiry into $N$ steps of $\Delta t = T/N$ years. At each step the price moves up by $u$ or down by $d$ , with the risk-neutral probability $p$ of an up move (Back, Liu and Loewenstein, binomial trees (secondary source)):

u = e^{\sigma\sqrt{\Delta t}}\qquad d = \frac1u\qquad p = \frac{e^{(r-q)\Delta t} - d}{u - d}

The calculator starts from the payoffs at expiry and works backwards, discounting the probability-weighted values at $e^{-r\Delta t}$ per step. For American exercise, each node before expiry takes the larger of the value of exercising now and the value of waiting (Back, Liu and Loewenstein, binomial trees (secondary source)) (Naranjo, American options (secondary source)). The tree needs $d < e^{(r-q)\Delta t} < u$ , otherwise $p$ is not a probability; the calculator then asks for more steps or a higher volatility.

Put–call parity. European prices on an asset with a continuous dividend yield must satisfy $C - P = Se^{-qT} - Ke^{-rT}$ (Naranjo, dividend-adjusted parity (secondary source)). American options only have bounds: without dividends and with a positive rate, $S - K \le C - P \le S - Ke^{-rT}$ , and the inequalities swap when the rate is negative (Naranjo, American options (secondary source)). The calculator checks both.

Worked example

1. An at-the-money call and put

A share at 100, a strike of 100, one year to expiry, a risk-free rate of 5%, volatility of 20% and no dividend:

d_1 = \frac{\ln(100/100) + (0.05 + \tfrac12 \times 0.2^2) \times 1}{0.2 \times \sqrt{1}} = 0.350000\qquad d_2 = 0.350000 - 0.2 = 0.150000

$N(d_1) = 0.636831$ , $N(d_2) = 0.559618$ and $e^{-0.05} = 0.951229$ , so:

c = 100 \times 0.636831 - 100 \times 0.951229 \times 0.559618 = 10.4506

p = 100 \times 0.951229 \times 0.440382 - 100 \times 0.363169 = 5.5735

Parity holds: 10.4506 − 5.5735 = 4.8771, and 100 − 100 × 0.951229 = 4.8771. The call’s delta is 0.6368, its gamma 0.018762, vega 0.3752 per percentage point of volatility, theta −0.0176 a day and rho 0.5323 per percentage point of the rate. The put has delta −0.3632, the same gamma and vega, theta −0.0045 a day and rho −0.4189.

2. The two-step tree of the binomial guide

The binomial option pricing guide values options on a share at 72 with a strike of 75 over two half-years, with $u = 1.25$ , $d = 0.80$ and a growth factor of 1.04 per step. The calculator builds the same tree from T = 1, 2 steps, a volatility of 31.55726% (ln 1.25 ÷ √0.5) and a rate of 7.84414% (2 ln 1.04). Then $p = (1.04 - 0.80)/(1.25 - 0.80) = 0.5333$ and the share ends at 112.50, 72.00 or 46.08.

The put pays 0.00, 3.00 and 28.92 at expiry. After one step it is worth 1.3462 where the share has risen to 90.00, and 14.5154 where it has fallen to 57.60 – if you must wait. An American holder at that lower node can exercise for 75 − 57.60 = 17.40, which is more, so the tree uses 17.40 there:

P_{\mathrm{Am}} = \frac{0.5333 \times 1.3462 + 0.4667 \times 17.40}{1.04} = 8.4980

The European put is worth 7.2036 and the call 9.8619, as in the guide. Choose American exercise and the put in the calculator: the diagram marks the exercised node with an E. With only two steps the tree is far from Black–Scholes–Merton, which gives 10.2518 for the call and 7.5936 for the European put.

3. A call on an asset with a high dividend yield

A share at 100, a strike of 90, one year, a rate of 3%, a dividend yield of 8% and volatility of 20%. Black–Scholes–Merton gives 9.9123 for the European call; a 100-step tree gives 9.9229 European and 11.3857 American. Early exercise is worth 1.4628 here. Holding the call instead of the share gives up the dividend yield, so waiting has a cost, and where the call is deep enough in the money the tree exercises. With 4 steps the whole tree fits in the diagram, and it marks 4 nodes with an E, all of them above the starting price. Without dividends there is no such cost, and an American call is never exercised early while the rate is positive (Naranjo, American options (secondary source)). Parity for the European prices still holds: 4.9715 against 4.9715.

What do the results mean for you?

The prices are per unit of the underlying. For a position of 100 calls, multiply the price and every Greek by 100: the at-the-money call would lose about 1.76 a day from time passing alone, and gain about 37.52 if volatility rose by one percentage point. Each Greek moves one input and holds the others fixed, so it is an estimate for small changes. Time decay is not linear and tends to speed up as expiry approaches (OIC, theta (secondary source)), so a daily theta does not stay the same until expiry.

The model price is what the option is worth under the model’s assumptions, not a market quote. Of the six inputs, volatility is the one you have to estimate (Back, Liu and Loewenstein, Black–Scholes formula (secondary source)), and each percentage point moves the price by about the vega shown. The option Greeks guide shows how to combine the Greeks into a hedge.

How close does the binomial tree get to Black–Scholes–Merton?

For European options the tree approaches the Black–Scholes–Merton price as the number of steps grows (Back, Liu and Loewenstein, binomial trees (secondary source)). It does not get there smoothly. The table, computed with this calculator, shows the at-the-money option from the first example:

Tree prices of the at-the-money option (S 100, K 100, one year, r 5%, σ 20%, no dividend)
Steps	European call	Call minus formula	European put	American put
1	12.1623	1.7117	7.2852	7.2852
2	9.5405	−0.9101	4.6634	5.7377
3	11.0439	0.5933	6.1668	6.4996
4	9.9705	−0.4801	5.0935	5.8828
5	10.8059	0.3554	5.9289	6.3678
10	10.2534	−0.1972	5.3764	6.0043
25	10.5210	0.0704	5.6439	6.1462
50	10.4107	−0.0399	5.5336	6.0737
100	10.4306	−0.0200	5.5536	6.0824
500	10.4466	−0.0040	5.5695	6.0888
1,000	10.4486	−0.0020	5.5715	6.0896

With few steps the tree price jumps from one side of the formula to the other: in this example odd numbers of steps land above it and even numbers below. At 100 steps the call is 0.0200 away, at 1,000 steps 0.0020. For a European option the formula is the precise answer; the tree is for American options and for working through a small tree by hand, as in the second example.

When is early exercise worth something?

An American option can be exercised at any step, so it is worth at least as much as the European one. Whether that right has value depends on the option:

Puts: early exercise can pay even without dividends (Naranjo, American options (secondary source)). In the at-the-money example the American put is worth 6.0824 against 5.5536 for the European put (100 steps).
Calls without dividends: while the rate is positive, exercising early never pays, so the American and European calls have the same value (Naranjo, American options (secondary source)).
Calls with a dividend yield: waiting gives up the yield, and a deep in-the-money call may be exercised early – 1.4628 of value in the third example.

Black–Scholes–Merton has no early exercise, so the calculator shows Greeks for the European options only; for the American tree it shows the hedge ratio after the first step.

Which option pricing mistakes are common?

Percentages as whole numbers. Volatility of 20% is 0.20 in the formula; 20 gives a nonsense price.
The wrong kind of rate. The formulas use a continuously compounded rate. If a question gives an annual rate $R$ , use $r = \ln(1 + R)$ : 5% a year becomes 4.87902%.
Days left as days. Time to expiry is in years: 90 days is 90 ÷ 365 = 0.2466.
The dividend yield left out. With a yield, $S$ becomes $Se^{-qT}$ in the price, in delta and in put–call parity.
Parity applied to American options. The equation holds for European options; American options only satisfy bounds (Naranjo, American options (secondary source)).
Mixed Greek units. A vega of 37.5240 per 1.00 of volatility is 0.3752 per percentage point; a theta of −6.4140 a year is −0.0176 a day. A put’s delta is negative (ASX, option Greeks (secondary source)).
Rounding from a normal table. With N(0.35) ≈ 0.6368 and N(0.15) ≈ 0.5596 from a table, the call comes to 10.4492 instead of 10.4506. A small gap in the third or fourth decimal is usually rounding, not an error in the method.

What are the limits of the calculator?

Constant inputs. The model keeps volatility and the rate fixed until expiry and assumes trading without costs. In the market, options with the same expiry but different strikes usually imply different volatilities – the volatility smile (Back, Liu and Loewenstein, Black–Scholes formula (secondary source)).
Dividends as a yield. A continuous yield suits an index or an asset with steady income. For a share paying a known cash dividend, parity uses the present value of the dividends instead: $C - P = S - D - Ke^{-rT}$ (Naranjo, dividend-adjusted parity (secondary source)). The forwards and put–call parity guide covers that case.
Plain options only. Barrier, Asian and other exotic options, implied volatility from a market price and Greeks of American options are not included.

Can you use these numbers in the exam?

This calculator supports study for the CFA® Program Level I, run by CFA Institute (CFA Institute, Level I exam), and the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP) (GARP, FRM Exam information). Our tools complement the exam bodies’ own material; they do not replace it. Which calculators you may use in the exam is set by CFA Institute for the CFA® Program and by GARP for the FRM® Exam, so check their current calculator rules; our page on which calculators are allowed in the exams sets out the models with their sources. Use this page to check your own working: the steps under the result show d₁, d₂, N(d₁), N(d₂), the tree parameters and the parity check with your numbers. The Part I formula sheet lists the option formulas next to the others.

What this tool covers

Last updated

30 October 2026

Based on

The Black–Scholes–Merton formulas and the Cox–Ross–Rubinstein binomial tree (see the sources)
Your own inputs; the default values are illustrative, not market data

Includes

European call and put prices with a continuous dividend yield
Delta, gamma, vega, theta and rho of the European options
A binomial tree with 1 to 1,000 steps, European and American exercise, and the value of early exercise
The put–call parity check, a diagram of the first four steps of the tree and the worked steps

Not included

Cash dividends paid on set dates, transaction costs and taxes
Implied volatility from a market price, and volatility that changes with strike or expiry
Exotic options such as barrier or Asian options, and Greeks of American options

Frequently asked questions

How do you calculate a Black–Scholes option price?

Work out d₁ = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T) and d₂ = d₁ − σ√T, then the call is S e^(−qT) N(d₁) − K e^(−rT) N(d₂). With S and K at 100, one year, r 5%, σ 20% and no dividend, d₁ = 0.35, d₂ = 0.15 and the call is worth 10.4506.

What is the difference between Black–Scholes–Merton and a binomial tree?

Black–Scholes–Merton is a formula for European options. A binomial tree splits the time to expiry into steps, lets the price move up or down at each one and values the option backwards from expiry, so it can price American options as well. For European options the tree approaches the formula as the steps increase.

How many steps does a binomial tree need?

It depends on the precision you need. For the one-year at-the-money call, 100 steps give 10.4306 and 1,000 steps 10.4486, against 10.4506 from the formula. With few steps the tree price jumps above and below the formula value.

Why can an American put be worth more than a European put?

It can be exercised before expiry, and for a put deep in the money, receiving the strike now can be worth more than waiting. In the example the American put is worth 6.0824 and the European put 5.5536 (100 steps). An American call on an asset without dividends is never exercised early while the rate is positive.

How does the calculator check put–call parity?

It compares C − P with S e^(−qT) − K e^(−rT). For European options the two must match: 4.8771 and 4.8771 in the example. American options only have bounds; without a dividend yield and with a positive rate, C − P lies between S − K and S − K e^(−rT).

In which units are the Greeks shown?

Delta and gamma per 1 of the underlying price, vega per percentage point of volatility, rho per percentage point of the rate and theta per calendar day. The formulas give vega and rho per 1.00 and theta per year, so the call’s theta of −6.4140 a year is shown as −0.0176 a day.

Can I use this online calculator in the exam?

No. It is a study aid that runs in a web page. Which calculators you may use in the exam is set by CFA Institute for the CFA® Program and by GARP for the FRM® Exam, so check their current calculator rules.

Does the calculator save my numbers?

No. The calculation runs in your browser. Your numbers appear only in the page address after the # sign, so you can share a result; they are not sent to us.

Sources

Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 9, §9.2 (the Black–Scholes formula for calls and puts with a dividend yield), §9.4 (delta, gamma, vega, theta and rho) and §9.7 (implied volatilities and the volatility smile). Accessed 30 October 2026.
Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 4, §§4.1–4.4 and Exercise 4.8: replication, risk-neutral pricing, backward induction, tree calibration and convergence. Accessed 29 October 2026.
Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): American Options — exercise rights, early-exercise decisions and binomial valuation. Accessed 29 October 2026.
Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): The Impact of Dividends — put-call parity adjusted for the present value of cash dividends. Accessed 29 October 2026.
Australian Securities Exchange (market education, secondary source): Option pricing in detail — Topics 3–4: the Greeks, position delta and equivalent stock positions. Accessed 29 October 2026.
Options Industry Council, OCC (market education, secondary source): Theta — daily sensitivity and non-linear time decay. Accessed 29 October 2026.
CFA Institute: CFA Program Level I exam: format and topic weights. Accessed 26 October 2026.
Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.

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