Black-Scholes Option Price Calculator

How to use it
  1. Enter the underlying price S and the strike price K.
  2. Enter the calendar days to expiration and the volatility, in percent a year. To use the volatility the market is pricing, find it from an option’s price with the implied volatility calculator.
  3. Enter the risk-free rate, in percent a year. Use a Treasury bill yield for the same term.
  4. If the underlying pays dividends, enter its dividend yield in percent a year. Leave it blank for none.
  5. The call and put prices, the greeks and what the prices imply appear side by side and update as you type. The chart shows each option’s value today and at expiration across a range of underlying prices.

To see it filled in, choose “Show an example”. The calculator prices European options, which can only be exercised at expiration. An American put, or a call just before a large dividend, can be worth more because it can be exercised early. For comparing options and for study; not financial advice.

Result

Enter the underlying price.

European exercise, rates and yield continuously compounded, time as calendar days over 365. Vega and rho are per 1 % change; theta is per calendar day.

Methodology, equations and examples

The Black-Scholes model prices a European option from five observable inputs and one estimate: the underlying price, the strike, the time to expiration, the risk-free rate, the dividend yield and the volatility σ. Volatility, the annualized standard deviation of the underlying's log returns, is the estimate. Enter a historical volatility to see what the option is worth if the past repeats, or the market's implied volatility to see what the market is pricing.

The Black-Scholes-Merton formula

With a continuous dividend yield q (Merton's extension), the prices of a European call C and put P are:

C = S e^(−qT) N(d₁) − K e^(−rT) N(d₂)

P = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁)

d₁ = [ln(S/K) + (r − q + σ²/2) T] / (σ √T)

d₂ = d₁ − σ √T

  • S – Underlying price
  • K – Strike price
  • T – Time to expiration in years, taken as calendar days / 365
  • r – Risk-free rate, continuously compounded
  • q – Dividend yield, continuously compounded
  • σ – Volatility, per year
  • N(x) – Cumulative standard normal distribution, the probability that a standard normal variable is less than x

The model assumes the underlying follows geometric Brownian motion with constant volatility, that the rate and yield are constant, and that there are no transaction costs. Real markets break each of these a little, which is why the implied volatility of options on the same stock differs by strike and expiration.

Example

A six-month option (T = 0.5, or 182.5 days) has a strike of $40. The stock is at $42, pays no dividend, has a volatility of 20 % a year, and the risk-free rate is 10 %. This is the Black-Scholes example in Hull, Options, Futures, and Other Derivatives.

d₁ = [ln(42/40) + (0.10 + 0.20²/2) × 0.5] / (0.20 × √0.5) = 0.7693

d₂ = 0.7693 − 0.20 × √0.5 = 0.6278

K e^(−rT) = $40 e^(−0.05) = $38.0492

C = $42 × N(0.7693) − $38.0492 × N(0.6278) = $42 × 0.7791 − $38.0492 × 0.7349 = $4.76

P = $38.0492 × N(−0.6278) − $42 × N(−0.7693) = $38.0492 × 0.2651 − $42 × 0.2209 = $0.81

The stock has to rise above $44.76 for the call buyer to profit at expiration, and to fall below $39.19 for the put buyer.

Put-call parity

A call and a put with the same strike and expiration are tied together by put-call parity, which holds whatever the volatility:

C − P = S e^(−qT) − K e^(−rT)

In the example, C − P = $4.7594 − $0.8086 = $3.9508, and S − K e^(−rT) = $42 − $38.0492 = $3.9508. The results show both sides, so a mistyped input or a pair of market prices that disagree is easy to spot.

The greeks

The greeks are the option's sensitivities. Delta is the change in option price per $1 change in the underlying, e^(−qT) N(d₁) for a call and −e^(−qT) N(−d₁) for a put. Gamma is the change in delta per $1, the same for the call and the put. Vega is the change in price per 1 percentage point of volatility, also the same for both. Theta is the change in price per calendar day that passes. Rho is the change in price per 1 percentage point of the risk-free rate.

In the example, the call's delta is 0.7791 and the put's is −0.2209, and they differ by exactly 1 because there is no dividend. Vega is $0.0881 for both, so a rise in volatility to 21 % would add about 9 cents to each. The call loses about 1.2 cents a day to time decay and the put about 0.2 cents.

What the prices imply

The results also give the risk-neutral chance of finishing in the money, N(d₂) for a call and N(−d₂) for a put, and the range of the price at expiration that the volatility implies. The expected move is S σ √T, one standard deviation, and the 68 % and 95 % ranges are S e^((r − q − σ²/2)T ± z σ √T), with z = 1 and 1.96. They are skewed upward, since a price can rise without limit but cannot fall below zero. These figures assume the underlying grows at the risk-free rate less the dividend yield, so treat them as what the model implies, not as a forecast.

In the example, the call has a 73.5 % chance of finishing in the money, the expected move is ±$5.94, and the 68 % range at expiration is $37.95 to $50.35.

The chart

The chart plots one option's value against the underlying price, from 40 % below the lower of S and K to 40 % above the higher. The solid line is the Black-Scholes value today; the dashed line is the payoff at expiration, max(0, S − K) for a call and max(0, K − S) for a put. The gap between them is the time value, which is largest near the strike and shrinks as expiration approaches. Choose Call or Put above the chart to switch between them.

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