Options are a type of derivative instrument because their price is linked to the price of the underlying security, such as stock. Buying an options contract grants the right, but not the obligation, to buy or sell an underlying asset at a set price (known as a strike) on/before a certain date. The main reason for the popularity of options is because they hedge away exposure to an asset's price moving in an undesirable way.

A European call/put option gives us the right (but again, no obligation) to buy/sell a certain asset on a certain expiry date (commonly denoted as T).

Some popular methods of options' valuation:

• Using analytic formulas
• Binomial tree approach
• Finite differences
• Monte Carlo simulations

European options are an exception in the sense that there exist an analytical formula for their valuation, which is not the case for more advanced derivatives, such as American or Exotic options.

To price options using Monte Carlo simulations, we use risk-neutral valuation, under which the fair value of a derivative is the expected value of its future payoff(s). In other words, we assume that the option premium grows at the same rate as the risk-free rate, which we use for discounting to the present value. For each of the simulated paths, we calculate the option's payoff at maturity, take the average of all the paths, and discount it to the present value.

In this recipe, we show how to code the closed-form solution to the Black-Scholes model and then use the simulation approach. For simplicity, we use fictitious input data, but real-life data could be used analogically.