Semi MonteCarlo (July, 2014)
Article Summary:
GGY has developed a new variance reduction method for MonteCarlo simulations for use with realworld and riskneutral scenario generators. The new method, called Semi MonteCarlo (SMC), is based on combining numeric integration with MonteCarlo simulation.
GGY has implemented the SMC method for a combined riskneutral HullWhite and Lognormal model generating scenarios containing a Yield curve, stochastic discount factor, and multiple equity markets, and plans to extend the method to other market models in the future.
In internal testing, SMC demonstrated Mean Square Error (MSE [1]) reduction in the range from 4 to 40 for different equity derivatives. Derivatives tested included vanilla European options and ratchets with various strikes and terms. For comparison: The straightforward application of wellknown low discrepancy Sobol sequences with randomization of the Sobol vector components order demonstrated MSE improvement in the range 1 to 10.
A significant advantage of the SMC approach is that it demonstrates good results when the number of scenarios is significantly lower than the dimension of the problem (number of individual values in the scenario generated). This is in sharp contrast with the Quasi MonteCarlo approach (Sobol sequences in particular) where effective MSE reduction appears only when the number of scenarios is large (traditionally Quasi MonteCarlo methods are evaluated by the asymptotic behavior of MSE reduction as the number of scenarios tends to infinity).
SMC can be considered a combination and farreaching extension of several known methods:
 Substitution of random numbers in MonteCarlo simulations with nodes of a numeric integration method;
 Brownian Bridge;
 Conditional MonteCarlo simulation.
The combination of numeric integration over some variables with MonteCarlo simulations over the other variables allows for a tradeoff between the variance produced by MonteCarlo and the bias coming from numerical integration error.
Within this general framework, there is a lot of flexibility in separating the variables and choosing a numeric integration method. The optimization depends on both the market model used to generate scenarios and the nature of the random variable (PV of cashflow in most cases).
The primary tradeoff made when using this method relates to the need to properly tune the parameters used to the class of problems considered. As a result, preliminary calculations must be performed based on each market model and application. Therefore, the potential total performance improvement is highest in cases of high dimension (long scenarios containing many economic parameters) with many cashflows (individual cells, model points or policies) to be averaged over the same set of scenarios.
If you are interested in learning more about Semi MonteCarlo, please contact us through our support portal at www.ggy.com/client.
Note:
[1] The overall error in this case should be measured by the Mean Square Error (MSE). In the case of MonteCarlo it coincides with the variance of the estimate; in the case of numeric integration, it coincides with the square of the difference between an estimate obtained from the numeric integration method and true value.
