Default event Latent Model. More...

#include <qle/models/defaultprobabilitylatentmodel.hpp>

Inheritance diagram for DefaultLatentModel< copulaPolicy >:

Collaboration diagram for DefaultLatentModel< copulaPolicy >:

Public Member Functions
	DefaultLatentModel (const std::vector< std::vector< Real > > &factorWeights, LatentModelIntegrationType::LatentModelIntegrationType integralType, const initTraits &ini=initTraits())

	DefaultLatentModel (const Handle< Quote > &mktCorrel, Size nVariables, LatentModelIntegrationType::LatentModelIntegrationType integralType, const initTraits &ini=initTraits())

void	resetBasket (const QuantLib::ext::shared_ptr< Basket > basket) const

Probability	conditionalDefaultProbability (Probability prob, Size iName, const std::vector< Real > &mktFactors) const

Probability	conditionalDefaultProbabilityInvP (Real invCumYProb, Size iName, const std::vector< Real > &m) const

Probability	probOfDefault (Size iName, const Date &d) const

Real	defaultCorrelation (const Date &d, Size iNamei, Size iNamej) const

Probability	probAtLeastNEvents (Size n, const Date &date) const

Protected Member Functions
void	update () override

Probability	conditionalDefaultProbability (const Date &date, Size iName, const std::vector< Real > &mktFactors) const

Probability	condProbProduct (Real invCumYProb1, Real invCumYProb2, Size iName1, Size iName2, const std::vector< Real > &mktFactors) const

Real	conditionalProbAtLeastNEvents (Size n, const Date &date, const std::vector< Real > &mktFactors) const
	Conditional probability of n default events or more. More...

const QuantLib::ext::shared_ptr< LMIntegration > &	integration () const override
	access to integration: More...

Protected Attributes
QuantLib::ext::shared_ptr< Basket >	basket_

QuantLib::ext::shared_ptr< LMIntegration >	integration_

Private Types
typedef copulaPolicy::initTraits	initTraits

Detailed Description

template<class copulaPolicy>
class QuantExt::DefaultLatentModel< copulaPolicy >

Default event Latent Model.

This is a model for joint default events based on a generic Latent Model. It models solely the default events in a portfolio, not making any reference to severities, exposures, etc... An implicit correspondence is stablished between the variables modelled and the names in the basket given by the basket and model variable access indices. The class is parametric on the Latent Model copula.

Definition at line 44 of file defaultprobabilitylatentmodel.hpp.

Member Typedef Documentation

◆ initTraits

typedef copulaPolicy::initTraits initTraits

private

Definition at line 62 of file defaultprobabilitylatentmodel.hpp.

Constructor & Destructor Documentation

◆ DefaultLatentModel() [1/2]

DefaultLatentModel	(	const std::vector< std::vector< Real > > &	factorWeights,
		LatentModelIntegrationType::LatentModelIntegrationType	integralType,
		const initTraits &	ini = `initTraits()`
	)

Parameters

factorWeights	Latent model independent factors weights for each variable.
integralType	Integration type.
ini	Copula initialization if any.

Warning:: Baskets with realized defaults not tested/WIP.

Definition at line 73 of file defaultprobabilitylatentmodel.hpp.

        : LatentModel<copulaPolicy>(factorWeights, ini),
          integration_(LatentModel<copulaPolicy>::IntegrationFactory::createLMIntegration(factorWeights[0].size(),
                                                                                          integralType)) {}

◆ DefaultLatentModel() [2/2]

DefaultLatentModel	(	const Handle< Quote > &	mktCorrel,
		Size	nVariables,
		LatentModelIntegrationType::LatentModelIntegrationType	integralType,
		const initTraits &	ini = `initTraits()`
	)

Definition at line 79 of file defaultprobabilitylatentmodel.hpp.

82 : LatentModel<copulaPolicy>(mktCorrel, nVariables, ini),

83 integration_(LatentModel<copulaPolicy>::IntegrationFactory::createLMIntegration(1, integralType)) {}

Member Function Documentation

◆ resetBasket()

void resetBasket ( const QuantLib::ext::shared_ptr< Basket > basket ) const

Definition at line 92 of file defaultprobabilitylatentmodel.hpp.

                                                                       {
        basket_ = basket;
        // in the future change 'size' to 'liveSize'
        QL_REQUIRE(basket_->size() == factorWeights_.size(), "Incompatible new basket and model sizes.");
    }

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◆ conditionalDefaultProbability() [1/2]

Probability conditionalDefaultProbability	(	Probability	prob,
		Size	iName,
		const std::vector< Real > &	mktFactors
	)		const

Returns the probability of default of a given name conditional on the realization of a given set of values of the model independent factors. The date at which the probability is given is implicit in the probability since theres not other time dependence in this model.

Parameters

prob	Unconditional probability of default.
iName	desired name.
mktFactors	Value of LM independent factors.

Warning:: Most often it is preferred to use the method below avoiding the cumulative inversion.

Definition at line 109 of file defaultprobabilitylatentmodel.hpp.

                                                                                                                     {
// we can be called from the outside (from an integrable loss model)
//   but we are called often at integration points. This or
//   consider a list of friends.
#if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(basket_, "No portfolio basket set.");
#endif
        /*Avoid redundant call to minimum value inversion (might be \infty),
        and this independently of the copula function.
        */
        if (prob < 1.e-10)
            return 0.; // use library macro...
        return conditionalDefaultProbabilityInvP(inverseCumulativeY(prob, iName), iName, mktFactors);
    }

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◆ update()

void update ( )

overrideprotected

Definition at line 125 of file defaultprobabilitylatentmodel.hpp.

                           {
        if (basket_)
            basket_->notifyObservers();
        LatentModel<copulaPolicy>::update();
    }

◆ conditionalDefaultProbabilityInvP()

Probability conditionalDefaultProbabilityInvP	(	Real	invCumYProb,
		Size	iName,
		const std::vector< Real > &	m
	)		const

Returns the probability of default of a given name conditional on the realization of a given set of values of the model independent factors. The date at which the probability is given is implicit in the probability since theres not other time dependent in this model. Same intention as above but provides a performance opportunity, if the integration is along the market factors (as usually is) avoids computing the inverse of the probability on each call.

Parameters

invCumYProb	Inverse cumul of the unconditional probability of default, has to follow the same copula law for results to be coherent
iName	desired name.
m	Value of LM independent factors.

Definition at line 144 of file defaultprobabilitylatentmodel.hpp.

                                                                                                                {
        Real sumMs = std::inner_product(factorWeights_[iName].begin(), factorWeights_[iName].end(), m.begin(), 0.);
        Real res = cumulativeZ((invCumYProb - sumMs) / idiosyncFctrs_[iName]);
#if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(res >= 0. && res <= 1., "conditional probability " << res << "out of range");
#endif
 
        return res;
    }

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◆ conditionalDefaultProbability() [2/2]

Probability conditionalDefaultProbability	(	const Date &	date,
		Size	iName,
		const std::vector< Real > &	mktFactors
	)		const

protected

Returns the probability of default of a given name conditional on the realization of a given set of values of the model independent factors.

Parameters

date	The date for the probability of default.
iName	desired name.
mktFactors	Value of LM independent factors.

Same intention as the above methods. Usage of this one is typically more expensive because most often the date we call this method with repeats itself and with this one the probability can not be cached outside the call.

Definition at line 167 of file defaultprobabilitylatentmodel.hpp.

                                                                                                                     {
        const QuantLib::ext::shared_ptr<Pool>& pool = basket_->pool();
        Probability pDefUncond =
            pool->get(pool->names()[iName]).defaultProbability(basket_->defaultKeys()[iName])->defaultProbability(date);
        return conditionalDefaultProbability(pDefUncond, iName, mktFactors);
    }

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◆ condProbProduct()

Probability condProbProduct	(	Real	invCumYProb1,
		Real	invCumYProb2,
		Size	iName1,
		Size	iName2,
		const std::vector< Real > &	mktFactors
	)		const

protected

Conditional default probability product, intermediate step in the correlation calculation.

Definition at line 175 of file defaultprobabilitylatentmodel.hpp.

                                                                         {
        return conditionalDefaultProbabilityInvP(invCumYProb1, iName1, mktFactors) *
               conditionalDefaultProbabilityInvP(invCumYProb2, iName2, mktFactors);
    }

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◆ conditionalProbAtLeastNEvents()

Real conditionalProbAtLeastNEvents	(	Size	n,
		const Date &	date,
		const std::vector< Real > &	mktFactors
	)		const

protected

Conditional probability of n default events or more.

Definition at line 246 of file defaultprobabilitylatentmodel.hpp.

                                                                                                    {
    QL_REQUIRE(basket_, "No portfolio basket set.");
 
    /* \todo
    This algorithm traverses all permutations starting form the
    lowest one. This is inneficient, there shouldnt be any need to
    go through the invalid ones. Use combinations of n elements.
 
    See integration in O'Kane for homogeneous ntds.
    */
    // first position with as many defaults as desired:
    Size poolSize = basket_->size(); // move to 'livesize'
    const QuantLib::ext::shared_ptr<Pool>& pool = basket_->pool();
 
    BigNatural limit = static_cast<BigNatural>(std::pow(2., (int)(poolSize)));
 
    // Precalc conditional probabilities
    std::vector<Probability> pDefCond;
    for (Size i = 0; i < poolSize; i++)
        pDefCond.push_back(conditionalDefaultProbability(
            pool->get(pool->names()[i]).defaultProbability(basket_->defaultKeys()[i])->defaultProbability(date), i,
            mktFactors));
 
    Probability probNEventsOrMore = 0.;
    for (BigNatural mask = static_cast<BigNatural>(std::pow(2., (int)(n)) - 1); mask < limit; mask++) {
        // cheap permutations
        boost::dynamic_bitset<> bsetMask(poolSize, mask);
        if (bsetMask.count() >= n) {
            Probability pConfig = 1;
            for (Size i = 0; i < bsetMask.size(); i++)
                pConfig *= (bsetMask[i] ? pDefCond[i] : (1. - pDefCond[i]));
            probNEventsOrMore += pConfig;
        }
    }
    return probNEventsOrMore;
}

◆ integration()

const QuantLib::ext::shared_ptr< LMIntegration > & integration ( ) const

overrideprotected

access to integration:

Definition at line 184 of file defaultprobabilitylatentmodel.hpp.

184{ return integration_; }

◆ probOfDefault()

Probability probOfDefault	(	Size	iName,
		const Date &	d
	)		const

Computes the unconditional probability of default of a given name. Trivial method for testing

Definition at line 190 of file defaultprobabilitylatentmodel.hpp.

                                                               {
        QL_REQUIRE(basket_, "No portfolio basket set.");
        const QuantLib::ext::shared_ptr<Pool>& pool = basket_->pool();
        // avoid repeating this in the integration:
        Probability pUncond =
            pool->get(pool->names()[iName]).defaultProbability(basket_->defaultKeys()[iName])->defaultProbability(d);
        if (pUncond < 1.e-10)
            return 0.;
 
        return integratedExpectedValue(QuantLib::ext::function<Real(const std::vector<Real>& v1)>(
            QuantLib::ext::bind(&DefaultLatentModel<copulaPolicy>::conditionalDefaultProbabilityInvP, this,
                        inverseCumulativeY(pUncond, iName), iName, QuantLib::ext::placeholders::_1)));
    }

◆ defaultCorrelation()

Real defaultCorrelation	(	const Date &	d,
		Size	iNamei,
		Size	iNamej
	)		const

Pearsons' default probability correlation. Users should consider specialization on the copula type for specific distributions since that might simplify the integrations, most importantly if this is to be used in calibration of observations for factor coefficients as it is expensive to integrate directly.

Definition at line 222 of file defaultprobabilitylatentmodel.hpp.

                                                                                                                 {
    QL_REQUIRE(basket_, "No portfolio basket set.");
 
    const QuantLib::ext::shared_ptr<Pool>& pool = basket_->pool();
    // unconditionals:
    Probability pi =
        pool->get(pool->names()[iNamei]).defaultProbability(basket_->defaultKeys()[iNamei])->defaultProbability(d);
    Probability pj =
        pool->get(pool->names()[iNamej]).defaultProbability(basket_->defaultKeys()[iNamej])->defaultProbability(d);
    Real pipj = pi * pj;
    Real invPi = inverseCumulativeY(pi, iNamei);
    Real invPj = inverseCumulativeY(pj, iNamej);
    // avoid repetitive calls when i=j?
    Real E1i1j; // joint default covariance term
    if (iNamei != iNamej) {
        E1i1j = integratedExpectedValue(QuantLib::ext::function<Real(const std::vector<Real>& v1)>(
            QuantLib::ext::bind(&DefaultLatentModel<CP>::condProbProduct, this, invPi, invPj, iNamei, iNamej, QuantLib::ext::placeholders::_1)));
    } else {
        E1i1j = pi;
    }
    return (E1i1j - pipj) / std::sqrt(pipj * (1. - pi) * (1. - pj));
}

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◆ probAtLeastNEvents()

Probability probAtLeastNEvents	(	Size	n,
		const Date &	date
	)		const

Returns the probaility of having a given or larger number of defaults in the basket portfolio at a given time.

Definition at line 214 of file defaultprobabilitylatentmodel.hpp.

                                                                   {
        return integratedExpectedValue(QuantLib::ext::function<Real(const std::vector<Real>& v1)>(QuantLib::ext::bind(
            &DefaultLatentModel<copulaPolicy>::conditionalProbAtLeastNEvents, this, n, QuantLib::ext::cref(date), QuantLib::ext::placeholders::_1)));
    }

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Member Data Documentation

◆ basket_

QuantLib::ext::shared_ptr<Basket> basket_

mutableprotected

Definition at line 58 of file defaultprobabilitylatentmodel.hpp.

◆ integration_

QuantLib::ext::shared_ptr<LMIntegration> integration_

protected

Definition at line 59 of file defaultprobabilitylatentmodel.hpp.

Public Member Functions

Protected Member Functions

Protected Attributes

Private Types

Detailed Description

Member Typedef Documentation

◆ initTraits

Constructor & Destructor Documentation

◆ DefaultLatentModel() [1/2]

◆ DefaultLatentModel() [2/2]

Member Function Documentation

◆ resetBasket()

◆ conditionalDefaultProbability() [1/2]

◆ update()

◆ conditionalDefaultProbabilityInvP()

◆ conditionalDefaultProbability() [2/2]

◆ condProbProduct()

◆ conditionalProbAtLeastNEvents()

◆ integration()

◆ probOfDefault()

◆ defaultCorrelation()

◆ probAtLeastNEvents()

Member Data Documentation

◆ basket_

◆ integration_