Synthetic Collateralized Debt Obligation. More...

#include <syntheticcdo.hpp>

Inheritance diagram for SyntheticCDO:

Collaboration diagram for SyntheticCDO:

Classes
class	arguments

class	engine
	CDO base engine. More...

class	results

Public Member Functions
	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

bool	isExpired () const override
	returns whether the instrument might have value greater than zero. More...

Rate	fairPremium () const

Rate	fairUpfrontPremium () const

Rate	premiumValue () const

Rate	protectionValue () const

Real	premiumLegNPV () const

Real	protectionLegNPV () const

Real	remainingNotional () const

Real	leverageFactor () const

const Date &	maturity () const
	Last protection date. More...

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

Public Member Functions inherited from Instrument
	Instrument ()

Real	NPV () const
	returns the net present value of the instrument. More...

Real	errorEstimate () const
	returns the error estimate on the NPV when available. More...

const Date &	valuationDate () const
	returns the date the net present value refers to. More...

template<typename T >
T	result (const std::string &tag) const
	returns any additional result returned by the pricing engine. More...

const std::map< std::string, ext::any > &	additionalResults () const
	returns all additional result returned by the pricing engine. More...

void	setPricingEngine (const ext::shared_ptr< PricingEngine > &)
	set the pricing engine to be used. More...

Public Member Functions inherited from LazyObject
	LazyObject ()

	~LazyObject () override=default

void	update () override

bool	isCalculated () const

void	forwardFirstNotificationOnly ()

void	alwaysForwardNotifications ()

void	recalculate ()

void	freeze ()

void	unfreeze ()

Public Member Functions inherited from Observable
	Observable ()

	Observable (const Observable &)

Observable &	operator= (const Observable &)

	Observable (Observable &&)=delete

Observable &	operator= (Observable &&)=delete

virtual	~Observable ()=default

void	notifyObservers ()

Public Member Functions inherited from Observer
	Observer ()=default

	Observer (const Observer &)

Observer &	operator= (const Observer &)

virtual	~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	update ()=0

virtual void	deepUpdate ()

Private Member Functions
void	setupExpired () const override

Private Attributes
ext::shared_ptr< Basket >	basket_

Protection::Side	side_

Leg	normalizedLeg_

Rate	upfrontRate_

Rate	runningRate_

const Real	leverageFactor_

DayCounter	dayCounter_

BusinessDayConvention	paymentConvention_

Real	premiumValue_

Real	protectionValue_

Real	upfrontPremiumValue_

Real	remainingNotional_

Size	error_

std::vector< Real >	expectedTrancheLoss_

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 Member Functions inherited from LazyObject
Protected Attributes inherited from Instrument
Real	NPV_

Real	errorEstimate_

Date	valuationDate_

std::map< std::string, ext::any >	additionalResults_

ext::shared_ptr< PricingEngine >	engine_

Protected Attributes inherited from LazyObject
bool	calculated_ = false

bool	frozen_ = false

bool	alwaysForward_

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.

Definition at line 104 of file syntheticcdo.hpp.

Constructor & Destructor Documentation

◆ SyntheticCDO()

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`
	)

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

Definition at line 36 of file syntheticcdo.cpp.

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Member Function Documentation

◆ basket()

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

Definition at line 127 of file syntheticcdo.hpp.

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◆ isExpired()

bool isExpired ( ) const

overridevirtual

returns whether the instrument might have value greater than zero.

Implements Instrument.

Definition at line 131 of file syntheticcdo.cpp.

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◆ fairPremium()

Rate fairPremium ( ) const

Definition at line 109 of file syntheticcdo.cpp.

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◆ fairUpfrontPremium()

Rate fairUpfrontPremium ( ) const

Definition at line 116 of file syntheticcdo.cpp.

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◆ premiumValue()

Rate premiumValue ( ) const

Definition at line 87 of file syntheticcdo.cpp.

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◆ protectionValue()

Rate protectionValue ( ) const

Definition at line 92 of file syntheticcdo.cpp.

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◆ premiumLegNPV()

Real premiumLegNPV ( ) const

Definition at line 97 of file syntheticcdo.cpp.

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◆ protectionLegNPV()

Real protectionLegNPV ( ) const

Definition at line 103 of file syntheticcdo.cpp.

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◆ remainingNotional()

Real remainingNotional ( ) const

Total outstanding tranche notional, not wiped out

Definition at line 138 of file syntheticcdo.cpp.

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◆ leverageFactor()

Real leverageFactor ( ) const

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

Definition at line 143 of file syntheticcdo.hpp.

◆ maturity()

const Date & maturity ( ) const

Last protection date.

Definition at line 147 of file syntheticcdo.hpp.

◆ 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.

Definition at line 229 of file syntheticcdo.cpp.

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◆ expectedTrancheLoss()

std::vector< Real > expectedTrancheLoss ( ) const

Expected tranche loss for all payment dates

Definition at line 121 of file syntheticcdo.cpp.

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◆ error()

Size error ( ) const

Definition at line 126 of file syntheticcdo.cpp.

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◆ setupArguments()

void setupArguments ( PricingEngine::arguments * ) const

overridevirtual

When a derived argument structure is defined for an instrument, this method should be overridden to fill it. This is mandatory in case a pricing engine is used.

Reimplemented from Instrument.

Definition at line 143 of file syntheticcdo.cpp.

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◆ fetchResults()

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

overridevirtual

When a derived result structure is defined for an instrument, this method should be overridden to read from it. This is mandatory in case a pricing engine is used.

Reimplemented from Instrument.

Definition at line 157 of file syntheticcdo.cpp.

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◆ setupExpired()

void setupExpired ( ) const

overrideprivatevirtual

This method must leave the instrument in a consistent state when the expiration condition is met.

Reimplemented from Instrument.

Definition at line 171 of file syntheticcdo.cpp.

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Definition at line 187 of file syntheticcdo.hpp.

Classes

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ SyntheticCDO()

Member Function Documentation

◆ basket()

◆ isExpired()

◆ fairPremium()

◆ fairUpfrontPremium()

◆ premiumValue()

◆ protectionValue()

◆ premiumLegNPV()

◆ protectionLegNPV()

◆ remainingNotional()

◆ leverageFactor()

◆ maturity()

◆ implicitCorrelation()

◆ expectedTrancheLoss()

◆ error()

◆ setupArguments()

◆ fetchResults()

◆ setupExpired()

Member Data Documentation

◆ basket_

◆ side_

◆ normalizedLeg_

◆ upfrontRate_

◆ runningRate_

◆ leverageFactor_

◆ dayCounter_

◆ paymentConvention_

◆ premiumValue_

◆ protectionValue_

◆ upfrontPremiumValue_

◆ remainingNotional_

◆ error_

◆ expectedTrancheLoss_