Price of a put option formula
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. 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. Call option C and put option P prices are calculated using the following 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. 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. If you want to use the Black-Scholes formulas in Excel and create an option pricing spreadsheet, see detailed guide here:. The Black-Scholes-Merton pricing formula is.
The other two variables are. It would be nice if we could simply carry out the additions, multiplications, divisions, etc. The situation is a little more difficult than that, however.
It is true we can calculate the numerator of the expression for d 1 , using scalar operations where appropriate, and probabilistic operations to add the last two terms together. Evaluating the price probabilistically could be a major challenge. A different way of presenting the same problem gives the answer without difficulty. The present value of the strike price is just Xe —rT , an expression that involves only one random variable, r , and can be readily computed.
To illustrate, Figure 1 shows the distributions of the present values of X and S T. The present value of the strike price X is narrow because there is not much uncertainty in the risk-free rate r.