Calibrating probability distributions with convex-concave-convex functions: application to CDO pricing

Abstract

This paper considers a class of functions referred to as convex-concave-convex (CCC) functions to calibrate unimodal or multimodal probability distributions. In discrete case, this class of functions can be expressed by a system of linear constraints and incorporated into an optimization problem. We use CCC functions for calibrating a risk-neutral probability distribution of obligors default intensities (hazard rates) in collateral debt obligations (CDO). The optimal distribution is calculated by maximizing the entropy function with no-arbitrage constraints given by bid and ask prices of CDO tranches. Such distribution reflects the views of market participants on the future market environments. We provide an explanation of why CCC functions may be applicable for capturing a non-data information about the considered distribution. The numerical experiments conducted on market quotes for the iTraxx index with different maturities and starting dates support our ideas and demonstrate that the proposed approach has stable performance. Distribution generalizations with multiple humps and their applications in credit risk are also discussed.

Appendix \(1\): running case study with portfolio safeguard (PSG)

PSG has several syntax formats for running optimization problems in MATLAB environment:

• Optimization subroutines for optimizing nonlinear functions. Subroutines (e.g., “riskprog”) use as a parameter the name of a nonlinear function (e.g. “entropyr”), which is optimized.

• General PSG format.

With PSG optimization language in general format, the problem solving typically involves three main stages:

• 1. Mathematical formulation of a problem with a meta-code using PSG nonlinear functions. Typically, a problem formulation involves 5–10 operators of a meta-code. See in the end of the Appendix 1 the PSG meta-code for Problem C($w_l,w_r$).

• 2. Preparation of data for the PSG functions in an appropriate format. For instance, the meansquare error function is defined by the matrix of loss scenarios. One of those matrices should be prepared if we use this function in the problem statement.

• 3. Solving the optimization problem with PSG using the predefined problem statement and data for PSG functions. The problem can be solved in several PSG environments, such as MATLAB environment and Run-File (Text) environment.

Further we present the PSG meta-code for solving Optimization Problem C($w_l$,$w_r$). The meta-code, data and solutions can be downloaded from the link at the bottom of this page⁴.

Meta-Code for Optimization Problem C($w_l,w_r$)

• 1 Problem: problem_CCC, type $=$ minimize

• 2 Objective: objective_h, linearize $=$ 1

• 3 entropyr_h(matrix_h)

• 4 Constraint: constraint_a, lower_bound $=$ vector_bl, upper_bound $=$ vector_b

• 5 linearmulti_a (matrix_a)

• 6 Constraint: constraint_aeq, lower_bound $=$ 1, upper_bound $=$ 1

• 7 linearmulti_aeq (matrix_aeq)

• 8 Box_of_Variables: lowerbounds $=$ 0

• 9 Solver: VAN, precision $=$ 5

Here is a brief description of the presented meta-code. We boldface the important parts of the code. The keyword minimize tells a solver that the Problem C($w_l,w_r$) is a minimization problem. The keyword Objective is used to define the objective function. The objective function (17), that is a Shannon entropy function, is defined in lines 2,3 with the keyword entropyr and the data matrix, located in the file matrix_h.txt. Each constraint starts with the keyword Constraint. The constraints (18), (19) and (22)–(25) are the system of linear inequalities, defined in lines 4,5 with the keyword linearmulti. The coefficients for these linear inequalities are given in the file matrix_a.txt. The probability distribution constraint (20) is defined in lines 6,7 with keyword linearmulti and the matrix of unit coefficients, located in the file matrix_aeq.txt. The Box_of_Variables in line 8 sets the non-negativity constraints (21).