Saddle point portfolio credit default loss model. More...

#include <saddlepointlossmodel.hpp>

Inheritance diagram for SaddlePointLossModel< CP >:

Collaboration diagram for SaddlePointLossModel< CP >:

Classes
class	SaddleObjectiveFunction

class	SaddlePercObjFunction

Public Member Functions
	SaddlePointLossModel (const ext::shared_ptr< ConstantLossLatentmodel< CP > > &m)

Real	percentile (const Date &d, Probability percentile) const override

Probability	probOverLoss (const Date &d, Real trancheLossFract) const override

std::map< Real, Probability >	lossDistribution (const Date &d) const override
	Full loss distribution. More...

Probability	probOverPortfLoss (const Date &d, Real loss) const

Real	expectedTrancheLoss (const Date &d) const override

Probability	probDensity (const Date &d, Real loss) const

std::vector< Real >	splitVaRLevel (const Date &date, Real loss) const override

Real	expectedShortfall (const Date &d, Probability percentile) const override
	Expected shortfall given a default loss percentile. More...

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 ()

Protected Member Functions
Real	CumulantGeneratingCond (const std::vector< Real > &invUncondProbs, Real lossFraction, const std::vector< Real > &mktFactor) const

Real	CumGen1stDerivativeCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const

Real	CumGen2ndDerivativeCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const

Real	CumGen3rdDerivativeCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const

Real	CumGen4thDerivativeCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const

ext::tuple< Real, Real, Real, Real >	CumGen0234DerivCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const

ext::tuple< Real, Real >	CumGen02DerivCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const

Real	CumulantGenerating (const Date &date, Real s) const

Real	CumGen1stDerivative (const Date &date, Real s) const

Real	CumGen2ndDerivative (const Date &date, Real s) const

Real	CumGen3rdDerivative (const Date &date, Real s) const

Real	CumGen4thDerivative (const Date &date, Real s) const

Real	findSaddle (const std::vector< Real > &invUncondProbs, Real lossLevel, const std::vector< Real > &mktFactor, Real accuracy=1.0e-3, Natural maxEvaluations=50) const

Probability	probOverLossCond (const std::vector< Real > &invUncondProbs, Real trancheLossFract, const std::vector< Real > &mktFactor) const

Probability	probOverLossPortfCond1stOrder (const std::vector< Real > &invUncondProbs, Real loss, const std::vector< Real > &mktFactor) const

Probability	probOverLossPortfCond (const std::vector< Real > &invUncondProbs, Real loss, const std::vector< Real > &mktFactor) const

Probability	probDensityCond (const std::vector< Real > &invUncondProbs, Real loss, const std::vector< Real > &mktFactor) const

std::vector< Real >	splitLossCond (const std::vector< Real > &invUncondProbs, Real loss, std::vector< Real > mktFactor) const

Real	expectedShortfallFullPortfolioCond (const std::vector< Real > &invUncondProbs, Real lossPerc, const std::vector< Real > &mktFactor) const

Real	expectedShortfallTrancheCond (const std::vector< Real > &invUncondProbs, Real lossPerc, Probability percentile, const std::vector< Real > &mktFactor) const

std::vector< Real >	expectedShortfallSplitCond (const std::vector< Real > &invUncondProbs, Real lossPerc, const std::vector< Real > &mktFactor) const

Real	conditionalExpectedLoss (const std::vector< Real > &invUncondProbs, const std::vector< Real > &mktFactor) const

Real	conditionalExpectedTrancheLoss (const std::vector< Real > &invUncondProbs, const std::vector< Real > &mktFactor) const

void	resetModel () override
	Concrete models do now any updates/inits they need on basket reset. More...

Protected Member Functions inherited from DefaultLossModel
	DefaultLossModel ()=default

virtual std::vector< Real >	splitESFLevel (const Date &d, Real loss) const
	Associated ESF fraction to each counterparty. More...

virtual Real	densityTrancheLoss (const Date &d, Real lossFraction) const
	Probability density of a given loss fraction of the basket notional. More...

virtual std::vector< Probability >	probsBeingNthEvent (Size n, const Date &d) const

virtual Real	defaultCorrelation (const Date &d, Size iName, Size jName) const
	Pearsons' default probability correlation. More...

virtual Probability	probAtLeastNEvents (Size n, const Date &d) const

virtual Real	expectedRecovery (const Date &, Size iName, const DefaultProbKey &) const

Protected Attributes
const ext::shared_ptr< ConstantLossLatentmodel< CP > >	copula_

Size	remainingSize_

std::vector< Real >	remainingNotionals_

Real	remainingNotional_

Real	attachRatio_

Real	detachRatio_

Protected Attributes inherited from DefaultLossModel
RelinkableHandle< Basket >	basket_

Detailed Description

template<class CP>
class QuantLib::SaddlePointLossModel< CP >

Saddle point portfolio credit default loss model.

: Default Loss model implementing the Saddle point expansion integrations on several default risk metrics. Codepence is dealt through a latent model making the integrals conditional to the latent model factor. Latent variables are integrated indirectly.

: See:

: Taking to the saddle by R.Martin, K.Thompson and C.Browne; RISK JUNE 2001; p.91

: The saddlepoint method and portfolio optionalities R.Martin in Risk December 2006

: VAR: who contributes and how much? R.Martin, K.Thompson and C.Browne RISK AUGUST 2001

: Shortfall: Who contributes and how much? R. J. Martin, Credit Suisse January 3, 2007

: Don't Fall from the Saddle: the Importance of Higher Moments of Credit Loss Distributions J.Annaert, C.Garcia Joao Batista, J.Lamoot, G.Lanine February 2006, Gent University

: Analytical techniques for synthetic CDOs and credit default risk measures A. Antonov, S. Mechkovy, and T. Misirpashaevz; NumeriX May 23, 2005

: Computation of VaR and VaR contribution in the Vasicek portfolio credit loss model: a comparative study X.Huang, C.W.Oosterlee, M.Mesters Journal of Credit Risk (75-96) Volume 3/ Number 3, Fall 2007

: Higher-order saddlepoint approximations in the Vasicek portfolio credit loss model X.Huang, C.W.Oosterlee, M.Mesters Journal of Computational Finance (93-113) Volume 11/Number 1, Fall 2007

: While more expensive, a high order expansion is used here; see the paper by Antonov et al for the terms retained.

: For a discussion of an alternative to fix the error at low loss levels (more relevant to pricing than risk metrics) see:

: The hybrid saddlepoint method for credit portfolios by A.Owen, A.McLeod and K.Thompson; in Risk, August 2009. This is not implemented here though (yet?...)

: For the more general context mathematical theory see: Saddlepoint approximations with applications by R.W. Butler, Cambridge series in statistical and probabilistic mathematics. 2007

Definition at line 100 of file saddlepointlossmodel.hpp.

Constructor & Destructor Documentation

◆ SaddlePointLossModel()

SaddlePointLossModel ( const ext::shared_ptr< ConstantLossLatentmodel< CP > > & m )

explicit

Definition at line 102 of file saddlepointlossmodel.hpp.

Member Function Documentation

◆ CumulantGeneratingCond()

Real CumulantGeneratingCond	(	const std::vector< Real > &	invUncondProbs,
		Real	lossFraction,
		const std::vector< Real > &	mktFactor
	)		const

protected

Returns the cumulant generating function (zero-th order expansion term) conditional to the mkt factor: \( K = \sum_j ln(1-p_j + p_j e^{N_j \times lgd_j \times s}) \)

Definition at line 566 of file saddlepointlossmodel.hpp.

◆ CumGen1stDerivativeCond()

Real CumGen1stDerivativeCond	(	const std::vector< Real > &	invUncondProbs,
		Real	saddle,
		const std::vector< Real > &	mktFactor
	)		const

protected

Returns the first derivative of the cumulant generating function (first order expansion term) conditional to the mkt factor: \( K1 = \sum_j \frac{p_j \times N_j \times LGD_j \times e^{N_j \times LGD_j \times s}} \ {1-p_j + p_j e^{N_j \times LGD_j \times s}} \) One of its properties is that its value at zero is the portfolio expected loss (in fractional units). Its value at infinity is the max attainable portfolio loss. To be understood conditional to the market factor.

Definition at line 587 of file saddlepointlossmodel.hpp.

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◆ CumGen2ndDerivativeCond()

Real CumGen2ndDerivativeCond	(	const std::vector< Real > &	invUncondProbs,
		Real	saddle,
		const std::vector< Real > &	mktFactor
	)		const

protected

Returns the second derivative of the cumulant generating function (first order expansion term) conditional to the mkt factor: \( K2 = \sum_j \frac{p_j \times (N_j \times LGD_j)^2 \times e^{N_j \times LGD_j \times s}} {1-p_j + p_j e^{N_j \times LGD_j \times s}} - (\frac{p_j \times N_j \times LGD_j \times e^{N_j \times LGD_j \times s}} {1-p_j + p_j e^{N_j \times LGD_j \times s}})^2 \)

Definition at line 610 of file saddlepointlossmodel.hpp.

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◆ CumGen3rdDerivativeCond()

Real CumGen3rdDerivativeCond	(	const std::vector< Real > &	invUncondProbs,
		Real	saddle,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 635 of file saddlepointlossmodel.hpp.

◆ CumGen4thDerivativeCond()

Real CumGen4thDerivativeCond	(	const std::vector< Real > &	invUncondProbs,
		Real	saddle,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 666 of file saddlepointlossmodel.hpp.

◆ CumGen0234DerivCond()

ext::tuple< Real, Real, Real, Real > CumGen0234DerivCond	(	const std::vector< Real > &	invUncondProbs,
		Real	saddle,
		const std::vector< Real > &	mktFactor
	)		const

protected

Returns the cumulant and second to fourth derivatives together. Included for optimization, most methods work on expansion of these terms. Alternatively use a local private buffer member?

Definition at line 699 of file saddlepointlossmodel.hpp.

◆ CumGen02DerivCond()

ext::tuple< Real, Real > CumGen02DerivCond	(	const std::vector< Real > &	invUncondProbs,
		Real	saddle,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 741 of file saddlepointlossmodel.hpp.

◆ CumulantGenerating()

Real CumulantGenerating	(	const Date &	date,
		Real	s
	)		const

protected

Returns the cumulant generating function (zero-th order expansion term) weighting the conditional value by the prob density of the market factor, called by integrations

Definition at line 387 of file saddlepointlossmodel.hpp.

◆ CumGen1stDerivative()

Real CumGen1stDerivative	(	const Date &	date,
		Real	s
	)		const

protected

Definition at line 403 of file saddlepointlossmodel.hpp.

◆ CumGen2ndDerivative()

Real CumGen2ndDerivative	(	const Date &	date,
		Real	s
	)		const

protected

Definition at line 419 of file saddlepointlossmodel.hpp.

◆ CumGen3rdDerivative()

Real CumGen3rdDerivative	(	const Date &	date,
		Real	s
	)		const

protected

Definition at line 435 of file saddlepointlossmodel.hpp.

◆ CumGen4thDerivative()

Real CumGen4thDerivative	(	const Date &	date,
		Real	s
	)		const

protected

Definition at line 451 of file saddlepointlossmodel.hpp.

◆ findSaddle()

Real findSaddle	(	const std::vector< Real > &	invUncondProbs,
		Real	lossLevel,
		const std::vector< Real > &	mktFactor,
		Real	accuracy = `1.0e-3`,
		Natural	maxEvaluations = `50`
	)		const

protected

Calculates the mkt-fct-conditional saddle point for the loss level given and the probability passed. The date is implicitly given through the probability. Performance requires to pass the probabilities for that date. Otherwise once we integrate this over the market factor we would be computing the same probabilities over and over. While this works fine here some models of the recovery rate might require the date.

The passed lossLevel is in total portfolio loss fractional units.

Definition at line 776 of file saddlepointlossmodel.hpp.

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◆ percentile()

Real percentile	(	const Date &	d,
		Probability	percentile
	)		const

overridevirtual

Returns the loss amount at the requested date for which the probability of lossing that amount or less is equal to the value passed.

Reimplemented from DefaultLossModel.

Definition at line 847 of file saddlepointlossmodel.hpp.

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◆ probOverLossCond()

Probability probOverLossCond	(	const std::vector< Real > &	invUncondProbs,
		Real	trancheLossFract,
		const std::vector< Real > &	mktFactor
	)		const

protected

Conditional (on the mkt factor) prob of a loss fraction of the the tranched portfolio.

The trancheLossFract parameter is the fraction over the tranche notional and must be in [0,1].

Definition at line 877 of file saddlepointlossmodel.hpp.

◆ probOverLossPortfCond1stOrder()

Probability probOverLossPortfCond1stOrder	(	const std::vector< Real > &	invUncondProbs,
		Real	loss,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 1013 of file saddlepointlossmodel.hpp.

◆ probOverLoss()

Probability probOverLoss	(	const Date &	d,
		Real	lossFraction
	)		const

overridevirtual

Probability of the tranche losing the same or more than the fractional amount given.

The passed lossFraction is a fraction of losses over the tranche notional (not the portfolio).

Reimplemented from DefaultLossModel.

Definition at line 467 of file saddlepointlossmodel.hpp.

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◆ lossDistribution()

std::map< Real, Probability > lossDistribution ( const Date & ) const

overridevirtual

Full loss distribution.

Reimplemented from DefaultLossModel.

Definition at line 896 of file saddlepointlossmodel.hpp.

◆ probOverLossPortfCond()

Probability probOverLossPortfCond	(	const std::vector< Real > &	invUncondProbs,
		Real	loss,
		const std::vector< Real > &	mktFactor
	)		const

protected

Probability of having losses in the portfolio due to default events equal or larger than a given absolute loss value on a given date conditional to the latent model factor. The integral expression on the expansion is the first order integration as presented in several references, see for instance; equation 8 in R.Martin, K.Thompson, and C. Browne 's 'Taking to the Saddle', Risk Magazine, June 2001, page 91

The passed loss is in absolute value.

Definition at line 919 of file saddlepointlossmodel.hpp.

◆ probOverPortfLoss()

Probability probOverPortfLoss	(	const Date &	d,
		Real	loss
	)		const

Definition at line 488 of file saddlepointlossmodel.hpp.

◆ expectedTrancheLoss()

Real expectedTrancheLoss ( const Date & d ) const

overridevirtual

Reimplemented from DefaultLossModel.

Definition at line 504 of file saddlepointlossmodel.hpp.

◆ probDensityCond()

Probability probDensityCond	(	const std::vector< Real > &	invUncondPs,
		Real	loss,
		const std::vector< Real > &	mktFactor
	)		const

protected

Probability density of having losses in the total portfolio (untranched) due to default events equal to a given value on a given date conditional to the latent model factor. Based on the integrals of the expected shortfall.

NOTICE THIS IS ON THE TOTAL PORTFOLIO -— UNTRANCHED Probability density of having losses in the portfolio due to default events equal to a given value on a given date conditional to the w latent model factor. Based on the integrals of the expected shortfall. See......refernce.

Definition at line 1082 of file saddlepointlossmodel.hpp.

◆ probDensity()

Probability probDensity	(	const Date &	d,
		Real	loss
	)		const

Definition at line 520 of file saddlepointlossmodel.hpp.

◆ splitLossCond()

std::vector< Real > splitLossCond	(	const std::vector< Real > &	invUncondProbs,
		Real	loss,
		std::vector< Real >	mktFactor
	)		const

protected

Definition at line 1127 of file saddlepointlossmodel.hpp.

◆ expectedShortfallFullPortfolioCond()

Real expectedShortfallFullPortfolioCond	(	const std::vector< Real > &	invUncondProbs,
		Real	lossPerc,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 1266 of file saddlepointlossmodel.hpp.

◆ expectedShortfallTrancheCond()

Real expectedShortfallTrancheCond	(	const std::vector< Real > &	invUncondProbs,
		Real	lossPerc,
		Probability	percentile,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 1230 of file saddlepointlossmodel.hpp.

◆ expectedShortfallSplitCond()

std::vector< Real > expectedShortfallSplitCond	(	const std::vector< Real > &	invUncondProbs,
		Real	lossPerc,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 1195 of file saddlepointlossmodel.hpp.

◆ splitVaRLevel()

std::vector< Real > splitVaRLevel	(	const Date &	date,
		Real	loss
	)		const

overridevirtual

Sensitivities of the individual names to a given portfolio loss value due to defaults. It returns ratios to the total structure notional, which aggregated add up to the requested loss value. Notice then that it refers to the total portfolio, not the tranched basket.

: see equation 8 in VAR: who contributes and how much? by R.Martin, K.Thompson, and C. Browne in Risk Magazine, August 2001

The passed loss is the loss amount level at which we want to request the sensitivity. Equivalent to a percentile.

Reimplemented from DefaultLossModel.

Definition at line 536 of file saddlepointlossmodel.hpp.

◆ expectedShortfall()

Real expectedShortfall	(	const Date &	d,
		Probability	percentile
	)		const

overridevirtual

Expected shortfall given a default loss percentile.

Reimplemented from DefaultLossModel.

Definition at line 1329 of file saddlepointlossmodel.hpp.

◆ conditionalExpectedLoss()

Real conditionalExpectedLoss	(	const std::vector< Real > &	invUncondProbs,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 1156 of file saddlepointlossmodel.hpp.

◆ conditionalExpectedTrancheLoss()

Real conditionalExpectedTrancheLoss	(	const std::vector< Real > &	invUncondProbs,
		const std::vector< Real > &	mktFactor
	)		const

protected

Definition at line 1174 of file saddlepointlossmodel.hpp.

◆ resetModel()

void resetModel ( )

overrideprotectedvirtual

Concrete models do now any updates/inits they need on basket reset.

Implements DefaultLossModel.

Definition at line 338 of file saddlepointlossmodel.hpp.

Member Data Documentation

◆ copula_

const ext::shared_ptr<ConstantLossLatentmodel<CP> > copula_

protected

Definition at line 348 of file saddlepointlossmodel.hpp.

◆ remainingSize_

Size remainingSize_

mutableprotected

Definition at line 350 of file saddlepointlossmodel.hpp.

◆ remainingNotionals_

std::vector<Real> remainingNotionals_

mutableprotected

Definition at line 351 of file saddlepointlossmodel.hpp.

◆ remainingNotional_

Real remainingNotional_

mutableprotected

Definition at line 352 of file saddlepointlossmodel.hpp.

◆ attachRatio_

Real attachRatio_

mutableprotected

Definition at line 354 of file saddlepointlossmodel.hpp.

◆ detachRatio_

Real detachRatio_

protected

Definition at line 354 of file saddlepointlossmodel.hpp.

Classes

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ SaddlePointLossModel()

Member Function Documentation

◆ CumulantGeneratingCond()

◆ CumGen1stDerivativeCond()

◆ CumGen2ndDerivativeCond()

◆ CumGen3rdDerivativeCond()

◆ CumGen4thDerivativeCond()

◆ CumGen0234DerivCond()

◆ CumGen02DerivCond()

◆ CumulantGenerating()

◆ CumGen1stDerivative()

◆ CumGen2ndDerivative()

◆ CumGen3rdDerivative()

◆ CumGen4thDerivative()

◆ findSaddle()

◆ percentile()

◆ probOverLossCond()

◆ probOverLossPortfCond1stOrder()

◆ probOverLoss()

◆ lossDistribution()

◆ probOverLossPortfCond()

◆ probOverPortfLoss()

◆ expectedTrancheLoss()

◆ probDensityCond()

◆ probDensity()

◆ splitLossCond()

◆ expectedShortfallFullPortfolioCond()

◆ expectedShortfallTrancheCond()

◆ expectedShortfallSplitCond()

◆ splitVaRLevel()

◆ expectedShortfall()

◆ conditionalExpectedLoss()

◆ conditionalExpectedTrancheLoss()

◆ resetModel()

Member Data Documentation

◆ copula_

◆ remainingSize_

◆ remainingNotionals_

◆ remainingNotional_

◆ attachRatio_

◆ detachRatio_