/usr/include/ql/math/statistics/gaussianstatistics.hpp is in libquantlib0-dev 1.1-2build1.
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/*
Copyright (C) 2003 Ferdinando Ametrano
Copyright (C) 2000, 2001, 2002, 2003 RiskMap srl
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file gaussianstatistics.hpp
\brief statistics tool for gaussian-assumption risk measures
*/
#ifndef quantlib_gaussian_statistics_h
#define quantlib_gaussian_statistics_h
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/statistics/generalstatistics.hpp>
namespace QuantLib {
//! Statistics tool for gaussian-assumption risk measures
/*! This class wraps a somewhat generic statistic tool and adds
a number of gaussian risk measures (e.g.: value-at-risk, expected
shortfall, etc.) based on the mean and variance provided by
the underlying statistic tool.
*/
template<class Stat>
class GenericGaussianStatistics : public Stat {
public:
typedef typename Stat::value_type value_type;
GenericGaussianStatistics() {}
GenericGaussianStatistics(const Stat& s) : Stat(s) {}
//! \name Gaussian risk measures
//@{
/*! returns the downside variance, defined as
\f[ \frac{N}{N-1} \times \frac{ \sum_{i=1}^{N}
\theta \times x_i^{2}}{ \sum_{i=1}^{N} w_i} \f],
where \f$ \theta \f$ = 0 if x > 0 and
\f$ \theta \f$ =1 if x <0
*/
Real gaussianDownsideVariance() const {
return gaussianRegret(0.0);
}
/*! returns the downside deviation, defined as the
square root of the downside variance.
*/
Real gaussianDownsideDeviation() const {
return std::sqrt(gaussianDownsideVariance());
}
/*! returns the variance of observations below target
\f[ \frac{\sum w_i (min(0, x_i-target))^2 }{\sum w_i}. \f]
See Dembo, Freeman "The Rules Of Risk", Wiley (2001)
*/
Real gaussianRegret(Real target) const;
/*! gaussian-assumption y-th percentile, defined as the value x
such that \f[ y = \frac{1}{\sqrt{2 \pi}}
\int_{-\infty}^{x} \exp (-u^2/2) du \f]
*/
Real gaussianPercentile(Real percentile) const;
Real gaussianTopPercentile(Real percentile) const;
//! gaussian-assumption Potential-Upside at a given percentile
Real gaussianPotentialUpside(Real percentile) const;
//! gaussian-assumption Value-At-Risk at a given percentile
Real gaussianValueAtRisk(Real percentile) const;
//! gaussian-assumption Expected Shortfall at a given percentile
/*! Assuming a gaussian distribution it
returns the expected loss in case that the loss exceeded
a VaR threshold,
\f[ \mathrm{E}\left[ x \;|\; x < \mathrm{VaR}(p) \right], \f]
that is the average of observations below the
given percentile \f$ p \f$.
Also know as conditional value-at-risk.
See Artzner, Delbaen, Eber and Heath,
"Coherent measures of risk", Mathematical Finance 9 (1999)
*/
Real gaussianExpectedShortfall(Real percentile) const;
//! gaussian-assumption Shortfall (observations below target)
Real gaussianShortfall(Real target) const;
//! gaussian-assumption Average Shortfall (averaged shortfallness)
Real gaussianAverageShortfall(Real target) const;
//@}
};
//! default gaussian statistic tool
typedef GenericGaussianStatistics<GeneralStatistics> GaussianStatistics;
//! Helper class for precomputed distributions
class StatsHolder {
public:
typedef Real value_type;
StatsHolder(Real mean,
Real standardDeviation)
: mean_(mean), standardDeviation_(standardDeviation) {}
~StatsHolder() {}
Real mean() const { return mean_; }
Real standardDeviation() const { return standardDeviation_; }
private:
Real mean_, standardDeviation_;
};
// inline definitions
template<class Stat>
inline
Real GenericGaussianStatistics<Stat>::gaussianRegret(Real target) const {
Real m = this->mean();
Real std = this->standardDeviation();
Real variance = std*std;
CumulativeNormalDistribution gIntegral(m, std);
NormalDistribution g(m, std);
Real firstTerm = variance + m*m - 2.0*target*m + target*target;
Real alfa = gIntegral(target);
Real secondTerm = m - target;
Real beta = variance*g(target);
Real result = alfa*firstTerm - beta*secondTerm;
return result/alfa;
}
/*! \pre percentile must be in range (0%-100%) extremes excluded */
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianPercentile(
Real percentile) const {
QL_REQUIRE(percentile>0.0,
"percentile (" << percentile << ") must be > 0.0");
QL_REQUIRE(percentile<1.0,
"percentile (" << percentile << ") must be < 1.0");
InverseCumulativeNormal gInverse(Stat::mean(),
Stat::standardDeviation());
return gInverse(percentile);
}
/*! \pre percentile must be in range (0%-100%) extremes excluded */
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianTopPercentile(
Real percentile) const {
return gaussianPercentile(1.0-percentile);
}
/*! \pre percentile must be in range [90%-100%) */
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianPotentialUpside(
Real percentile) const {
QL_REQUIRE(percentile<1.0 && percentile>=0.9,
"percentile (" << percentile << ") out of range [0.9, 1)");
Real result = gaussianPercentile(percentile);
// potential upside must be a gain, i.e., floored at 0.0
return std::max<Real>(result, 0.0);
}
/*! \pre percentile must be in range [90%-100%) */
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianValueAtRisk(
Real percentile) const {
QL_REQUIRE(percentile<1.0 && percentile>=0.9,
"percentile (" << percentile << ") out of range [0.9, 1)");
Real result = gaussianPercentile(1.0-percentile);
// VAR must be a loss
// this means that it has to be MIN(dist(1.0-percentile), 0.0)
// VAR must also be a positive quantity, so -MIN(*)
return -std::min<Real>(result, 0.0);
}
/*! \pre percentile must be in range [90%-100%) */
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianExpectedShortfall(
Real percentile) const {
QL_REQUIRE(percentile<1.0 && percentile>=0.9,
"percentile (" << percentile << ") out of range [0.9, 1)");
Real m = this->mean();
Real std = this->standardDeviation();
InverseCumulativeNormal gInverse(m, std);
Real var = gInverse(1.0-percentile);
NormalDistribution g(m, std);
Real result = m - std*std*g(var)/(1.0-percentile);
// expectedShortfall must be a loss
// this means that it has to be MIN(result, 0.0)
// expectedShortfall must also be a positive quantity, so -MIN(*)
return -std::min<Real>(result, 0.0);
}
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianShortfall(
Real target) const {
CumulativeNormalDistribution gIntegral(this->mean(),
this->standardDeviation());
return gIntegral(target);
}
template<class Stat>
inline Real GenericGaussianStatistics<Stat>::gaussianAverageShortfall(
Real target) const {
Real m = this->mean();
Real std = this->standardDeviation();
CumulativeNormalDistribution gIntegral(m, std);
NormalDistribution g(m, std);
return ( (target-m) + std*std*g(target)/gIntegral(target) );
}
}
#endif
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