Black Scholes Calculator: Call and Put Prices With d1 and d2

How to use the black scholes calculator: call and put prices with d1 and d2

1. Enter the stock price and the strike price.
2. Enter days to expiration. The calculator converts days to years by dividing by 365.
3. Enter implied volatility, the risk free rate and the dividend yield as annual percentages.
4. Read the call price, put price, d1, d2, N(d1) and N(d2) in the results panel.
5. Check the Greeks table, the put call parity check and the American comparison below it.

What is the Black Scholes model?

The Black Scholes model is a formula for the fair value of a European option, one that can be exercised only at expiration. Fischer Black and Myron Scholes published it in 1973, and Robert Merton extended it the same year. The version with a continuous dividend yield, used here, is often called Black Scholes Merton.

The model assumes the stock price follows a lognormal random walk with constant volatility, that the interest rate and dividend yield are constant, and that trading is continuous with no transaction costs. Those assumptions are simplifications, but the formula remains the common reference point for option pricing, and this Black Scholes model calculator applies it to any set of inputs.

Black Scholes formula for calls and puts

S is the stock price, K the strike, T the time to expiration in years (days ÷ 365), r the continuously compounded risk free rate, q the continuous dividend yield and σ the implied volatility. N(x) is the standard normal cumulative distribution function, the probability that a standard normal variable is below x.

N(d2) is the risk neutral probability that the call finishes in the money, and N(d1) is the call's delta when there is no dividend yield. Calls and puts are tied together by put call parity, C minus P = S × e^(minus q × T) minus K × e^(minus r × T), and the calculator checks that the difference is zero. The formulas are:

d1 = [ln(S ÷ K) + (r minus q + σ² ÷ 2) × T] ÷ (σ × √T)
d2 = d1 minus σ × √T
Call = S × e^(minus q × T) × N(d1) minus K × e^(minus r × T) × N(d2)
Put = K × e^(minus r × T) × N(minus d2) minus S × e^(minus q × T) × N(minus d1)
N(minus x) = 1 minus N(x)

Worked example: $100 stock, $100 strike, 45 days

With the default inputs (S = 100, K = 100, 45 days so T = 45 ÷ 365 = 0.1233, σ = 30%, r = 4.1%, q = 0%), the calculator returns a call price of $4.4467 and a put price of $3.9425, with d1 = 0.1007, d2 = negative 0.0047, N(d1) = 0.5401 and N(d2) = 0.4981.

The Greeks table shows a call delta of 0.5401 and a put delta of negative 0.4599, gamma of 0.0377 for both, theta of negative 0.0523 per day for the call and negative 0.0411 for the put, and vega of 0.1394 per volatility point. Put call parity holds: 4.4467 minus 3.9425 = 0.5042, which equals 100 minus 99.4958. Step by step, the formula works like this:

• σ × √T = 0.30 × 0.3511 = 0.1053
• ln(S ÷ K) = ln(1) = 0, and (r + σ² ÷ 2) × T = (0.041 + 0.045) × 0.1233 = 0.0106
• d1 = 0.0106 ÷ 0.1053 = 0.1007, and d2 = 0.1007 minus 0.1053 = negative 0.0047
• N(d1) = 0.5401, N(d2) = 0.4981, N(minus d1) = 0.4599 and N(minus d2) = 0.5019
• Discounted strike: K × e^(minus r × T) = 100 × 0.994958 = 99.4958
• Call = 100 × N(d1) minus 99.4958 × N(d2) = 54.0088 minus 49.5621 = $4.4467
• Put = 99.4958 × N(minus d2) minus 100 × N(minus d1) = 49.9337 minus 45.9912 = $3.9425

How to calculate Black Scholes in Excel

You can build a Black Scholes option pricing calculator in Excel with four built in functions: LN, EXP, SQRT and NORM.S.DIST. Put the inputs in cells B1 to B6 and enter volatility, the rate and the yield as decimals, so 4.1% is 0.041. NORM.S.DIST(x,TRUE) returns the cumulative standard normal N(x). Excel 2007 and earlier use NORMSDIST(x) instead, and Google Sheets accepts the same formulas.

With the default inputs, B8 returns 0.1007, B9 returns negative 0.0047, B10 returns 4.4467 and B11 returns 3.9425, matching this calculator. A sheet like this gives European values only. To price American options in a spreadsheet you need a binomial tree, which takes far more cells. The cell layout and formulas are:

B1 stock price: 100
B2 strike: 100
B3 days to expiration: 45
B4 volatility: 0.30
B5 risk free rate: 0.041
B6 dividend yield: 0
B7 time in years: =B3/365
B8 d1: =(LN(B1/B2)+(B5-B6+B4^2/2)*B7)/(B4*SQRT(B7))
B9 d2: =B8-B4*SQRT(B7)
B10 call price: =B1*EXP(-B6*B7)*NORM.S.DIST(B8,TRUE)-B2*EXP(-B5*B7)*NORM.S.DIST(B9,TRUE)
B11 put price: =B2*EXP(-B5*B7)*NORM.S.DIST(-B9,TRUE)-B1*EXP(-B6*B7)*NORM.S.DIST(-B8,TRUE)

Why Black Scholes misprices American puts

Black Scholes underprices American puts because it assumes the option can be exercised only at expiration. The holder of an American put can exercise early and receive the strike price now. When the put is deep enough in the money, the interest earned on that cash is worth more than the put's remaining time value, so early exercise is optimal and the American put is worth more than the European one.

In the default example the American Bjerksund and Stensland put is worth $3.97 against $3.9425 for Black Scholes, a gap of about 2.3 cents. The gap grows for deep in the money puts, longer expirations and higher interest rates. A deep in the money European put can even be worth less than its intrinsic value, which an American put never is.

For calls, Black Scholes is exact when the stock pays no dividend, because exercising such a call early only throws away time value. With a dividend due before expiration, an American call can be worth more than the European value, since exercising just before the ex dividend date captures the dividend. Using a European model on American options also pushes implied volatility too high for puts, which is why CalcOptions prices listed options with American models and keeps Black Scholes as a benchmark.

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Educational tool, not financial advice. Calculations are theoretical and depend on your inputs; real fills, fees, early assignment and dividends can change results. Options involve risk and are not suitable for every investor.