Key words: finite difference, greeks, black scholes, option, pricing, simulation, volatility, quantitative finance, financial engineering, analysis, derivatives, delta, theta, central difference method
finite difference

When it comes to simulations, one of the first steps is to choose a method that would actually run through the hypothetical time steps or simply perform certain calculations and bring us to the final price.
The decision can be based on the answer to the question: “Is the asset price path dependant or not?” or “is there a possibility of early exercise?”
An example of path-dependant asset price is the premium of a barrier option.

Monte-Carlo simulation is the most popular approach to simulate asset prices, especially since it allows us to model numerous assets simultaneously.
On the other hand, there can be a problem with modeling early option exercise (in case of American options) and its accuracy depends on the number of simulated runs.
Certain problems can use a method which is similar to the binomial pricing and is also grid-based: the finite-difference method.

The finite difference method is based on calculating the option values at each point on the grid surface to such variables as time steps and underlying price movements that are imposed by moving along the finite-difference grid (see illustration below).

Unlike the binomial grid, where the step size is proportional to the volatility of the asset movements, the finite-difference grid has fixed steps.
The option’s value in any point of the grid is thus a function of time (t) and underlying asset price (S).
If the grid is made up of N asset steps in the asset values and M time steps, we can represent the value of the option as:

And speaking of option pricing, the finite-difference method is an excellent way to solve the discrete Black-Scholes equation:

It allows us to simultaneously model the Greeks (Theta, Gamma and Delta) and derive the option value by plugging them into the equation above.
Differentiating along the grid is done using the central difference method (which is more accurate than the forward or backward differences):




The last arguments in all these equations represent the error of the approximations. Calculating the option value at a single point of the grid will require knowledge of values in other points within the vicinity. If we reorganize the Black-Scholes equation by plugging in the formulae into the derivative arguments, we could get the value of option at time M + 1 given the underlying price at N by writing:

Charting the option values on an Asset Price vs Option Price basis would result in an option payoff diagram.
Statistics show that this approach is more accurate than Monte-Carlo simulations, but it is slower at higher dimensions (normally when there are more than 4 assets).