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Black-Scholes Formula (d1, d2, Call Price, Put Price, Greeks)

This page explains the Black-Scholes formulas for d1, d2, call option price, put option price, and formulas for the most common option Greeks (delta, gamma, theta, vega, and rho).

This is illustrated by the following equation

Call option (C) and put option (P) prices are calculated using the following formulas:

$$ C = S0 \space e^{-qt} * N(d1)-X \space e^{-rt} *N(d2) $$
$$ P = X \space e^{-rt} * N(-d2)-S0 \space e^{-qt} *N(-d1) $$

… where N(x) is the standard normal cumulative distribution function.

The formulas for d1 and d2 are:

$$ d1 = {ln(\frac {S0}{X})+t(r-q+\frac {\alpha ^2}{2}) \above 1pt \alpha \sqrt t} $$
$$ d2 = d1- \alpha \sqrt t $$

Original Black-Scholes vs. Merton’s Formulas

In the original Black-Scholes model, which doesn’t account for dividends, the equations are the same as above except:

  • There is just S0 in place of S0 e-qt
  • There is no q in the formula for d1

Therefore, if dividend yield is zero, then e-qt = 1 and the models are identical.

Black-Scholes Formulas for Option Greeks

Below you can find formulas for the most commonly used option Greeks. Some of the Greeks (gamma and vega) are the same for calls and puts. Other Greeks (delta, theta, and rho) are different. The difference between the formulas for calls and puts are often very small – usually a minus sign here and there. It is very easy to make a mistake.

In several formulas you can see the term:

$$ \frac {1}{ \sqrt 2 \pi } * e \raise 5pt \frac {-d1^2}{2} $$

… which is the standard normal probability density function.

Delta

$$ call \space delta = e^{-qt} * N(d1) $$
$$ put \space delta = e^{-qt} * (N(d1)-1) $$

Gamma

$$Gamma = \frac {e^{-qt}}{S0 \space \sigma \sqrt t } * \frac {1}{\sqrt 2 \pi} * e^{\frac {-d1^2}{2}}$$

Theta

$$Call \space theta = \frac 1T \bigg (- \bigg (\frac {S_0 \space \sigma \space e^{-qt}}{2 \sqrt t} * \frac {1}{\sqrt 2 \pi} * e^{\frac {{-d_1}^2}{2}} \bigg ) -r \space X \space e^{-rt} N(d_2)+q \space S_0 \space e^{-qt} N(d_1) \bigg ) $$
$$Put \space theta = \frac 1T \bigg (- \bigg (\frac {S_0 \space \sigma \space e^{-qt}}{2 \sqrt t} * \frac {1}{\sqrt 2 \pi} * e^{\frac {{-d_1}^2}{2}} \bigg ) +r \space X \space e^{-rt} N(d_2)+q \space S_0 \space e^{-qt} N(d_1) \bigg ) $$

… where T is the number of days per year (calendar or trading days, depending on what you are using).

Vega

$$Vega = \frac {1}{100} {S_0 \space e^{-qt}}{\sqrt t} * \frac {1}{\sqrt 2 \pi} * e^{\frac {{-d_1}^2}{2}} $$

Rho

$$Call \space rho = \frac {1}{100} {X \space t \space e^{-rt}} * N(d_2) $$
$$Put \space rho = -\frac {1}{100} {X \space t \space e^{-rt}} * N(-d_2) $$
Variables
S0 = underlying price ($$$ per share)
X = strike price ($$$ per share)
σ = volatility (% p.a.)
r = continuously compounded risk-free interest rate (% p.a.)
q = continuously compounded dividend yield (% p.a.)
t = time to expiration (% of year)
Strike Price ($):
Stock Price ($):
Volatility: (%):
Interest Rate (%):
Days to Expiration (#):




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