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/qldefines.hpp>
 
#ifndef QL_PATCH_SOLARIS
 
#include <ql/instrument.hpp>
#include <ql/default.hpp>
#include <ql/optional.hpp>
#include <ql/time/schedule.hpp>
 
#include <ql/experimental/credit/basket.hpp>
#include <ql/cashflows/fixedratecoupon.hpp>
 
namespace QuantLib {
 
    class YieldTermStructure;
 
    //! 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.
 
      \todo Investigate and fix cases \f$ E_{i+1} < E_i. \f$
    */
    class SyntheticCDO : public Instrument {
    public:
        class arguments;
        class results;
        class engine;
 
        // Review: No accrual settlement flag. No separate upfront payment date.
        // Review: Forward start case.
        /*! If the notional exceeds the basket inception tranche
            notional, the cdo is leveraged by that factor.
 
            \todo: allow for extra payment flags, arbitrary upfront
                   payment date...
        */
        SyntheticCDO (const ext::shared_ptr<Basket>& basket,
                      Protection::Side side,
                      Schedule schedule,
                      Rate upfrontRate,
                      Rate runningRate,
                      const DayCounter& dayCounter,
                      BusinessDayConvention paymentConvention,
                      ext::optional<Real> notional = ext::nullopt);
 
        const ext::shared_ptr<Basket>& basket() const { return basket_; }
 
        bool isExpired() const override;
        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;
        /*! The number of times the contract contains the portfolio tranched
                notional.
        */
        Real leverageFactor() const {
            return leverageFactor_;
        }
        //! Last protection date.
        const Date& maturity() const {
            return ext::dynamic_pointer_cast<FixedRateCoupon>(
                normalizedLeg_.back())->accrualEndDate();
        }
        /*! The Gaussian Copula LHP implied correlation that makes the
            contract zero value. This is for a flat correlation along
            time and portfolio loss level.
        */
        Real implicitCorrelation(const std::vector<Real>& recoveries,
            const Handle<YieldTermStructure>& discountCurve,
            Real targetNPV = 0.,
            Real accuracy = 1.0e-3) const;
 
        /*!
          Expected tranche loss for all payment dates
         */
        std::vector<Real> expectedTrancheLoss() const;
        Size error () const;
 
        void setupArguments(PricingEngine::arguments*) const override;
        void fetchResults(const PricingEngine::results*) const override;
 
      private:
        void setupExpired() const override;
 
        ext::shared_ptr<Basket> basket_;
        Protection::Side side_;
        Leg normalizedLeg_;
 
        Rate upfrontRate_;
        Rate runningRate_;
        const Real leverageFactor_;
        DayCounter dayCounter_;
        BusinessDayConvention paymentConvention_;
 
        mutable Real premiumValue_;
        mutable Real protectionValue_;
        mutable Real upfrontPremiumValue_;
        mutable Real remainingNotional_;
        mutable Size error_;
        mutable std::vector<Real> expectedTrancheLoss_;
    };
 
    class SyntheticCDO::arguments : public virtual PricingEngine::arguments {
    public:
        arguments() : side(Protection::Side(-1)),
                      upfrontRate(Null<Real>()),
                      runningRate(Null<Real>()) {}
        void validate() const override;
 
        ext::shared_ptr<Basket> basket;
        Protection::Side side;
        Leg normalizedLeg;
 
        Rate upfrontRate;
        Rate runningRate;
        Real leverageFactor;
        DayCounter dayCounter;
        BusinessDayConvention paymentConvention;
    };
 
    class SyntheticCDO::results : public Instrument::results {
    public:
      void reset() override;
      Real premiumValue;
      Real protectionValue;
      Real upfrontPremiumValue;
      Real remainingNotional;
      Real xMin, xMax;
      Size error;
      /* Expected tranche losses affecting this tranche coupons. Notice this
      number might be below the actual basket losses, since the cdo protection
      might start after basket inception (forward start CDO)*/
      std::vector<Real> expectedTrancheLoss;
    };
 
 
    //! CDO base engine
    class SyntheticCDO::engine :
        public GenericEngine<SyntheticCDO::arguments,
                             SyntheticCDO::results> { };
 
}
 
#endif
 
#endif