lossdistribution.hpp

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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
 
/*
 Copyright (C) 2008 Roland Lichters
 
 This file is part of QuantLib, a free-software/open-source library
 for financial quantitative analysts and developers - http://quantlib.org/
 
 QuantLib is free software: you can redistribute it and/or modify it
 under the terms of the QuantLib license.  You should have received a
 copy of the license along with this program; if not, please email
 <quantlib-dev@lists.sf.net>. The license is also available online at
 <http://quantlib.org/license.shtml>.
 
 This program is distributed in the hope that it will be useful, but WITHOUT
 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 FOR A PARTICULAR PURPOSE.  See the license for more details.
*/
 
/*! \file lossdistribution.hpp
    \brief Loss distributions and probability of n defaults
*/
 
#ifndef quantlib_loss_distribution_hpp
#define quantlib_loss_distribution_hpp
 
#include <ql/math/distributions/binomialdistribution.hpp>
#include <ql/experimental/credit/distribution.hpp>
#include <ql/experimental/credit/onefactorcopula.hpp>
 
namespace QuantLib {
 
    //! Probability formulas and algorithms
    class LossDist {
    public:
      LossDist() = default;
      virtual ~LossDist() = default;
 
      virtual Distribution operator()(const std::vector<Real>& volumes,
                                      const std::vector<Real>& probabilities) const = 0;
      virtual Size buckets() const = 0;
      virtual Real maximum() const = 0;
 
      /*! Binomial probability of n defaults using prob[0]
       */
      static Real binomialProbabilityOfNEvents(int n, std::vector<Real>& p);
 
      /*! Binomial probability of at least n defaults using prob[0]
       */
      static Real binomialProbabilityOfAtLeastNEvents(int n, std::vector<Real>& p);
      /*! Probability of exactly n default events
        Xiaofong Ma, "Numerical Methods for the Valuation of Synthetic
        Collateralized Debt Obligations", PhD Thesis,
        Graduate Department of Computer Science, University of Toronto, 2007
        http://www.cs.toronto.edu/pub/reports/na/ma-07-phd.pdf (formula 2.1)
      */
      static std::vector<Real> probabilityOfNEvents(std::vector<Real>& p);
 
      static Real probabilityOfNEvents(int n, std::vector<Real>& p);
 
      /*! Probability of at least n defaults
       */
      static Real probabilityOfAtLeastNEvents(int n, std::vector<Real>& p);
    }; 
 
    //! Probability of N events 
    class ProbabilityOfNEvents {
    public:
        explicit ProbabilityOfNEvents (int n) : n_(n) {}
        Real operator()(std::vector<Real> p) const;
    private:
        Size n_;
    };
 
    //! Probability of at least N events 
    class ProbabilityOfAtLeastNEvents {
    public:
        explicit ProbabilityOfAtLeastNEvents (int n) : n_(n) {}
        Real operator()(std::vector<Real> p) const;
    private:
        Size n_;
    };
 
    //! Probability of at least N events 
    class BinomialProbabilityOfAtLeastNEvents {
    public:
        explicit BinomialProbabilityOfAtLeastNEvents(int n) : n_(n) {}
        Real operator()(std::vector<Real> p) const;
 
      private:
        int n_;
    };
 
    //! Binomial loss distribution
    class LossDistBinomial : public LossDist {
    public:
        LossDistBinomial (Size nBuckets, Real maximum)
            : nBuckets_(nBuckets), maximum_(maximum) {}
        Distribution operator()(Size n, Real volume, Real probability) const;
        Distribution operator()(const std::vector<Real>& volumes,
                                const std::vector<Real>& probabilities) const override;
        Size buckets() const override { return nBuckets_; }
        Real maximum() const override { return maximum_; }
        Real volume() const { return volume_; }
        Size size () const { return n_; }
        std::vector<Real> probability() const { return probability_; }
        std::vector<Real> excessProbability() const { return excessProbability_; }
    private:
        Size nBuckets_;
        Real maximum_;
        mutable Real volume_;
        mutable Size n_;
        mutable std::vector<Real> probability_;
        mutable std::vector<Real> excessProbability_;
    };
 
    //! Loss Distribution for Homogeneous Pool
    /*! Loss Distribution for Homogeneous Pool
 
      Loss distribution for equal volumes but varying probabilities of 
      default.
 
      The method builds the exact loss distribution for a homogeneous pool
      of underlyings iteratively by computing the convolution of the given
      loss distribution with the "loss distribution" of an additional credit
      following 
      
      Xiaofong Ma, "Numerical Methods for the Valuation of Synthetic
      Collateralized Debt Obligations", PhD Thesis, 
      Graduate Department of Computer Science, University of Toronto, 2007  
      http://www.cs.toronto.edu/pub/reports/na/ma-07-phd.pdf (formula 2.1)
 
      avoiding numerical instability of the algorithm by
 
      John Hull and Alan White, "Valuation of a CDO and nth to default CDS 
      without Monte Carlo simulation", Journal of Derivatives 12, 2, 2004 
     */
    class LossDistHomogeneous : public LossDist {
    public:
      LossDistHomogeneous(Size nBuckets, Real maximum) : nBuckets_(nBuckets), maximum_(maximum) {}
      Distribution operator()(Real volume, const std::vector<Real>& probabilities) const;
      Distribution operator()(const std::vector<Real>& volumes,
                              const std::vector<Real>& probabilities) const override;
      Size buckets() const override { return nBuckets_; }
      Real maximum() const override { return maximum_; }
      Size size() const { return n_; }
      Real volume() const { return volume_; }
      std::vector<Real> probability() const { return probability_; }
      std::vector<Real> excessProbability() const { return excessProbability_; }
    private:
        Size nBuckets_;
        Real maximum_;
        mutable Size n_ = 0;
        mutable Real volume_ = 0.0;
        mutable std::vector<Real> probability_;
        mutable std::vector<Real> excessProbability_;
    };
 
    //! Loss distribution with Hull-White bucketing 
    /*! Loss distribution with Hull-White bucketing 
 
      Loss distribution for varying volumes and probabilities of default, 
      independence assumed.
 
      The implementation of the loss distribution follows 
 
      John Hull and Alan White, "Valuation of a CDO and nth to default CDS 
      without Monte Carlo simulation", Journal of Derivatives 12, 2, 2004. 
    */
    class LossDistBucketing : public LossDist {
    public:
        LossDistBucketing (Size nBuckets, Real maximum, 
                           Real epsilon = 1e-6)
            : nBuckets_(nBuckets), maximum_(maximum), epsilon_(epsilon) {}
        Distribution operator()(const std::vector<Real>& volumes,
                                const std::vector<Real>& probabilities) const override;
        Size buckets() const override { return nBuckets_; }
        Real maximum() const override { return maximum_; }
 
      private:
        int locateTargetBucket (Real loss, Size i0 = 0) const;
 
        Size nBuckets_;
        Real maximum_;
        Real epsilon_;
    };
 
    //! Loss distribution with Monte Carlo simulation
    /*!
      Loss distribution for varying volumes and probabilities of default
      via Monte Carlo simulation of independent default events.
    */
    class LossDistMonteCarlo : public LossDist {
    public:
        LossDistMonteCarlo (Size nBuckets, Real maximum, Size simulations,
                            long seed = 42, Real epsilon = 1e-6)
            : nBuckets_(nBuckets), maximum_(maximum), 
              simulations_(simulations), seed_(seed), epsilon_(epsilon) {}
        Distribution operator()(const std::vector<Real>& volumes,
                                const std::vector<Real>& probabilities) const override;
        Size buckets() const override { return nBuckets_; }
        Real maximum() const override { return maximum_; }
 
      private:
        Size nBuckets_;
        Real maximum_;
        Size simulations_;
        long seed_;
        Real epsilon_;
    };
 
}
 
#endif