/**
* Copyright (C) 2011 - present by OpenGamma Inc. and the OpenGamma group of companies
*
* Please see distribution for license.
*/
package com.opengamma.analytics.financial.interestrate.payments.provider;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import com.opengamma.analytics.financial.interestrate.PresentValueSABRSensitivityDataBundle;
import com.opengamma.analytics.financial.interestrate.payments.derivative.CapFloorCMS;
import com.opengamma.analytics.financial.interestrate.payments.derivative.Payment;
import com.opengamma.analytics.financial.interestrate.swap.derivative.SwapFixedCoupon;
import com.opengamma.analytics.financial.model.option.definition.SABRInterestRateParameters;
import com.opengamma.analytics.financial.model.option.pricing.analytic.formula.EuropeanVanillaOption;
import com.opengamma.analytics.financial.model.option.pricing.analytic.formula.SABRExtrapolationRightFunction;
import com.opengamma.analytics.financial.model.volatility.smile.function.SABRFormulaData;
import com.opengamma.analytics.financial.provider.calculator.discounting.ParRateCurveSensitivityDiscountingCalculator;
import com.opengamma.analytics.financial.provider.calculator.discounting.ParRateDiscountingCalculator;
import com.opengamma.analytics.financial.provider.description.interestrate.SABRSwaptionProviderInterface;
import com.opengamma.analytics.financial.provider.sensitivity.multicurve.MulticurveSensitivity;
import com.opengamma.analytics.financial.provider.sensitivity.multicurve.MultipleCurrencyMulticurveSensitivity;
import com.opengamma.analytics.math.MathException;
import com.opengamma.analytics.math.function.Function1D;
import com.opengamma.analytics.math.integration.RungeKuttaIntegrator1D;
import com.opengamma.util.ArgumentChecker;
import com.opengamma.util.money.Currency;
import com.opengamma.util.money.MultipleCurrencyAmount;
import com.opengamma.util.tuple.DoublesPair;
/**
* Class used to compute the price of a CMS cap/floor by swaption replication on a SABR formula with extrapolation.
* Reference: Hagan, P. S. (2003). Convexity conundrums: Pricing CMS swaps, caps, and floors. Wilmott Magazine, March, pages 38--44.
* OpenGamma implementation note: Replication pricing for linear and TEC format CMS, Version 1.2, March 2011.
* OpenGamma implementation note for the extrapolation: Smile extrapolation, version 1.2, May 2011.
*/
public class CapFloorCMSSABRExtrapolationRightReplicationMethod extends CapFloorCMSSABRReplicationAbstractMethod {
/**
* The cut-off strike. The smile is extrapolated above that level.
*/
private final double _cutOffStrike;
/**
* The tail thickness parameter.
*/
private final double _mu;
/**
* The par rate calculator.
*/
private static final ParRateDiscountingCalculator PRDC = ParRateDiscountingCalculator.getInstance();
/**
* The par rate sensitivity calculator.
*/
private static final ParRateCurveSensitivityDiscountingCalculator PRCSDC = ParRateCurveSensitivityDiscountingCalculator.getInstance();
/**
* Default constructor of the CMS cap/floor replication method. The default integration interval is 1.00 (100%).
* @param cutOffStrike The cut-off strike.
* @param mu The tail thickness parameter.
*/
public CapFloorCMSSABRExtrapolationRightReplicationMethod(final double cutOffStrike, final double mu) {
super(1.0);
_mu = mu;
_cutOffStrike = cutOffStrike;
}
/**
* Default constructor of the CMS cap/floor replication method. The default integration interval is 1.00 (100%).
* @param cutOffStrike The cut-off strike.
* @param mu The tail thickness parameter.
* @param integrationInterval Integration range.
*/
public CapFloorCMSSABRExtrapolationRightReplicationMethod(final double cutOffStrike, final double mu, final double integrationInterval) {
super(integrationInterval);
_mu = mu;
_cutOffStrike = cutOffStrike;
}
/**
* Compute the present value of a CMS cap/floor by replication in SABR framework with extrapolation on the right.
* @param cmsCapFloor The CMS cap/floor.
* @param sabrData The SABR data bundle.
* @return The present value.
*/
@Override
public MultipleCurrencyAmount presentValue(final CapFloorCMS cmsCapFloor, final SABRSwaptionProviderInterface sabrData) {
ArgumentChecker.notNull(cmsCapFloor, "CMA cap/floor");
ArgumentChecker.notNull(sabrData, "SABR swaption provider");
final Currency ccy = cmsCapFloor.getCurrency();
final SABRInterestRateParameters sabrParameter = sabrData.getSABRParameter();
final SwapFixedCoupon<? extends Payment> underlyingSwap = cmsCapFloor.getUnderlyingSwap();
final double forward = underlyingSwap.accept(PRDC, sabrData.getMulticurveProvider());
final double discountFactorTp = sabrData.getMulticurveProvider().getDiscountFactor(ccy, cmsCapFloor.getPaymentTime());
final double maturity = underlyingSwap.getFixedLeg().getNthPayment(underlyingSwap.getFixedLeg().getNumberOfPayments() - 1).getPaymentTime() - cmsCapFloor.getSettlementTime();
final DoublesPair expiryMaturity = new DoublesPair(cmsCapFloor.getFixingTime(), maturity);
final double alpha = sabrParameter.getAlpha(expiryMaturity);
final double beta = sabrParameter.getBeta(expiryMaturity);
final double rho = sabrParameter.getRho(expiryMaturity);
final double nu = sabrParameter.getNu(expiryMaturity);
final SABRFormulaData sabrPoint = new SABRFormulaData(alpha, beta, rho, nu);
final CMSIntegrant integrant = new CMSIntegrant(cmsCapFloor, sabrPoint, forward, _cutOffStrike, _mu);
final double strike = cmsCapFloor.getStrike();
final double factor = discountFactorTp / integrant.h(forward) * integrant.g(forward);
final double strikePart = factor * integrant.k(strike) * integrant.bs(strike);
final double absoluteTolerance = 1.0 / (factor * Math.abs(cmsCapFloor.getNotional()) * cmsCapFloor.getPaymentYearFraction());
final double relativeTolerance = 1E-10;
final RungeKuttaIntegrator1D integrator = new RungeKuttaIntegrator1D(absoluteTolerance, relativeTolerance, getNbIteration());
double integralPart;
try {
if (cmsCapFloor.isCap()) {
integralPart = discountFactorTp * integrator.integrate(integrant, strike, strike + getIntegrationInterval());
} else {
integralPart = discountFactorTp * integrator.integrate(integrant, 0.0, strike);
}
} catch (final Exception e) {
throw new MathException(e);
}
final double priceCMS = (strikePart + integralPart) * cmsCapFloor.getNotional() * cmsCapFloor.getPaymentYearFraction();
return MultipleCurrencyAmount.of(cmsCapFloor.getCurrency(), priceCMS);
}
/**
* Computes the present value sensitivity to the yield curves of a CMS cap/floor by replication in the SABR framework with extrapolation on the right.
* @param cmsCapFloor The CMS cap/floor.
* @param sabrData The SABR data bundle. The SABR function need to be the Hagan function.
* @return The present value sensitivity to curves.
*/
@Override
public MultipleCurrencyMulticurveSensitivity presentValueCurveSensitivity(final CapFloorCMS cmsCapFloor, final SABRSwaptionProviderInterface sabrData) {
ArgumentChecker.notNull(cmsCapFloor, "CMA cap/floor");
ArgumentChecker.notNull(sabrData, "SABR swaption provider");
final Currency ccy = cmsCapFloor.getCurrency();
final SABRInterestRateParameters sabrParameter = sabrData.getSABRParameter();
final SwapFixedCoupon<? extends Payment> underlyingSwap = cmsCapFloor.getUnderlyingSwap();
final double forward = underlyingSwap.accept(PRDC, sabrData.getMulticurveProvider());
final double discountFactor = sabrData.getMulticurveProvider().getDiscountFactor(ccy, cmsCapFloor.getPaymentTime());
final double strike = cmsCapFloor.getStrike();
final double maturity = underlyingSwap.getFixedLeg().getNthPayment(underlyingSwap.getFixedLeg().getNumberOfPayments() - 1).getPaymentTime() - cmsCapFloor.getSettlementTime();
final DoublesPair expiryMaturity = new DoublesPair(cmsCapFloor.getFixingTime(), maturity);
final double alpha = sabrParameter.getAlpha(expiryMaturity);
final double beta = sabrParameter.getBeta(expiryMaturity);
final double rho = sabrParameter.getRho(expiryMaturity);
final double nu = sabrParameter.getNu(expiryMaturity);
final SABRFormulaData sabrPoint = new SABRFormulaData(alpha, beta, rho, nu);
// Common
final CMSIntegrant integrantPrice = new CMSIntegrant(cmsCapFloor, sabrPoint, forward, _cutOffStrike, _mu);
final CMSDeltaIntegrant integrantDelta = new CMSDeltaIntegrant(cmsCapFloor, sabrPoint, forward, _cutOffStrike, _mu);
final double factor = discountFactor / integrantDelta.h(forward) * integrantDelta.g(forward);
final double absoluteTolerance = 1.0 / (factor * Math.abs(cmsCapFloor.getNotional()) * cmsCapFloor.getPaymentYearFraction());
final double relativeTolerance = 1E-10;
final RungeKuttaIntegrator1D integrator = new RungeKuttaIntegrator1D(absoluteTolerance, relativeTolerance, getNbIteration());
// Price
final double[] bs = integrantDelta.bsbsp(strike);
@SuppressWarnings("synthetic-access")
final double[] n = integrantDelta.nnp(forward);
final double strikePartPrice = discountFactor * integrantDelta.k(strike) * n[0] * bs[0];
double integralPartPrice;
try {
if (cmsCapFloor.isCap()) {
integralPartPrice = discountFactor * integrator.integrate(integrantPrice, strike, strike + getIntegrationInterval());
} else {
integralPartPrice = discountFactor * integrator.integrate(integrantPrice, 0.0, strike);
}
} catch (final Exception e) {
throw new RuntimeException(e);
}
final double price = (strikePartPrice + integralPartPrice) * cmsCapFloor.getNotional() * cmsCapFloor.getPaymentYearFraction();
// Delta
final double strikePart = discountFactor * integrantDelta.k(strike) * (n[1] * bs[0] + n[0] * bs[1]);
double integralPart;
try {
if (cmsCapFloor.isCap()) {
integralPart = discountFactor * integrator.integrate(integrantDelta, strike, strike + getIntegrationInterval());
} else {
integralPart = discountFactor * integrator.integrate(integrantDelta, 0.0, strike);
}
} catch (final Exception e) {
throw new RuntimeException(e);
}
final double deltaS0 = (strikePart + integralPart) * cmsCapFloor.getNotional() * cmsCapFloor.getPaymentYearFraction();
final double deltaPD = price / discountFactor;
final double sensiDF = -cmsCapFloor.getPaymentTime() * discountFactor * deltaPD;
final List<DoublesPair> list = new ArrayList<>();
list.add(new DoublesPair(cmsCapFloor.getPaymentTime(), sensiDF));
final Map<String, List<DoublesPair>> resultMapDsc = new HashMap<>();
resultMapDsc.put(sabrData.getMulticurveProvider().getName(ccy), list);
final MulticurveSensitivity dscDp = MulticurveSensitivity.ofYieldDiscounting(resultMapDsc);
final MulticurveSensitivity forwardDp = cmsCapFloor.getUnderlyingSwap().accept(PRCSDC, sabrData.getMulticurveProvider());
return MultipleCurrencyMulticurveSensitivity.of(ccy, dscDp.plus(forwardDp.multipliedBy(deltaS0)));
}
/**
* Computes the present value sensitivity to the SABR parameters of a CMS cap/floor by replication in SABR framework with extrapolation on the right.
* @param cmsCapFloor The CMS cap/floor.
* @param sabrData The SABR data bundle. The SABR function need to be the Hagan function.
* @return The present value sensitivity to SABR parameters.
*/
@Override
public PresentValueSABRSensitivityDataBundle presentValueSABRSensitivity(final CapFloorCMS cmsCapFloor, final SABRSwaptionProviderInterface sabrData) {
ArgumentChecker.notNull(cmsCapFloor, "CMA cap/floor");
ArgumentChecker.notNull(sabrData, "SABR swaption provider");
final Currency ccy = cmsCapFloor.getCurrency();
final SABRInterestRateParameters sabrParameter = sabrData.getSABRParameter();
final SwapFixedCoupon<? extends Payment> underlyingSwap = cmsCapFloor.getUnderlyingSwap();
final double forward = underlyingSwap.accept(PRDC, sabrData.getMulticurveProvider());
final double discountFactorTp = sabrData.getMulticurveProvider().getDiscountFactor(ccy, cmsCapFloor.getPaymentTime());
final double strike = cmsCapFloor.getStrike();
final double maturity = underlyingSwap.getFixedLeg().getNthPayment(underlyingSwap.getFixedLeg().getNumberOfPayments() - 1).getPaymentTime() - cmsCapFloor.getSettlementTime();
final DoublesPair expiryMaturity = new DoublesPair(cmsCapFloor.getFixingTime(), maturity);
final double alpha = sabrParameter.getAlpha(expiryMaturity);
final double beta = sabrParameter.getBeta(expiryMaturity);
final double rho = sabrParameter.getRho(expiryMaturity);
final double nu = sabrParameter.getNu(expiryMaturity);
final SABRFormulaData sabrPoint = new SABRFormulaData(alpha, beta, rho, nu);
final CMSVegaIntegrant integrantVega = new CMSVegaIntegrant(cmsCapFloor, sabrPoint, forward, _cutOffStrike, _mu);
final double factor = discountFactorTp / integrantVega.h(forward) * integrantVega.g(forward);
final SABRExtrapolationRightFunction sabrExtrapolation = new SABRExtrapolationRightFunction(forward, sabrPoint, _cutOffStrike, cmsCapFloor.getFixingTime(), _mu);
final EuropeanVanillaOption option = new EuropeanVanillaOption(strike, cmsCapFloor.getFixingTime(), cmsCapFloor.isCap());
final double factor2 = factor * integrantVega.k(strike);
final double[] strikePartPrice = new double[4];
sabrExtrapolation.priceAdjointSABR(option, strikePartPrice);
for (int loopvega = 0; loopvega < 4; loopvega++) {
strikePartPrice[loopvega] *= factor2;
}
final double absoluteTolerance = 1.0 / (factor * Math.abs(cmsCapFloor.getNotional()) * cmsCapFloor.getPaymentYearFraction());
final double relativeTolerance = 1E-3;
final RungeKuttaIntegrator1D integrator = new RungeKuttaIntegrator1D(absoluteTolerance, relativeTolerance, getNbIteration());
final double[] integralPart = new double[4];
final double[] totalSensi = new double[4];
for (int loopparameter = 0; loopparameter < 4; loopparameter++) {
integrantVega.setParameterIndex(loopparameter);
try {
if (cmsCapFloor.isCap()) {
integralPart[loopparameter] = discountFactorTp * integrator.integrate(integrantVega, strike, strike + getIntegrationInterval());
} else {
integralPart[loopparameter] = discountFactorTp * integrator.integrate(integrantVega, 0.0, strike);
}
} catch (final Exception e) {
throw new RuntimeException(e);
}
totalSensi[loopparameter] = (strikePartPrice[loopparameter] + integralPart[loopparameter]) * cmsCapFloor.getNotional() * cmsCapFloor.getPaymentYearFraction();
}
final PresentValueSABRSensitivityDataBundle sensi = new PresentValueSABRSensitivityDataBundle();
sensi.addAlpha(expiryMaturity, totalSensi[0]);
sensi.addBeta(expiryMaturity, totalSensi[1]);
sensi.addRho(expiryMaturity, totalSensi[2]);
sensi.addNu(expiryMaturity, totalSensi[3]);
return sensi;
}
/**
* Computes the present value sensitivity to the strike of a CMS cap/floor by replication in SABR framework with extrapolation on the right.
* @param cmsCapFloor The CMS cap/floor.
* @param sabrData The SABR data bundle. The SABR function need to be the Hagan function.
* @return The present value sensitivity to strike.
*/
@Override
public double presentValueStrikeSensitivity(final CapFloorCMS cmsCapFloor, final SABRSwaptionProviderInterface sabrData) {
ArgumentChecker.notNull(cmsCapFloor, "CMA cap/floor");
ArgumentChecker.notNull(sabrData, "SABR swaption provider");
final Currency ccy = cmsCapFloor.getCurrency();
final SABRInterestRateParameters sabrParameter = sabrData.getSABRParameter();
final SwapFixedCoupon<? extends Payment> underlyingSwap = cmsCapFloor.getUnderlyingSwap();
final double forward = underlyingSwap.accept(PRDC, sabrData.getMulticurveProvider());
final double discountFactor = sabrData.getMulticurveProvider().getDiscountFactor(ccy, cmsCapFloor.getPaymentTime());
final double strike = cmsCapFloor.getStrike();
final double maturity = underlyingSwap.getFixedLeg().getNthPayment(underlyingSwap.getFixedLeg().getNumberOfPayments() - 1).getPaymentTime() - cmsCapFloor.getSettlementTime();
final DoublesPair expiryMaturity = new DoublesPair(cmsCapFloor.getFixingTime(), maturity);
final double alpha = sabrParameter.getAlpha(expiryMaturity);
final double beta = sabrParameter.getBeta(expiryMaturity);
final double rho = sabrParameter.getRho(expiryMaturity);
final double nu = sabrParameter.getNu(expiryMaturity);
final SABRFormulaData sabrPoint = new SABRFormulaData(alpha, beta, rho, nu);
final CMSStrikeIntegrant integrant = new CMSStrikeIntegrant(cmsCapFloor, sabrPoint, forward, _cutOffStrike, _mu);
final double factor = discountFactor * integrant.g(forward) / integrant.h(forward);
final double absoluteTolerance = 1.0E-9;
final double relativeTolerance = 1.0E-5;
final RungeKuttaIntegrator1D integrator = new RungeKuttaIntegrator1D(absoluteTolerance, relativeTolerance, getNbIteration());
final SABRExtrapolationRightFunction sabrExtrapolation = new SABRExtrapolationRightFunction(forward, sabrPoint, _cutOffStrike, cmsCapFloor.getFixingTime(), _mu);
final EuropeanVanillaOption option = new EuropeanVanillaOption(strike, cmsCapFloor.getFixingTime(), cmsCapFloor.isCap());
final double[] kpkpp = integrant.kpkpp(strike);
double firstPart;
double thirdPart;
if (cmsCapFloor.isCap()) {
firstPart = -kpkpp[0] * integrant.bs(strike);
thirdPart = integrator.integrate(integrant, strike, strike + getIntegrationInterval());
} else {
firstPart = 3 * kpkpp[0] * integrant.bs(strike);
thirdPart = integrator.integrate(integrant, 0.0, strike);
}
final double secondPart = integrant.k(strike) * sabrExtrapolation.priceDerivativeStrike(option);
return cmsCapFloor.getNotional() * cmsCapFloor.getPaymentYearFraction() * factor * (firstPart + secondPart + thirdPart);
}
/**
* Inner class to implement the integration used in price replication.
*/
private class CMSIntegrant extends Function1D<Double, Double> {
protected static final double EPS = 1E-10;
private final int _nbFixedPeriod;
private final int _nbFixedPaymentYear;
private final double _tau;
private final double _delta;
private final double _eta;
private final double _timeToExpiry;
private final double _strike;
private final double _forward;
private final double _factor;
private final SABRExtrapolationRightFunction _sabrExtrapolation;
private final boolean _isCall;
/**
* Gets the _nbFixedPeriod field.
* @return the _nbFixedPeriod
*/
public int getNbFixedPeriod() {
return _nbFixedPeriod;
}
/**
* Gets the _nbFixedPaymentYear field.
* @return the _nbFixedPaymentYear
*/
public int getNbFixedPaymentYear() {
return _nbFixedPaymentYear;
}
/**
* Gets the _tau field.
* @return the _tau
*/
public double getTau() {
return _tau;
}
/**
* Gets the _eta field.
* @return the _eta
*/
public double getEta() {
return _eta;
}
/**
* Gets the _timeToExpiry field.
* @return the _timeToExpiry
*/
public double getTimeToExpiry() {
return _timeToExpiry;
}
/**
* Gets the _sabrExtrapolation field.
* @return the _sabrExtrapolation
*/
public SABRExtrapolationRightFunction getSabrExtrapolation() {
return _sabrExtrapolation;
}
/**
* Gets the _isCall field.
* @return the _isCall
*/
public boolean isCall() {
return _isCall;
}
/**
* Gets the _strike field.
* @return the _strike
*/
public double getStrike() {
return _strike;
}
/**
* Constructor.
* @param cmsCap The CMS cap/floor.
* @param sabrParameter The SABR parameters.
* @param forward The forward.
* @param cutOffStrike The cut-off strike.
* @param mu The tail thickness parameter.
*/
public CMSIntegrant(final CapFloorCMS cmsCap, final SABRFormulaData sabrPoint, final double forward, final double cutOffStrike, final double mu) {
_nbFixedPeriod = cmsCap.getUnderlyingSwap().getFixedLeg().getPayments().length;
_nbFixedPaymentYear = (int) Math.round(1.0 / cmsCap.getUnderlyingSwap().getFixedLeg().getNthPayment(0).getPaymentYearFraction());
_tau = 1.0 / _nbFixedPaymentYear;
_delta = cmsCap.getPaymentTime() - cmsCap.getSettlementTime();
_eta = -_delta;
_timeToExpiry = cmsCap.getFixingTime();
_forward = forward;
_sabrExtrapolation = new SABRExtrapolationRightFunction(forward, sabrPoint, cutOffStrike, _timeToExpiry, mu);
_isCall = cmsCap.isCap();
_strike = cmsCap.getStrike();
_factor = g(_forward) / h(_forward);
}
@Override
public Double evaluate(final Double x) {
final double[] kD = kpkpp(x);
// Implementation note: kD[0] contains the first derivative of k; kD[1] the second derivative of k.
return _factor * (kD[1] * (x - _strike) + 2.0 * kD[0]) * bs(x);
}
/**
* The approximation of the discount factor as function of the swap rate.
* @param x The swap rate.
* @return The discount factor.
*/
double h(final double x) {
return Math.pow(1.0 + _tau * x, _eta);
}
/**
* The cash annuity.
* @param x The swap rate.
* @return The annuity.
*/
double g(final double x) {
if (x >= EPS) {
final double periodFactor = 1 + x / _nbFixedPaymentYear;
final double nPeriodDiscount = Math.pow(periodFactor, -_nbFixedPeriod);
return 1.0 / x * (1.0 - nPeriodDiscount);
}
return ((double) _nbFixedPeriod) / _nbFixedPaymentYear;
}
/**
* The factor used in the strike part and in the integration of the replication.
* @param x The swap rate.
* @return The factor.
*/
double k(final double x) {
double g;
double h;
if (x >= EPS) {
final double periodFactor = 1 + x / _nbFixedPaymentYear;
final double nPeriodDiscount = Math.pow(periodFactor, -_nbFixedPeriod);
g = 1.0 / x * (1.0 - nPeriodDiscount);
h = Math.pow(1.0 + _tau * x, _eta);
} else {
g = ((double) _nbFixedPeriod) / _nbFixedPaymentYear;
h = 1.0;
}
return h / g;
}
/**
* The first and second derivative of the function k.
* @param x The swap rate.
* @return The derivative (first element is the first derivative, second element is second derivative.
*/
protected double[] kpkpp(final double x) {
final double periodFactor = 1 + x / _nbFixedPaymentYear;
final double nPeriodDiscount = Math.pow(periodFactor, -_nbFixedPeriod);
/**
* The value of the annuity and its first and second derivative.
*/
double g, gp, gpp;
if (x >= EPS) {
g = 1.0 / x * (1.0 - nPeriodDiscount);
gp = -g / x + _nbFixedPeriod / x / _nbFixedPaymentYear * nPeriodDiscount / periodFactor;
gpp = 2.0 / (x * x) * g - 2.0 * _nbFixedPeriod / (x * x) / _nbFixedPaymentYear * nPeriodDiscount / periodFactor - (_nbFixedPeriod + 1.0) * _nbFixedPeriod / x
/ (_nbFixedPaymentYear * _nbFixedPaymentYear) * nPeriodDiscount / (periodFactor * periodFactor);
} else {
// Implementation comment: When x is (almost) 0, useful for CMS swaps which are priced as CMS cap of strike 0.
g = ((double) _nbFixedPeriod) / _nbFixedPaymentYear;
gp = -_nbFixedPeriod / 2.0 * (_nbFixedPeriod + 1.0) / (_nbFixedPaymentYear * _nbFixedPaymentYear);
gpp = _nbFixedPeriod / 2.0 * (_nbFixedPeriod + 1.0) * (1.0 + (_nbFixedPeriod + 2.0) / 3.0) / (_nbFixedPaymentYear * _nbFixedPaymentYear * _nbFixedPaymentYear);
}
final double h = Math.pow(1.0 + _tau * x, _eta);
final double hp = _eta * _tau * h / periodFactor;
final double hpp = (_eta - 1.0) * _tau * hp / periodFactor;
final double kp = hp / g - h * gp / (g * g);
final double kpp = hpp / g - 2 * hp * gp / (g * g) - h * (gpp / (g * g) - 2 * (gp * gp) / (g * g * g));
return new double[] {kp, kpp};
}
/**
* The Black-Scholes formula with numeraire 1 as function of the strike.
* @param strike The strike.
* @return The Black-Scholes formula.
*/
double bs(final double strike) {
final EuropeanVanillaOption option = new EuropeanVanillaOption(strike, _timeToExpiry, _isCall);
return _sabrExtrapolation.price(option);
}
}
private class CMSDeltaIntegrant extends CMSIntegrant {
private final double[] _nnp;
public CMSDeltaIntegrant(final CapFloorCMS cmsCap, final SABRFormulaData sabrPoint, final double forward, final double cutOffStrike, final double mu) {
super(cmsCap, sabrPoint, forward, cutOffStrike, mu);
_nnp = nnp(forward);
}
@Override
public Double evaluate(final Double x) {
final double[] kD = kpkpp(x);
// Implementation note: kD[0] contains the first derivative of k; kD[1] the second derivative of k.
final double[] bs = bsbsp(x);
return (kD[1] * (x - getStrike()) + 2.0 * kD[0]) * (_nnp[1] * bs[0] + _nnp[0] * bs[1]);
}
/**
* The Black-Scholes formula and its derivative with respect to the forward.
* @param strike The strike.
* @return The Black-Scholes formula and its derivative.
*/
double[] bsbsp(final double strike) {
final double[] result = new double[2];
final EuropeanVanillaOption option = new EuropeanVanillaOption(strike, getTimeToExpiry(), isCall());
result[0] = getSabrExtrapolation().price(option);
result[1] = getSabrExtrapolation().priceDerivativeForward(option);
return result;
}
private double[] nnp(final double x) {
final double[] result = new double[2];
final double[] ggp = ggp(x);
final double[] hhp = hhp(x);
result[0] = ggp[0] / hhp[0];
result[1] = ggp[1] / hhp[0] - ggp[0] * hhp[1] / (hhp[0] * hhp[0]);
return result;
}
private double[] ggp(final double x) {
final double[] result = new double[2];
if (x >= EPS) {
final double periodFactor = 1 + x / getNbFixedPaymentYear();
final double nPeriodDiscount = Math.pow(periodFactor, -getNbFixedPeriod());
result[0] = 1.0 / x * (1.0 - nPeriodDiscount);
result[1] = -result[0] / x + getTau() * getNbFixedPeriod() / x * nPeriodDiscount / periodFactor;
} else {
result[0] = getNbFixedPeriod() * getTau();
result[1] = -getNbFixedPeriod() * (getNbFixedPeriod() + 1.0) * getTau() * getTau() / 2.0;
}
return result;
}
private double[] hhp(final double x) {
final double[] result = new double[2];
result[0] = Math.pow(1.0 + getTau() * x, getEta());
result[1] = getEta() * getTau() * result[0] / (1 + x * getTau());
return result;
}
}
private class CMSVegaIntegrant extends CMSIntegrant {
/**
* The index of the sensitivity computed.
*/
private int _parameterIndex;
/**
* Constructor.
* @param cmsCap The CMS cap/floor.
* @param sabrParameter The SABR parameters.
* @param forward The forward.
* @param cutOffStrike The cut-off strike.
* @param mu The tail thickness parameter.
*/
public CMSVegaIntegrant(final CapFloorCMS cmsCap, final SABRFormulaData sabrPoint, final double forward, final double cutOffStrike, final double mu) {
super(cmsCap, sabrPoint, forward, cutOffStrike, mu);
}
/**
* Sets the index of the sensitivity computed.
* @param parameterIndex The index.
*/
public void setParameterIndex(final int parameterIndex) {
this._parameterIndex = parameterIndex;
}
@SuppressWarnings("synthetic-access")
@Override
public Double evaluate(final Double x) {
final double[] kD = super.kpkpp(x);
// Implementation note: kD[0] contains the first derivative of k; kD[1] the second derivative of k.
final EuropeanVanillaOption option = new EuropeanVanillaOption(x, super._timeToExpiry, super._isCall);
final double[] priceDerivativeSABR = new double[4];
getSabrExtrapolation().priceAdjointSABR(option, priceDerivativeSABR);
return super._factor * (kD[1] * (x - super._strike) + 2.0 * kD[0]) * priceDerivativeSABR[_parameterIndex];
}
}
private class CMSStrikeIntegrant extends CMSIntegrant {
/**
* @param cmsCap
* @param sabrParameter
* @param forward
*/
public CMSStrikeIntegrant(final CapFloorCMS cmsCap, final SABRFormulaData sabrPoint, final double forward, final double cutOffStrike, final double mu) {
super(cmsCap, sabrPoint, forward, cutOffStrike, mu);
}
@Override
public Double evaluate(final Double x) {
final double[] kD = super.kpkpp(x);
// Implementation note: kD[0] contains the first derivative of k; kD[1] the second derivative of k.
return -kD[1] * bs(x);
}
} // End CMSStrikeIntegrant
}