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syntheticcdo.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 syntheticcdo.hpp
    \brief Synthetic Collateralized Debt Obligation and pricing engines
*/

#ifndef quantlib_synthetic_cdo_hpp
#define quantlib_synthetic_cdo_hpp

#include <ql/instrument.hpp>
#include <ql/default.hpp>
#include <ql/experimental/credit/issuer.hpp>
#include <ql/time/schedule.hpp>
#include <ql/termstructures/yieldtermstructure.hpp>
#include <ql/termstructures/defaulttermstructure.hpp>

#include <ql/experimental/credit/basket.hpp>
#include <ql/experimental/credit/lossdistribution.hpp>
#include <ql/experimental/credit/onefactorcopula.hpp>

namespace QuantLib {

    //! Synthetic Collateralized Debt Obligation
    /*!
      The instrument prices a mezzanine CDO tranche with loss given default between
      attachment point \f$ D_1\f$ and detachment point \f$ D_2 > D_1 \f$.

      For purchased protection, the instrument value is given by the difference
      of the protection value \f$ V_1 \f$ and premium value \f$ V_2 \f$,

      \f[ V = V_1 - V_2. \f]

      The protection leg is priced as follows:

      - Build the probability distribution for volume of defaults \f$ L \f$ (before
      recovery) or Loss Given Default \f$ LGD = (1-r)\,L \f$ at times/dates
      \f$ t_i, i=1, ..., N\f$ (premium schedule times with
      intermediate steps)

      - Determine the expected value \f$ E_i = E_{t_i}\,\left[Pay(LGD)\right] \f$
      of the protection payoff \f$ Pay(LGD) \f$  at each time \f$ t_i\f$ where
      \f[
      Pay(L) = min (D_1, LGD) - min (D_2, LGD) = \left\{
      \begin{array}{lcl}
      \displaystyle 0 &;& LGD < D_1 \\
      \displaystyle LGD - D_1 &;& D_1 \leq LGD \leq D_2 \\
      \displaystyle D_2 - D_1 &;& LGD > D_2
      \end{array}
      \right.
      \f]

      - The protection value is then calculated as
      \f[ V_1 \:=\: \sum_{i=1}^N (E_i - E_{i-1}) \cdot  d_i \f]
      where \f$ d_i\f$ is the discount factor at time/date \f$ t_i \f$

      The premium is paid on the protected notional amount, initially
      \f$ D_2 - D_1. \f$ This notional amount is reduced by the expected protection
      payments \f$ E_i \f$ at times \f$ t_i, \f$ so that the premium value is
      calculated as

      \f[
      V_2 = m \, \cdot \sum_{i=1}^N \,(D_2 - D_1 - E_i) \cdot \Delta_{i-1,i}\,d_i
      \f]

      where \f$ m \f$ is the premium rate, \f$ \Delta_{i-1, i}\f$ is the day
      count fraction between date/time \f$ t_{i-1}\f$ and \f$ t_i.\f$

      The construction of the portfolio loss distribution \f$ E_i \f$ is
      based on the probability bucketing algorithm described in

      <strong>
      John Hull and Alan White, "Valuation of a CDO and nth to default CDS
      without Monte Carlo simulation", Journal of Derivatives 12, 2, 2004
      </strong>

      The pricing algorithm allows for varying notional amounts and
      default termstructures of the underlyings.

      \ingroup credit

      \todo Investigate and fix cases \f$ E_{i+1} < E_i. \f$
    */
00100     class SyntheticCDO : public Instrument {
    public:
        class arguments;
        class results;
        class engine;

        SyntheticCDO (const boost::shared_ptr<Basket> basket,
                      Protection::Side side,
                      const Schedule& schedule,
                      Rate upfrontRate,
                      Rate runningRate,
                      const DayCounter& dayCounter,
                      BusinessDayConvention paymentConvention,
                      const Handle<YieldTermStructure>& yieldTS);

        boost::shared_ptr<Basket> basket() const { return basket_; }

        bool isExpired () const;
        Rate fairPremium() const;
        Rate fairUpfrontPremium() const;
        Rate premiumValue () const;
        Rate protectionValue () const;
        Real premiumLegNPV() const;
        Real protectionLegNPV() const;
        /*!
          Total outstanding tranche notional, not wiped out
        */
        Real remainingNotional() const;
        /*!
          Expected tranche loss for all payment dates
         */
        std::vector<Real> expectedTrancheLoss() const;
        Size error () const;

        void setupArguments(PricingEngine::arguments*) const;
        void fetchResults(const PricingEngine::results*) const;

    private:
        void setupExpired() const;

        boost::shared_ptr<Basket> basket_;
        Protection::Side side_;
        Schedule schedule_;
        Rate upfrontRate_;
        Rate runningRate_;
        DayCounter dayCounter_;
        BusinessDayConvention paymentConvention_;
        Handle<YieldTermStructure> yieldTS_;

        mutable Rate fairPremium_;
        mutable Real premiumValue_;
        mutable Real protectionValue_;
        mutable Real upfrontPremiumValue_;
        mutable Real remainingNotional_;
        mutable Size error_;
        mutable std::vector<Real> expectedTrancheLoss_;
    };

00158     class SyntheticCDO::arguments : public virtual PricingEngine::arguments {
    public:
        arguments() : side(Protection::Side(-1)),
                      upfrontRate(Null<Real>()),
                      runningRate(Null<Real>()) {}
        void validate() const;

        boost::shared_ptr<Basket> basket;
        Protection::Side side;
        Schedule schedule;
        Rate upfrontRate;
        Rate runningRate;
        DayCounter dayCounter;
        BusinessDayConvention paymentConvention;
        Handle<YieldTermStructure> yieldTS;
    };

00175     class SyntheticCDO::results : public Instrument::results {
    public:
        void reset();
        Real premiumValue;
        Real protectionValue;
        Real upfrontPremiumValue;
        Real remainingNotional;
        Real xMin, xMax;
        Size error;
        std::vector<Real> expectedTrancheLoss;
    };

}

#endif

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