collateralized debt obligation More...

#include <cdo.hpp>

Inheritance diagram for CDO:

Collaboration 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. More...

Rate	fairPremium () const

Rate	premiumValue () const

Rate	protectionValue () const

Size	error () const

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...

virtual void	setupArguments (PricingEngine::arguments *) const

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

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

void	performCalculations () const override

Real	expectedTrancheLoss (Date d) const

Private Attributes
Real	attachment_

Real	detachment_

std::vector< Real >	nominals_

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_

Period	integrationStep_

std::vector< Real >	lgds_

Real	nominal_

Real	lgd_

Real	xMax_

Real	xMin_

Size	error_

Real	premiumValue_

Real	protectionValue_

Real	upfrontPremiumValue_

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

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 98 of file cdo.hpp.

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

Definition at line 28 of file cdo.cpp.

Here is the call graph for this function:

Member Function Documentation

◆ nominal()

Real nominal ( ) const

Definition at line 136 of file cdo.hpp.

◆ lgd()

Real lgd ( ) const

Definition at line 137 of file cdo.hpp.

◆ attachment()

Real attachment ( ) const

Definition at line 138 of file cdo.hpp.

◆ detachment()

Real detachment ( ) const

Definition at line 139 of file cdo.hpp.

◆ nominals()

std::vector< Real > nominals ( )

Definition at line 140 of file cdo.hpp.

◆ size()

Size size ( )

Definition at line 141 of file cdo.hpp.

◆ isExpired()

bool isExpired ( ) const

overridevirtual

returns whether the instrument might have value greater than zero.

Implements Instrument.

Definition at line 84 of file cdo.cpp.

Here is the call graph for this function:

◆ fairPremium()

Rate fairPremium ( ) const

Definition at line 203 of file cdo.cpp.

Here is the call graph for this function:

◆ premiumValue()

Rate premiumValue ( ) const

Definition at line 188 of file cdo.cpp.

Here is the call graph for this function:

◆ protectionValue()

Rate protectionValue ( ) const

Definition at line 193 of file cdo.cpp.

Here is the call graph for this function:

◆ error()

Size error ( ) const

Definition at line 198 of file cdo.cpp.

Here is the call graph for this function:

◆ 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 90 of file cdo.cpp.

Here is the call graph for this function:

◆ performCalculations()

void performCalculations ( ) const

overrideprivatevirtual

This method must implement any calculations which must be (re)done in order to calculate the desired results.

Implements LazyObject.

Definition at line 118 of file cdo.cpp.

Here is the call graph for this function:

◆ expectedTrancheLoss()

Real expectedTrancheLoss ( Date d ) const

private

Definition at line 95 of file cdo.cpp.

Here is the call graph for this function:

Here is the caller graph for this function:

Member Data Documentation

◆ attachment_

Real attachment_

private

Definition at line 154 of file cdo.hpp.

◆ detachment_

Real detachment_

private

Definition at line 155 of file cdo.hpp.

◆ nominals_

std::vector<Real> nominals_

private

Definition at line 156 of file cdo.hpp.

◆ basket_

std::vector<Handle<DefaultProbabilityTermStructure> > basket_

private

Definition at line 157 of file cdo.hpp.

◆ copula_

Handle<OneFactorCopula> copula_

private

Definition at line 158 of file cdo.hpp.

◆ protectionSeller_

bool protectionSeller_

private

Definition at line 159 of file cdo.hpp.

◆ premiumSchedule_

Schedule premiumSchedule_

private

Definition at line 161 of file cdo.hpp.

◆ premiumRate_

Rate premiumRate_

private

Definition at line 162 of file cdo.hpp.

◆ dayCounter_

DayCounter dayCounter_

private

Definition at line 163 of file cdo.hpp.

◆ recoveryRate_

Rate recoveryRate_

private

Definition at line 164 of file cdo.hpp.

◆ upfrontPremiumRate_

Rate upfrontPremiumRate_

private

Definition at line 165 of file cdo.hpp.

◆ yieldTS_

Handle<YieldTermStructure> yieldTS_

private

Definition at line 166 of file cdo.hpp.

◆ nBuckets_

Size nBuckets_

private

Definition at line 167 of file cdo.hpp.

◆ integrationStep_

Period integrationStep_

private

Definition at line 168 of file cdo.hpp.

◆ lgds_

std::vector<Real> lgds_

private

Definition at line 170 of file cdo.hpp.

◆ nominal_

Real nominal_

private

Definition at line 172 of file cdo.hpp.

◆ lgd_

Real lgd_

private

Definition at line 173 of file cdo.hpp.

◆ xMax_

Real xMax_

private

Definition at line 174 of file cdo.hpp.

◆ xMin_

Real xMin_

private

Definition at line 175 of file cdo.hpp.

◆ error_

Size error_

mutableprivate

Definition at line 177 of file cdo.hpp.

◆ premiumValue_

Real premiumValue_

mutableprivate

Definition at line 179 of file cdo.hpp.

◆ protectionValue_

Real protectionValue_

mutableprivate

Definition at line 180 of file cdo.hpp.

◆ upfrontPremiumValue_

Real upfrontPremiumValue_

mutableprivate

Definition at line 181 of file cdo.hpp.

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ CDO()

Member Function Documentation

◆ nominal()

◆ lgd()

◆ attachment()

◆ detachment()

◆ nominals()

◆ size()

◆ isExpired()

◆ fairPremium()

◆ premiumValue()

◆ protectionValue()

◆ error()

◆ setupExpired()

◆ performCalculations()

◆ expectedTrancheLoss()

Member Data Documentation

◆ attachment_

◆ detachment_

◆ nominals_

◆ basket_

◆ copula_

◆ protectionSeller_

◆ premiumSchedule_

◆ premiumRate_

◆ dayCounter_

◆ recoveryRate_

◆ upfrontPremiumRate_

◆ yieldTS_

◆ nBuckets_

◆ integrationStep_

◆ lgds_

◆ nominal_

◆ lgd_

◆ xMax_

◆ xMin_

◆ error_

◆ premiumValue_

◆ protectionValue_

◆ upfrontPremiumValue_