analytic Heston-model engine based on Fourier transform More...

#include <analytichestonengine.hpp>

Inheritance diagram for AnalyticHestonEngine:

Collaboration diagram for AnalyticHestonEngine:

Classes
class	AP_Helper

class	Integration

class	OptimalAlpha

Public Types
enum	ComplexLogFormula { Gatheral , BranchCorrection , AndersenPiterbarg , AndersenPiterbargOptCV , AsymptoticChF , AngledContour , AngledContourNoCV , OptimalCV }

Public Types inherited from Observer
typedef set_type::iterator	iterator

Public Member Functions
	AnalyticHestonEngine (const ext::shared_ptr< HestonModel > &model, Real relTolerance, Size maxEvaluations)

	AnalyticHestonEngine (const ext::shared_ptr< HestonModel > &model, Size integrationOrder=144)

	AnalyticHestonEngine (const ext::shared_ptr< HestonModel > &model, ComplexLogFormula cpxLog, const Integration &itg, Real andersenPiterbargEpsilon=1e-25, Real alpha=-0.5)

void	calculate () const override

std::complex< Real >	chF (const std::complex< Real > &z, Time t) const

std::complex< Real >	lnChF (const std::complex< Real > &z, Time t) const

Size	numberOfEvaluations () const

Real	priceVanillaPayoff (const ext::shared_ptr< PlainVanillaPayoff > &payoff, const Date &maturity) const

Real	priceVanillaPayoff (const ext::shared_ptr< PlainVanillaPayoff > &payoff, Time maturity) const

Public Member Functions inherited from GenericModelEngine< HestonModel, VanillaOption::arguments, VanillaOption::results >
	GenericModelEngine (Handle< HestonModel > model=Handle< HestonModel >())

	GenericModelEngine (const ext::shared_ptr< HestonModel > &model)

Public Member Functions inherited from GenericEngine< ArgumentsType, ResultsType >
PricingEngine::arguments *	getArguments () const override

const PricingEngine::results *	getResults () const override

void	reset () override

void	update () override

Public Member Functions inherited from PricingEngine
	~PricingEngine () override=default

virtual arguments *	getArguments () const =0

virtual const results *	getResults () const =0

virtual void	reset ()=0

virtual void	calculate () const =0

Public Member Functions inherited from Observable
	Observable ()=default

	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 ()

Static Public Member Functions
static void	doCalculation (Real riskFreeDiscount, Real dividendDiscount, Real spotPrice, Real strikePrice, Real term, Real kappa, Real theta, Real sigma, Real v0, Real rho, const TypePayoff &type, const Integration &integration, ComplexLogFormula cpxLog, const AnalyticHestonEngine *enginePtr, Real &value, Size &evaluations)

static ComplexLogFormula	optimalControlVariate (Time t, Real v0, Real kappa, Real theta, Real sigma, Real rho)

Protected Member Functions
virtual std::complex< Real >	addOnTerm (Real phi, Time t, Size j) const

Private Member Functions
Real	priceVanillaPayoff (const ext::shared_ptr< PlainVanillaPayoff > &payoff, Time maturity, Real fwd) const

Private Attributes
Size	evaluations_

const ComplexLogFormula	cpxLog_

const ext::shared_ptr< Integration >	integration_

const Real	andersenPiterbargEpsilon_

const Real	alpha_

Additional Inherited Members
Protected Attributes inherited from GenericModelEngine< HestonModel, VanillaOption::arguments, VanillaOption::results >
Handle< HestonModel >	model_

Protected Attributes inherited from GenericEngine< ArgumentsType, ResultsType >
ArgumentsType	arguments_

ResultsType	results_

Detailed Description

analytic Heston-model engine based on Fourier transform

Integration detail: Two algebraically equivalent formulations of the complex logarithm of the Heston model exist. Gatherals [2005] (also Duffie, Pan and Singleton [2000], and Schoutens, Simons and Tistaert[2004]) version does not cause discoutinuities whereas the original version (e.g. Heston [1993]) needs some sort of "branch correction" to work properly. Gatheral's version does also work with adaptive integration routines and should be preferred over the original Heston version.

References:

Heston, Steven L., 1993. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The review of Financial Studies, Volume 6, Issue 2, 327-343.

A. Sepp, Pricing European-Style Options under Jump Diffusion Processes with Stochastic Volatility: Applications of Fourier Transform (http://math.ut.ee/~spartak/papers/stochjumpvols.pdf)

R. Lord and C. Kahl, Why the rotation count algorithm works, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=921335

H. Albrecher, P. Mayer, W.Schoutens and J. Tistaert, The Little Heston Trap, http://www.schoutens.be/HestonTrap.pdf

J. Gatheral, The Volatility Surface: A Practitioner's Guide, Wiley Finance

F. Le Floc'h, Fourier Integration and Stochastic Volatility Calibration, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2362968

L. Andersen, and V. Piterbarg, 2010, Interest Rate Modeling, Volume I: Foundations and Vanilla Models, Atlantic Financial Press London.

L. Andersen and M. Lake, 2018 Robust High-Precision Option Pricing by Fourier Transforms: Contour Deformations and Double-Exponential Quadrature, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3231626

Tests:: the correctness of the returned value is tested by reproducing results available in web/literature and comparison with Black pricing.

Definition at line 91 of file analytichestonengine.hpp.

Member Enumeration Documentation

◆ ComplexLogFormula

enum ComplexLogFormula

Enumerator
Gatheral
BranchCorrection
AndersenPiterbarg
AndersenPiterbargOptCV
AsymptoticChF
AngledContour
AngledContourNoCV
OptimalCV

Definition at line 100 of file analytichestonengine.hpp.

Constructor & Destructor Documentation

◆ AnalyticHestonEngine() [1/3]

AnalyticHestonEngine	(	const ext::shared_ptr< HestonModel > &	model,
		Real	relTolerance,
		Size	maxEvaluations
	)

Definition at line 675 of file analytichestonengine.cpp.

◆ AnalyticHestonEngine() [2/3]

AnalyticHestonEngine	(	const ext::shared_ptr< HestonModel > &	model,
		Size	integrationOrder = `144`
	)

Definition at line 661 of file analytichestonengine.cpp.

◆ AnalyticHestonEngine() [3/3]

AnalyticHestonEngine	(	const ext::shared_ptr< HestonModel > &	model,
		ComplexLogFormula	cpxLog,
		const Integration &	itg,
		Real	andersenPiterbargEpsilon = `1e-25`,
		Real	alpha = `-0.5`
	)

Definition at line 689 of file analytichestonengine.cpp.

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Member Function Documentation

◆ calculate()

void calculate ( ) const

overridevirtual

Implements PricingEngine.

Reimplemented in AnalyticHestonHullWhiteEngine.

Definition at line 863 of file analytichestonengine.cpp.

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◆ chF()

std::complex< Real > chF	(	const std::complex< Real > &	z,
		Time	t
	)		const

Definition at line 580 of file analytichestonengine.cpp.

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◆ lnChF()

std::complex< Real > lnChF	(	const std::complex< Real > &	z,
		Time	t
	)		const

Definition at line 623 of file analytichestonengine.cpp.

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◆ numberOfEvaluations()

Size numberOfEvaluations ( ) const

Definition at line 723 of file analytichestonengine.cpp.

◆ doCalculation()

void doCalculation	(	Real	riskFreeDiscount,
		Real	dividendDiscount,
		Real	spotPrice,
		Real	strikePrice,
		Real	term,
		Real	kappa,
		Real	theta,
		Real	sigma,
		Real	v0,
		Real	rho,
		const TypePayoff &	type,
		const Integration &	integration,
		ComplexLogFormula	cpxLog,
		const AnalyticHestonEngine *	enginePtr,
		Real &	value,
		Size &	evaluations
	)

static

Definition at line 1070 of file analytichestonengine.cpp.

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◆ priceVanillaPayoff() [1/3]

Real priceVanillaPayoff	(	const ext::shared_ptr< PlainVanillaPayoff > &	payoff,
		const Date &	maturity
	)		const

Definition at line 727 of file analytichestonengine.cpp.

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◆ priceVanillaPayoff() [2/3]

Real priceVanillaPayoff	(	const ext::shared_ptr< PlainVanillaPayoff > &	payoff,
		Time	maturity
	)		const

Definition at line 739 of file analytichestonengine.cpp.

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◆ optimalControlVariate()

AnalyticHestonEngine::ComplexLogFormula optimalControlVariate	(	Time	t,
		Real	v0,
		Real	kappa,
		Real	theta,
		Real	sigma,
		Real	rho
	)

static

Definition at line 710 of file analytichestonengine.cpp.

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◆ addOnTerm()

std::complex< Real > addOnTerm	(	Real	phi,
		Time	t,
		Size	j
	)		const

protectedvirtual

Reimplemented in AnalyticH1HWEngine, AnalyticHestonHullWhiteEngine, BatesEngine, BatesDetJumpEngine, BatesDoubleExpEngine, and BatesDoubleExpDetJumpEngine.

Definition at line 307 of file analytichestonengine.hpp.

◆ priceVanillaPayoff() [3/3]

Real priceVanillaPayoff	(	const ext::shared_ptr< PlainVanillaPayoff > &	payoff,
		Time	maturity,
		Real	fwd
	)		const

private

Definition at line 750 of file analytichestonengine.cpp.

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Member Data Documentation

◆ evaluations_

Size evaluations_

mutableprivate

Definition at line 191 of file analytichestonengine.hpp.

◆ cpxLog_

const ComplexLogFormula cpxLog_

private

Definition at line 192 of file analytichestonengine.hpp.

◆ integration_

const ext::shared_ptr<Integration> integration_

private

Definition at line 193 of file analytichestonengine.hpp.

◆ andersenPiterbargEpsilon_

const Real andersenPiterbargEpsilon_

private

Definition at line 194 of file analytichestonengine.hpp.

◆ alpha_

const Real alpha_

private

Definition at line 194 of file analytichestonengine.hpp.

Classes

Public Types

Public Member Functions

Static Public Member Functions

Protected Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Member Enumeration Documentation

◆ ComplexLogFormula

Constructor & Destructor Documentation

◆ AnalyticHestonEngine() [1/3]

◆ AnalyticHestonEngine() [2/3]

◆ AnalyticHestonEngine() [3/3]

Member Function Documentation

◆ calculate()

◆ chF()

◆ lnChF()

◆ numberOfEvaluations()

◆ doCalculation()

◆ priceVanillaPayoff() [1/3]

◆ priceVanillaPayoff() [2/3]

◆ optimalControlVariate()

◆ addOnTerm()

◆ priceVanillaPayoff() [3/3]

Member Data Documentation

◆ evaluations_

◆ cpxLog_

◆ integration_

◆ andersenPiterbargEpsilon_

◆ alpha_