QuantLib: SaddlePointLossModel< CP > Class Template Reference

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

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

Inheritance diagram for SaddlePointLossModel< CP >:

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.
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.
Public Member Functions inherited from Observable
	Observable (const Observable &)
Observable &	operator= (const Observable &)
	Observable (Observable &&)=delete
Observable &	operator= (Observable &&)=delete
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
std::tuple< Real, Real, Real, Real >	CumGen0234DerivCond (const std::vector< Real > &invUncondProbs, Real saddle, const std::vector< Real > &mktFactor) const
std::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.
virtual std::vector< Real >	splitESFLevel (const Date &d, Real loss) const
	Associated ESF fraction to each counterparty.
virtual Real	densityTrancheLoss (const Date &d, Real lossFraction) const
	Probability density of a given loss fraction of the basket notional.
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.
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

Member Function Documentation

◆ CumulantGeneratingCond()

template<class CP>

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: \( K = \sum_j ln(1-p_j + p_j e^{N_j \times lgd_j \times s}) \)

◆ CumGen1stDerivativeCond()

template<class CP>

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.

◆ CumGen2ndDerivativeCond()

template<class CP>

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: \( K2 = \sum_j \frac{p_j \times (N_j \times LGD_j)^2 \times e^{N_j \times LGD_j \times s}} {1-p_j + p_j e^{N_j \times LGD_j \times s}} - (\frac{p_j \times N_j \times LGD_j \times e^{N_j \times LGD_j \times s}} {1-p_j + p_j e^{N_j \times LGD_j \times s}})^2 \)

◆ CumGen0234DerivCond()

template<class CP>

std::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?

◆ CumulantGenerating()

template<class CP>

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

◆ findSaddle()

template<class CP>

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.

◆ percentile()

template<class CP>

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.

◆ probOverLossCond()

template<class CP>

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].

◆ probOverLoss()

template<class CP>

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.

◆ lossDistribution()

template<class CP>

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

overridevirtual

Full loss distribution.

Reimplemented from DefaultLossModel.

◆ probOverLossPortfCond()

template<class CP>

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.

◆ expectedTrancheLoss()

template<class CP>

Real expectedTrancheLoss ( const Date & d ) const

overridevirtual

Reimplemented from DefaultLossModel.

◆ probDensityCond()

template<class CP>

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.

◆ splitVaRLevel()

template<class CP>

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.

◆ expectedShortfall()

template<class CP>

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

overridevirtual

Expected shortfall given a default loss percentile.

Reimplemented from DefaultLossModel.

◆ resetModel()

template<class CP>

void resetModel ( )

overrideprotectedvirtual

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

Implements DefaultLossModel.

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Member Function Documentation

◆ CumulantGeneratingCond()

◆ CumGen1stDerivativeCond()

◆ CumGen2ndDerivativeCond()

◆ CumGen0234DerivCond()

◆ CumulantGenerating()

◆ findSaddle()

◆ percentile()

◆ probOverLossCond()

◆ probOverLoss()

◆ lossDistribution()

◆ probOverLossPortfCond()

◆ expectedTrancheLoss()

◆ probDensityCond()

◆ splitVaRLevel()

◆ expectedShortfall()

◆ resetModel()