Semi Monte-Carlo (July, 2014)
GGY has developed a new variance reduction method for Monte-Carlo simulations for use with real-world and risk-neutral scenario generators. The new method, called Semi Monte-Carlo (SMC), is based on combining numeric integration with Monte-Carlo simulation.
GGY has implemented the SMC method for a combined risk-neutral Hull-White 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 ) 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 well-known 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 Monte-Carlo approach (Sobol sequences in particular) where effective MSE reduction appears only when the number of scenarios is large (traditionally Quasi Monte-Carlo 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 far-reaching extension of several known methods:
- Substitution of random numbers in Monte-Carlo simulations with nodes of a numeric integration method;
- Brownian Bridge;
- Conditional Monte-Carlo simulation.
The combination of numeric integration over some variables with Monte-Carlo simulations over the other variables allows for a tradeoff between the variance produced by Monte-Carlo 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 cash-flows (individual cells, model points or policies) to be averaged over the same set of scenarios.
If you are interested in learning more about Semi Monte-Carlo, please contact us through our support portal at www.ggy.com/client.
 The overall error in this case should be measured by the Mean Square Error (MSE). In the case of Monte-Carlo 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.