The core of the program is BasketCDS class, which is performing the actual Monte Carlo process. This class is using NAGCopula class for creating correlated uniform random numbers needed for all simulations. It also uses CreditCurve class for retrieving credit-related data for calculating default times. Finally, it holds a delegate method for desired discounting method. For storing the calculated default and premium leg values, delegate methods are used to send these values to Statistics class, which then stores the both leg values into its own data structures. In essence, BasketCDS class is doing the calculation and Statistics class is responsible for storing the calculated values and reporting its own set of desired (and configured) statistics when requested.
As already mentioned, CreditCurve class is holding all the credit-related information (CDS curve, hazard rates, recovery rate). DiscountCurve class holds the information on zero-coupon curve. In this class, discounting and interpolation methods are not hard-coded inside the class. Instead, the class is holding two delegate methods for this purpose. The actual implementations for the both methods are stored in static MathTools class, which is nothing more than just collection of methods for different types of mathematical operations. Some matrix operations are needed for handling matrices when calculating default times in BasketCDS class. For this purpose, static MatrixTools class is providing collection of methods for different types of matrix operations. Needless to say, new method implementations can be provided for concrete methods used by these delegates.
Source market data (co-variance matrix, CDS curve and zero-coupon curve) for this example program has been completely hard-coded. However, with the examples presented in here or in here, the user should be able to interface this program back to Excel, if so desired.
MONTE CARLO PROCESS
The client example program is calculating five fair spreads for all kth-to-default basket CDS for all five reference names in the basket. In a nutshell, Monte Carlo procedure to calculate kth-to-default BDS spread is the following.
- Simulate five correlated uniformly distributed random numbers by using Gaussian or Student Copula model.
- Transform simulated five uniform random numbers to default times by using inverse CDF of exponential distribution.
- By using term structure of hazard rates, define the exact default times for all five names.
- Calculate the value for default leg and premium leg.
- Store the both calculated leg values.
- Repeat steps from one to five N times.
- Finally, calculate BDS spread by dividing average of all stored default leg values with average of all stored premium leg values.
In order to run this program, one needs to create a new C# console project and copyPaste the program given below into a new cs file. It is also assumed, that the user is a registrated NAG licence holder. Other users have to provide their own implementation for Copula needed in this program or use my implementation. With the second approach, a bit more time will be spent in adjusting this program for all the changes required.
Results for one test run for the both Copula models is presented in the following console screenshot.
Observed results are behaving as theory is suggesting. BDS prices are decreasing as more reference names are needed for triggering the actual default. Also, tail-effect created by the use of Student Copula model is captured as higher prices after 1st-to-default BDS. Computational help provided by NAG G05 library is just amazing. After you outsource the handling of Copula model for external library, there is not so much to program either.
That is all I wanted to share with you this time. Thanks for reading my blog. Mike.