# Case Study: Convex-Concave-Convex Distributions in Application to CDO Pricing

** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

**Problem 1a: problem_hull_meansquare**

Minimize Meansquare(matrix_h) (minimize sum of meansquare error functions specified by matrix_meansquare1 and matrix_meansquare2)

Meansquare(matrix_meansquare1) +

0.0189*Meansquare(matrix_meansquare2)

subject to

Linearmulti(matrix_aeq) = 1 (constraint: system of linear equations specified by matrix_aeq)

Box constraints (constraints on individual decision variables specified by point_lb and point_ub )

——————————————————————–

Meansquare = Meansquare Error

Box constraints = constraints on individual decision variables

——————————————————————– Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 1,000 | 6 | 8.94 e-10 | 2.86 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 1,000 | 6 | 8.94 e-10 | 2.82 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 1,000 | 6 | 8.94 e-10 | 2.82 |

**Problem 1b: problem_hull_variance**

Minimize Variance(matrix_h) (minimize variance specified by matrix_h)

subject to

Linearmulti(matrix_aeq) = 1 (constraint: system of linear equations specified by matrix_aeq)

Box constraints (constraints on individual decision variables specified by point_lb and point_ub )

——————————————————————–

Variance = Variance

Box constraints = constraints on individual decision variables

——————————————————————– Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 1,000 | 1,000 | 6.45 e-08 | 0.64 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 1,000 | 1,000 | 6.45 e-08 | 0.62 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 1,000 | 1,000 | 6.45 e-08 | 0.62 |

** Problem 2: Problem_CCC **

Minimize Entropyr (matrix_h) (minimize entropyr specified by matrix_h)

subject to

Vector_bl ≤ Linearmulti(matrix_a) ≤ vector_b (constraint: system of linear inequalities specified by matrix_a)

Linearmulti(matrix_aeq) = 1 (constraint: system of linear equations specified by matrix_aeq)

Box constraints (constraints on individual decision variables specified by point_lb and point_ub )

——————————————————————–

Entropyr = Relative Entropy

Box constraints = probabilities should be non-negative

——————————————————————–

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 100 | 112 | -3.47386 | 0.57 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 100 | 112 | -3.47386 | 0.76 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 100 | 112 | -3.47386 | 0.76 |

**CASE STUDY SUMMARY**

This case study estimates default distributions of a basket of credit default swaps using prices of CDO tranches. We calibrated probabilities with two approached in the so called “implied copula” model proposed by Hull and White (2006). The theory and the case study presented here is described in paper Veremyev, Tsyurmasto, Uryasev, Rockafellar (2012).

We solved two calibration problems:

– The fist calibration problem is a reproduction of the case study in Hull and White (2006). We solved the same optimization problem two times, using different PSG nonlinear functions: Problem 1a with the meansquare function and Problem 1b with the variance function.

– The second calibration problem (Problem 2) is based on “the maximum entropy principle.” We maximized the entropy of a distribution, subject to no-arbitrage constraints. The probabilistic distribution belongs to the class of Convex-Concave-Convex (CCC) distributions. The CCC distribution is specified by a system of linear inequalities for the discrete probability atoms.

This web page contains codes and data Problems 1a, 1b and 2 in PSG Run-File Text Environment.

Also, we posted MATLAB files to run this Case Study in the PSG MATLAB Environment. The link Matlab_codes_and_data_for_Case_Study Study refers to the zipped folder containing files and folders: file CS_Implied_Copula_Hull.m, file CS_Implied_Copula_CCC.m, folder “CDO_data”, folder “Data_Genearation”.

To optimize Hull model, run CS_Implied_Copula_Hull.m. To optimize CCC-distribution model, run CS_Implied_Copula_CCC.m . Data for solving optimization problems are already prepared in the files matrix_A_hull_mid.mat and matrix_A_100_5y.mat .

The MATLAB files for generating data, CDO_cashflow_generator.m, CDO_cashflow_generator_bin.m, CDO_matrix_generator_bid_ask.m are placed to the folder “Data_Genearation”. To generate data, run MATLAB file CDO_matrix_generator_bid_ask.m.

Folder “CDO_data” contains raw iTraxx data.

**References**

• Hull, J., White, A. (2006). Valuing Credit Derivatives Using an Implied Copula Approach. The Journal of Derivatives 14, 2, 8-28.

• Veremyev, A., Tsyurmasto, P., Uryasev, S. Rockafellar, R.T. (2012). Convex-Concave-Convex Distributions in Application to CDO Pricing. Research Report 2012-1, ISE Dept., University of Florida, June 2012.