If the option is European, it can only be used exercised at the maturity date.

If the option is American, it can be used at any date up to and including the maturity date. We use the following notation: Price of the underlying, eg stock price.

Risk free interest rate. Standard deviation of the underlying asset, eg stock. At maturity, a call option is worth.

In trading of options, a number of partial derivatives of the option price formula is important. Delta The first derivative of the option price with respect to the underlying is called the delta of the option price. It is the derivative most people will run into, since it is important in hedging of options.

Gamma The second derivative of the option price wrt the underlying stock. These are equal for puts and calls.

Black–Scholes equation - Wikipedia

Theta The partial with respect to time-to-maturity. Here is the algorithm that calculates all the above derivatives.

In calculation of the option pricing formulas, in particular the Black Scholes formula, the only unknown is the standard deviation of the underlying stock.

A common problem in option pricing is to find the implied volatility, given the observed price quoted in the market.

Options Pricing: Black-Scholes Model

For example, given , the price of a call option, the following equation should be solved for the value of. Computer algorithm, implied volatility, bisections.

Instead of this simple bracketing, which is actually pretty fast, and will almost always find the solution, we can use the Newton-Raphson formula for finding the root of an equation in a single variable.