Black-Scholes Inputs
There are six parameters which affect option prices:
S0 = underlying price ($$$ per share)
X = strike price ($$$ per share)
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.
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 call option price](https://www.macroption.com/images/formulas/black-scholes-call-price-formula.png)
![Black-Scholes formula for put option price Black-Scholes formula for put option price](https://www.macroption.com/images/formulas/black-scholes-put-price-formula.png)
… 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 d1](https://www.macroption.com/images/formulas/black-scholes-d1-formula.png)
![Black-Scholes formula for d2 Black-Scholes formula for d2](https://www.macroption.com/images/formulas/black-scholes-d2-formula.png)
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. 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.
In several formulas you can see the term:
![Standard normal probability density function Standard normal probability density function](https://www.macroption.com/images/formulas/black-scholes-standard-normal-probability-density-function.png)
… which is the standard normal probability density function.
Delta
![Black-Scholes formula for call option delta Black-Scholes formula for call option delta](https://www.macroption.com/images/formulas/black-scholes-call-delta-formula.png)
![Black-Scholes formula for put option delta Black-Scholes formula for put option delta](https://www.macroption.com/images/formulas/black-scholes-put-delta-formula.png)
Gamma
![Black-Scholes formula for gamma Black-Scholes formula for gamma](https://www.macroption.com/images/formulas/black-scholes-gamma-formula.png)
Theta
![Black-Scholes formula for call option theta Black-Scholes formula for call option theta](https://www.macroption.com/images/formulas/black-scholes-call-theta-formula.png)
![Black-Scholes formula for put option theta Black-Scholes formula for put option theta](https://www.macroption.com/images/formulas/black-scholes-put-theta-formula.png)
… 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 Black-Scholes formula for vega](https://www.macroption.com/images/formulas/black-scholes-vega-formula.png)
Rho
![Black-Scholes formula for call option rho Black-Scholes formula for call option rho](https://www.macroption.com/images/formulas/black-scholes-call-rho-formula.png)
![Black-Scholes formula for put option rho Black-Scholes formula for put option rho](https://www.macroption.com/images/formulas/black-scholes-put-rho-formula.png)