Synthetic Collateralized Debt Obligation. More...

#include <qle/instruments/syntheticcdo.hpp>

Inheritance diagram for SyntheticCDO:

Classes
class	arguments

class	engine
	CDO base engine. More...

class	results

Public Member Functions
	SyntheticCDO (const QuantLib::ext::shared_ptr< QuantExt::Basket > &basket, Protection::Side side, const Schedule &schedule, Rate upfrontRate, Rate runningRate, const DayCounter &dayCounter, BusinessDayConvention paymentConvention, bool settlesAccrual=true, const QuantLib::CreditDefaultSwap::ProtectionPaymentTime protectionPaymentTime=QuantLib::CreditDefaultSwap::ProtectionPaymentTime::atDefault, Date protectionStart=Date(), Date upfrontDate=Date(), boost::optional< Real > notional=boost::none, Real recoveryRate=Null< Real >(), const DayCounter &lastPeriodDayCounter=DayCounter())

const QuantLib::ext::shared_ptr< QuantExt::Basket > &	basket () const

bool	isExpired () const override

Rate	fairPremium () const

Rate	fairUpfrontPremium () const

Rate	premiumValue () const

Rate	protectionValue () const

Real	premiumLegNPV () const

Real	protectionLegNPV () const

Real	recoveryRate () const
	Returns the recovery rate for fixed recovery CDO, otherwise returns Null<Real>()

Real	remainingNotional () const

Real	leverageFactor () const

const Date &	maturity () const
	Last protection date.

Real	implicitCorrelation (const std::vector< Real > &recoveries, const Handle< YieldTermStructure > &discountCurve, Real targetNPV=0., Real accuracy=1.0e-3) const

std::vector< Real >	expectedTrancheLoss () const

Size	error () const

void	setupArguments (PricingEngine::arguments *) const override

void	fetchResults (const PricingEngine::results *) const override

Detailed Description

Synthetic 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

◆ SyntheticCDO()

SyntheticCDO	(	const QuantLib::ext::shared_ptr< QuantExt::Basket > &	basket,
		Protection::Side	side,
		const Schedule &	schedule,
		Rate	upfrontRate,
		Rate	runningRate,
		const DayCounter &	dayCounter,
		BusinessDayConvention	paymentConvention,
		bool	settlesAccrual = `true`,
		const QuantLib::CreditDefaultSwap::ProtectionPaymentTime	protectionPaymentTime = `QuantLib::CreditDefaultSwap::ProtectionPaymentTime::atDefault`,
		Date	protectionStart = `Date()`,
		Date	upfrontDate = `Date()`,
		boost::optional< Real >	notional = `boost::none`,
		Real	recoveryRate = `Null< Real >()`,
		const DayCounter &	lastPeriodDayCounter = `DayCounter()`
	)

If the notional exceeds the basket inception tranche notional, the cdo is leveraged by that factor.

Member Function Documentation

◆ remainingNotional()

Real remainingNotional ( ) const

Total outstanding tranche notional, not wiped out

◆ leverageFactor()

Real leverageFactor ( ) const

The number of times the contract contains the portfolio tranched notional.

◆ implicitCorrelation()

Real implicitCorrelation	(	const std::vector< Real > &	recoveries,
		const Handle< YieldTermStructure > &	discountCurve,
		Real	targetNPV = `0.`,
		Real	accuracy = `1.0e-3`
	)		const

The Gaussian Copula LHP implied correlation that makes the contract zero value. This is for a flat correlation along time and portfolio loss level.

◆ expectedTrancheLoss()

std::vector<Real> expectedTrancheLoss ( ) const

Expected tranche loss for all payment dates

Classes

Public Member Functions