CDO Class Reference

collateralized debt obligation More...

#include <ql/experimental/credit/cdo.hpp>

Inheritance diagram for CDO:

List of all members.

Public Member Functions

 CDO (Real attachment, Real detachment, const std::vector< Real > &nominals, const std::vector< Handle< DefaultProbabilityTermStructure > > &basket, const Handle< OneFactorCopula > &copula, bool protectionSeller, const Schedule &premiumSchedule, Rate premiumRate, const DayCounter &dayCounter, Rate recoveryRate, Rate upfrontPremiumRate, const Handle< YieldTermStructure > &yieldTS, Size nBuckets, const Period &integrationStep=Period(10, Years))
Real nominal ()
Real lgd ()
Real attachment ()
Real detachment ()
std::vector< Realnominals ()
Size size ()
bool isExpired () const
 returns whether the instrument might have value greater than zero.
Rate fairPremium () const
Rate premiumValue () const
Rate protectionValue () const
Size error () const

Detailed Description

collateralized debt obligation

The instrument prices a mezzanine CDO tranche with loss given default between attachment point $ D_1$ and detachment point $ D_2 > D_1 $.

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

\[ V = V_1 - V_2. \]

The protection leg is priced as follows:

  • Build the probability distribution for volume of defaults $ L $ (before recovery) or Loss Given Default $ LGD = (1-r)\,L $ at times/dates $ t_i, i=1, ..., N$ (premium schedule times with intermediate steps)
  • Determine the expected value $ E_i = E_{t_i}\,\left[Pay(LGD)\right] $ of the protection payoff $ Pay(LGD) $ at each time $ t_i$ where

    \[ Pay(L) = min (D_1, LGD) - min (D_2, LGD) = \left\{ \begin{array}{lcl} \displaystyle 0 &;& LGD < D_1 \\ \displaystyle LGD - D_1 &;& D_1 \leq LGD \leq D_2 \\ \displaystyle D_2 - D_1 &;& LGD > D_2 \end{array} \right. \]

  • The protection value is then calculated as

    \[ V_1 \:=\: \sum_{i=1}^N (E_i - E_{i-1}) \cdot d_i \]

    where $ d_i$ is the discount factor at time/date $ t_i $

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

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

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

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

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

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

Possible enhancements:
Investigate and fix cases $ E_{i+1} < E_i. $

Constructor & Destructor Documentation

CDO ( Real  attachment,
Real  detachment,
const std::vector< Real > &  nominals,
const std::vector< Handle< DefaultProbabilityTermStructure > > &  basket,
const Handle< OneFactorCopula > &  copula,
bool  protectionSeller,
const Schedule premiumSchedule,
Rate  premiumRate,
const DayCounter dayCounter,
Rate  recoveryRate,
Rate  upfrontPremiumRate,
const Handle< YieldTermStructure > &  yieldTS,
Size  nBuckets,
const Period integrationStep = Period(10, Years) 
)
Parameters:
attachmentfraction of the LGD where protection starts
detachmentfraction of the LGD where protection ends
nominalsvector of basket nominal amounts
basketdefault basket represented by a vector of default term structures that allow computing single name default probabilities depending on time
copulaone-factor copula
protectionSellersold protection if set to true, purchased otherwise
premiumScheduleschedule for premium payments
premiumRateannual premium rate, e.g. 0.05 for 5% p.a.
dayCounterday count convention for the premium rate
recoveryRaterecovery rate as a fraction
upfrontPremiumRatepremium as a tranche notional fraction
yieldTSyield term structure handle
nBucketsnumber of distribution buckets
integrationSteptime step for integrating over one premium period; if larger than premium period length, a single step is taken