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 Formula Parameters

According to the Black-Scholes option pricing model (its Merton’s extension that accounts for dividends), there are 6 parameters which affect option prices:

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)

Note: In many resources you can find different symbols for some of these parameters. For example, strike price is often denoted K (here I use X), underlying price is often denoted S (without the zero), and time to expiration is often denoted T – t (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.

Black-Scholes Call and Put Option Price Formulas

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

Black-Scholes formula for call option price Black-Scholes formula for put option price

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

The formulas for d1 and d2 are:

Black-Scholes formula for d1 Black-Scholes formula for d2

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:

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:

Standard normal probability density function

… which is the standard normal probability density function.

Delta

Black-Scholes formula for call option delta Black-Scholes formula for put option delta

Gamma

Black-Scholes formula for gamma

Theta

Black-Scholes formula for call option theta Black-Scholes formula for 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 formula for vega

Rho

Black-Scholes formula for call option rho Black-Scholes formula for put option rho

Black-Scholes Formulas in Excel

If you want to use the Black-Scholes formulas in Excel and create an option pricing spreadsheet, see detailed guide here:

Black-Scholes Excel Formulas and How to Create a Simple Option Pricing Spreadsheet

Option Greeks Excel Formulas

Or get a ready-made Excel calculator here:

Black-Scholes Calculator + Guide


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