# Hybrid Monte Carlo methods for high-dimensional problems in finance

**Project level:** PhD, Masters

Partial Differential Equation (PDE) and Monte Carlo (MC) are the two major computational approaches in finance. The PDE approach is a very robust and efficient pricing approach for problems having less than four dimensions. Applying the PDE approach to problems of more than three dimensions is generally not feasible to, due to the “curse of dimensionality”, i.e., the complexity of the PDE approach increases exponentially with the dimensionality. The MC approach, on the other hand, is very suitable for high-dimensional problems, due to the fact that the complexity of MC methods increases only linearly with respect to the number of dimensions. However, this approach suffers from simulation errors, and cannot easily handle complex contract features, such as early exercise. Motivated by the important advantages of each approach, there are number of available projects that focus on developing hybrid computational approach that combines the MC and PDE approach for high-dimensional problems under jump-diffusion models in finance. Typical applications include cross-currency options pricing, spread options under stochastic volatility and interest rate models.