CDO Class Reference

collateralized debt obligation More...

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

Inheritance diagram for CDO:

Public Member Functions
	CDO (Real attachment, Real detachment, std::vector< Real > nominals, const std::vector< Handle< DefaultProbabilityTermStructure > > &basket, Handle< OneFactorCopula > copula, bool protectionSeller, Schedule premiumSchedule, Rate premiumRate, DayCounter dayCounter, Rate recoveryRate, Rate upfrontPremiumRate, Handle< YieldTermStructure > yieldTS, Size nBuckets, const Period &integrationStep=Period(10, Years))
Real	nominal () const
Real	lgd () const
Real	attachment () const
Real	detachment () const
std::vector< Real >	nominals ()
Size	size ()
bool	isExpired () const override
	returns whether the instrument might have value greater than zero.
Rate	fairPremium () const
Rate	premiumValue () const
Rate	protectionValue () const
Size	error () const
Public Member Functions inherited from Instrument
Real	NPV () const
	returns the net present value of the instrument.
Real	errorEstimate () const
	returns the error estimate on the NPV when available.
const Date &	valuationDate () const
	returns the date the net present value refers to.
template<typename T>
T	result (const std::string &tag) const
	returns any additional result returned by the pricing engine.
const std::map< std::string, ext::any > &	additionalResults () const
	returns all additional result returned by the pricing engine.
void	setPricingEngine (const ext::shared_ptr< PricingEngine > &)
	set the pricing engine to be used.
virtual void	setupArguments (PricingEngine::arguments *) const
virtual void	fetchResults (const PricingEngine::results *) const
Public Member Functions inherited from LazyObject
void	update () override
bool	isCalculated () const
void	forwardFirstNotificationOnly ()
void	alwaysForwardNotifications ()
void	recalculate ()
void	freeze ()
void	unfreeze ()
Public Member Functions inherited from Observable
	Observable (const Observable &)
Observable &	operator= (const Observable &)
	Observable (Observable &&)=delete
Observable &	operator= (Observable &&)=delete
void	notifyObservers ()
Public Member Functions inherited from Observer
	Observer (const Observer &)
Observer &	operator= (const Observer &)
std::pair< iterator, bool >	registerWith (const ext::shared_ptr< Observable > &)
void	registerWithObservables (const ext::shared_ptr< Observer > &)
Size	unregisterWith (const ext::shared_ptr< Observable > &)
void	unregisterWithAll ()
virtual void	deepUpdate ()

Additional Inherited Members
Public Types inherited from Observer
typedef set_type::iterator	iterator
Protected Member Functions inherited from Instrument
void	calculate () const override
void	performCalculations () const override
Protected Attributes inherited from Instrument
Real	NPV_
Real	errorEstimate_
Date	valuationDate_
std::map< std::string, ext::any >	additionalResults_
ext::shared_ptr< PricingEngine >	engine_
bool	calculated_ = false
bool	frozen_ = false
bool	alwaysForward_

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.

Constructor & Destructor Documentation

CDO()

CDO	(	Real	attachment,
		Real	detachment,
		std::vector< Real >	nominals,
		const std::vector< Handle< DefaultProbabilityTermStructure > > &	basket,
		Handle< OneFactorCopula >	copula,
		bool	protectionSeller,
		Schedule	premiumSchedule,
		Rate	premiumRate,
		DayCounter	dayCounter,
		Rate	recoveryRate,
		Rate	upfrontPremiumRate,
		Handle< YieldTermStructure >	yieldTS,
		Size	nBuckets,
		const Period &	integrationStep = Period(10, Years) )

Parameters

attachment	fraction of the LGD where protection starts
detachment	fraction of the LGD where protection ends
nominals	vector of basket nominal amounts
basket	default basket represented by a vector of default term structures that allow computing single name default probabilities depending on time
copula	one-factor copula
protectionSeller	sold protection if set to true, purchased otherwise
premiumSchedule	schedule for premium payments
premiumRate	annual premium rate, e.g. 0.05 for 5% p.a.
dayCounter	day count convention for the premium rate
recoveryRate	recovery rate as a fraction
upfrontPremiumRate	premium as a tranche notional fraction
yieldTS	yield term structure handle
nBuckets	number of distribution buckets
integrationStep	time step for integrating over one premium period; if larger than premium period length, a single step is taken

Member Function Documentation

isExpired()

bool isExpired ( ) const

overridevirtual

returns whether the instrument might have value greater than zero.

Implements Instrument.