Numerical solution for derivative models using finite difference methods and how this can be used with Monte Carlo simulation

Abstract: Derivative models often come in the form of stochastic differential equations. From these equations a partial differential equation (PDE) can be derived. By discretizing the PDE the numerical solution is obtained on a form where the value of the derivative can be seen as a probabilistic weighting of future values. These probabilities can be used to simulate trajectories of the under- lying assets. This connection between the finite difference scheme and the simulation is rather unique for this pricing method and can be very useful. The probability weights can be forced to have a perfect probability interpre- tation in some cases, meaning they are positive and less than one, but in other cases we will end up with negative weights meaning we somehow have to simulate using negative probabilities. This paper presents how to price and simulate options with these methods in a few different situations and how to solve some of the problems that may come up.