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).
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:
... where N(x) is the standard normal cumulative distribution function:
The formulas for d1 and d2 are:
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:
... where T is the number of days per year (calendar or trading days, depending on what you are using).
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).
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.