Solver SDK Platform - Monte Carlo Simulation

The Solver SDK Platform includes complete facilities for creating models in C++, C#, VB.NET, Visual Basic, Java, and MATLAB with uncertain variables and functions, running Monte Carlo simulations, and collecting statistics from the Monte Carlo samples.  You could pay $1,200 or more for other software libraries that provide only probability distribution modeling and Monte Carlo simulation -- but in the Solver SDK Platform, all this power is included at no extra cost.

• Sampling Methods
• Probability Distributions
• Distribution Fitting
• Correlated Distributions
• Statistical Results
• Risk Measures
• Example Source Code

Sampling Methods

The Solver SDK Platform includes four different, high quality random number generators, covering the full spectrum of tradeoffs between long periods and statistical independence of the samples:

• Park-Miller 'Minimal' Generator with Bayes-Durham shuffle and safeguards. This generator has a period of 2³¹-2, and very good statistical independence of samples.
• Combined Multiple Recursive Generator (CMRG) of L'Ecuyer. This generator has a period of 2¹⁹¹, and excellent statistical independence of samples within its period.
• Well Equidistributed Long-period Linear (WELL1024) generator of Panneton, L'Ecuyer and Matsumoto. This very new generator combines a long period of 2¹⁰²⁴ with very good statistical independence.
• Mersenne Twister generator of Matsumoto and Nishimura. This generator has the longest period of 2¹⁹⁹³⁷-1, but the samples are not as 'equidistributed' as for the WELL1024 and CMRG generators.

Monte Carlo samples are generated from a wide range of probability distributions, using any of three methods:

• Standard Monte Carlo
• Latin Hypercube
• Sobol Numbers

Sobol numbers are an innovation in the Solver SDK Platform that's not found in other software for Monte Carlo simulation.  They are widely used by application developers in quantitative finance.  For low to moderate dimensional problems, Sobol numbers offer the "best of both worlds" -- the speed of Standard Monte Carlo with the "coverage" of Latin Hypercube sampling.

Probability Distributions

The Solver SDK Platform provides a complete set of analytic probability distributions.  And you can specify shifting and truncation to customize your probability distributions.

You can easily create an instance of a Distribution object, with the properties of any of these probability distributions.  By simply accessing properties of this object, you can obtain the probability density (PDF) or cumulative density (CDF) function, or analytic values for the moments of the distribution, based on its type and parameters.

Distribution Fitting

The Solver SDK Platform makes it easy to fit an analytic distribution and its parameters to sample data.  You can specify the distribution type and ask the SDK to find the best-fitting parameters, or you can just supply the sample data, specify continuous or discrete, and let the SDK automatically choose the best-fitting distribution type and the best parameters.  This is illustrated in the Example Source Code.  The SDK can fit 22 different distributions (from the list of 38 above), including both continuous and discrete distributions.

Correlated Distributions

The Solver SDK Platform makes it easy to create correlated input distributions, by creating a DoubleMatrix object that specifies rank correlations between two or more distributions.  You simply assign this correlation matrix to the appropriate property of the Model object.  This is illustrated in the Example Source Code.

To consistently specify correlations among multiple distributions, a rank correlation matrix must be positive semidefinite (PSD).  Users often have "desired correlations" among the key distributions, but insufficient information to fill out the matrix.  The Solver SDK Platform includes methods to test a DoubleMatrix for positive semidefiniteness (IsPSD) and to transform a non-PSD matrix into a "nearest" matrix that is positive semidefinite (MakePSD).  Unlike other software, the SDK can find a PSD matrix that leaves your "desired correlations" among key distributions nearly unchanged.

Statistical Results

In the Solver SDK Platform, you can obtain any of the statistics listed below for both uncertain variables and uncertain functions, by simply accessing the appropriate property or method of a Statistics object embedded in each Variable and Function object.  You can obtain confidence intervals (CI) for the mean or standard deviation, or the number of Monte Carlo trials required to obtain a result within your specified confidence interval.

100 percentile values (0 to 99) are computed for each variable and function.  In addition, you can obtain the observed correlation in the Monte Carlo sample between any two uncertain functions, or between an uncertain function and an uncertain variable, by accessing a property of the Function object.

Risk Measures

The Solver SDK Platform goes beyond computation of "standard moments" to compute several risk measures popular in quantitative finance applications.  Again you can obtain these values by simply accessing the appropriate property or method of a Statistics object embedded in each Variable and Function object.

• Mean Absolute Deviation
• Semivariance or Lower Partial Moment
• Semideviation (square root or pth root)
• Value at Risk
• Conditional Value at Risk

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