Examples comparing low discrepancy to Monte Carlo methods
We compare the efficiency of methods using low discrepancy sequences, as implemented in FinDer, to that of Monte Carlo for a number of high dimensional financial problems. We use our methods to price derivatives and to calculate VaR.
In the pricing examples, the horizontal axis of the graphs denotes the number of paths of the simulation. The vertical axis denotes the price of the financial instrument or the relative error in its price. For the horizontal axis we may use a logarithmic scale to focus on the speed of the convergence at the beginning of a simulation. A relative error of 0.01 corresponds to 1% error.
The dimension (path length) of a problem we denote by d. Thus, a 1000 path simulation for a problem for which d=360 is using 1000 different tuples each having 360 components.
We show graphs of relatively long simulations to emphasize the fact low discrepancy methods will not only reach the answer fast but will maintain their performance, while the Monte Carlo solution may fluctuate a lot.
The graphs of the VaR examples are somewhat different. We compare the true VaR with the estimates produced by Monte Carlo and low discrepancy methods after a certain number of paths has been simulated. In particular, the horizontal axis shows a probability interval and the vertical axis denotes the corresponding VaR of a portfolio.