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).

Black-Scholes Inputs

According to the Black-Scholes option pricing model (its Merton's extension that accounts for dividends), there are six parameters which affect option prices:

S = underlying price ($$$ per share)
K = strike price ($$$ 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)

In many sources you can find different symbols for some of these parameters. For example, strike price (here K) is often denoted X, underlying price (here S) is often denoted S0, and time to expiration (here t) is often denoted T – t (as difference between expiration and now).

In the original Black and Scholes paper (The Pricing of Options and Corporate Liabilities, 1973) the parameters were denoted x (underlying price), c (strike price), v (volatility), r (interest rate), and t* – t (time to expiration). Dividend yield was only added by Merton in Theory of Rational Option Pricing, 1973.

Call and Put Option Price Formulas

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

Black-Scholes call price formula Black-Scholes put price formula

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

Standard normal cumulative distribution function (CDF)

The formulas for d1 and d2 are:

Black-Scholes d1 formula Black-Scholes d2 formula

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 S in place of Se-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 Greeks Formulas

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. Differences between the Greek formulas for calls and puts are often very small – usually a minus sign here and there. It is very easy to make a mistake.

Besides the already familiar N(d1), some of the Greek formulas (namely gamma, theta, and vega) use the term N'(d1) – with an apostrophe after N, indicating a derivative. This is the standard normal probability density function:

Standard normal probability density function

Delta

Black-Scholes call delta formula Black-Scholes put delta formula

Gamma

Black-Scholes gamma formula

Theta

Black-Scholes call option theta Black-Scholes put option theta

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

Vega

Black-Scholes vega formula

Note: Divide by 100 to get the resulting vega as option price change for one percentage point change in volatility (if you don't, it is for 100 percentage points change in volatility; same logic applies to rho below).

Rho

Black-Scholes call rho formula Black-Scholes put rho formula

Black-Scholes Formulas in Excel

All these formulas for option prices and Greeks are relatively easy to implement in Excel (the most advanced functions you will need are NORM.DIST, EXP and LN). You can continue to the Black-Scholes Excel Tutorial, where I have demonstrated the Excel calculations step-by-step (first part is for option prices, second part for Greeks).

Or you can get a ready-made Black-Scholes Excel Calculator.

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