@Test
/** Test the present value using the method with the direct formula with extrapolation. */
public void presentValueAboveCutOff() {
CurrencyAmount methodPrice = METHOD.presentValue(CAP_HIGH_LONG, SABR_BUNDLE);
final double df =
CURVES.getCurve(FUNDING_CURVE_NAME).getDiscountFactor(CAP_HIGH_LONG.getPaymentTime());
final double forward = CAP_HIGH_LONG.accept(PRC, CURVES);
final double maturity =
CAP_HIGH_LONG.getFixingPeriodEndTime() - CAP_LONG.getFixingPeriodStartTime();
final DoublesPair expiryMaturity = new DoublesPair(CAP_HIGH_LONG.getFixingTime(), maturity);
final double alpha = SABR_PARAMETERS.getAlpha(expiryMaturity);
final double beta = SABR_PARAMETERS.getBeta(expiryMaturity);
final double rho = SABR_PARAMETERS.getRho(expiryMaturity);
final double nu = SABR_PARAMETERS.getNu(expiryMaturity);
final SABRFormulaData sabrParam = new SABRFormulaData(alpha, beta, rho, nu);
final SABRExtrapolationRightFunction sabrExtrapolation =
new SABRExtrapolationRightFunction(
forward, sabrParam, CUT_OFF_STRIKE, CAP_HIGH_LONG.getFixingTime(), MU);
final EuropeanVanillaOption option =
new EuropeanVanillaOption(
CAP_HIGH_LONG.getStrike(), CAP_HIGH_LONG.getFixingTime(), CAP_HIGH_LONG.isCap());
final double expectedPrice =
sabrExtrapolation.price(option)
* CAP_HIGH_LONG.getNotional()
* CAP_HIGH_LONG.getPaymentYearFraction()
* df;
assertEquals(
"Cap/floor: SABR with extrapolation pricing", expectedPrice, methodPrice.getAmount(), 1E-2);
methodPrice = METHOD.presentValue(CAP_HIGH_LONG, SABR_BUNDLE);
assertEquals(
"Cap/floor: SABR with extrapolation pricing", expectedPrice, methodPrice.getAmount(), 1E-2);
}
/**
* 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 SABRInterestRateDataBundle sabrData) {
final SABRInterestRateParameters sabrParameter = sabrData.getSABRParameter();
final SwapFixedCoupon<? extends Payment> underlyingSwap = cmsCapFloor.getUnderlyingSwap();
final double forward = underlyingSwap.accept(PRC, sabrData);
final double discountFactor =
sabrData
.getCurve(underlyingSwap.getFixedLeg().getNthPayment(0).getFundingCurveName())
.getDiscountFactor(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);
}
/** Check C2 smoothness of Benaim-Dodgson-Kainth extrapolation */
@Test
public void BDKSmoothnessGeneralTest() {
double eps = 1.0e-5;
double expiry = 1.5;
double forward = 1.1;
int nStrikes = 10;
double[] strikes = new double[nStrikes];
double[] impliedVols =
new double[] {0.97, 0.92, 0.802, 0.745, 0.781, 0.812, 0.8334, 0.878, 0.899, 0.9252};
for (int i = 0; i < nStrikes; ++i) {
strikes[i] = forward * (0.85 + i * 0.05);
}
double muLow =
strikes[0]
* BlackFormulaRepository.dualDelta(forward, strikes[0], expiry, impliedVols[0], false)
/ BlackFormulaRepository.price(forward, strikes[0], expiry, impliedVols[0], false);
double muHigh =
-strikes[nStrikes - 1]
* BlackFormulaRepository.dualDelta(
forward, strikes[nStrikes - 1], expiry, impliedVols[nStrikes - 1], true)
/ BlackFormulaRepository.price(
forward, strikes[nStrikes - 1], expiry, impliedVols[nStrikes - 1], true);
SmileExtrapolationFunctionSABRProvider extrapBDK =
new BenaimDodgsonKainthExtrapolationFunctionProvider(muLow, muHigh);
SmileInterpolatorSABRWithExtrapolation interpBDK =
new SmileInterpolatorSABRWithExtrapolation(
new SABRBerestyckiVolatilityFunction(), extrapBDK);
InterpolatedSmileFunction funcBDK =
new InterpolatedSmileFunction(interpBDK, forward, strikes, expiry, impliedVols);
List<SABRFormulaData> modelParams =
(new SmileInterpolatorSABR())
.getFittedModelParameters(forward, strikes, expiry, impliedVols);
SABRExtrapolationLeftFunction sabrLeftExtrapolation =
new SABRExtrapolationLeftFunction(
forward,
modelParams.get(0),
strikes[0],
expiry,
muLow,
new SABRHaganVolatilityFunction());
SABRExtrapolationRightFunction sabrRightExtrapolation =
new SABRExtrapolationRightFunction(
forward,
modelParams.get(nStrikes - 3),
strikes[nStrikes - 1],
expiry,
muHigh,
new SABRHaganVolatilityFunction());
/*
* left interpolation
*/
{
// Checking underlying extrapolation
double boundaryValue =
sabrLeftExtrapolation.price(new EuropeanVanillaOption(strikes[0], expiry, false));
double CutoffUp = strikes[0] + eps;
double CutoffDw = strikes[0] - eps;
double optionPriceExt =
sabrLeftExtrapolation.price(new EuropeanVanillaOption(CutoffDw, expiry, false));
double optionPriceInt =
sabrLeftExtrapolation.price(new EuropeanVanillaOption(CutoffUp, expiry, false));
assertEquals(boundaryValue, optionPriceExt, eps);
assertEquals(boundaryValue, optionPriceInt, eps);
double optionPriceExtDw =
sabrLeftExtrapolation.price(new EuropeanVanillaOption(CutoffDw - eps, expiry, false));
double firstExt = (1.5 * boundaryValue + 0.5 * optionPriceExtDw - 2.0 * optionPriceExt) / eps;
double optionPriceIntUp =
sabrLeftExtrapolation.price(new EuropeanVanillaOption(CutoffUp + eps, expiry, false));
double firstInt = (2.0 * optionPriceInt - 0.5 * optionPriceIntUp - 1.5 * boundaryValue) / eps;
assertEquals(firstInt, firstExt, eps);
double secondExt = (boundaryValue + optionPriceExtDw - 2.0 * optionPriceExt) / eps / eps;
double secondInt = (optionPriceIntUp + boundaryValue - 2.0 * optionPriceInt) / eps / eps;
assertEquals(secondInt, secondExt, Math.abs(secondInt) * 1.0e-3);
// Checking volatility function
double volInt = funcBDK.getVolatility(CutoffUp);
double volExt = funcBDK.getVolatility(CutoffDw);
double volBoundary = funcBDK.getVolatility(strikes[0]);
assertEquals(volBoundary, volInt, eps);
assertEquals(volBoundary, volExt, eps);
double volExtDw = funcBDK.getVolatility(CutoffDw - eps);
double volFirstExt = (1.5 * volBoundary + 0.5 * volExtDw - 2.0 * volExt) / eps;
double volIntUp = funcBDK.getVolatility(CutoffUp + eps);
double volFirstInt = (2.0 * volInt - 0.5 * volIntUp - 1.5 * volBoundary) / eps;
assertEquals(volFirstInt, volFirstExt, eps);
double volSecondExt = (volBoundary + volExtDw - 2.0 * volExt) / eps / eps;
double volSecondInt = (volIntUp + volBoundary - 2.0 * volInt) / eps / eps;
assertEquals(volSecondInt, volSecondExt, Math.abs(volSecondInt) * 1.0e-3);
}
/*
* right interpolation
*/
{
// Checking underlying extrapolation
double boundaryValue =
sabrRightExtrapolation.price(
new EuropeanVanillaOption(strikes[nStrikes - 1], expiry, true));
double CutoffUp = strikes[nStrikes - 1] + eps;
double CutoffDw = strikes[nStrikes - 1] - eps;
double optionPriceExt =
sabrRightExtrapolation.price(new EuropeanVanillaOption(CutoffUp, expiry, true));
double optionPriceInt =
sabrRightExtrapolation.price(new EuropeanVanillaOption(CutoffDw, expiry, true));
assertEquals(boundaryValue, optionPriceExt, eps);
assertEquals(boundaryValue, optionPriceInt, eps);
double optionPriceExtUp =
sabrRightExtrapolation.price(new EuropeanVanillaOption(CutoffUp + eps, expiry, true));
double firstExt = (2.0 * optionPriceExt - 0.5 * optionPriceExtUp - 1.5 * boundaryValue) / eps;
double optionPriceIntDw =
sabrRightExtrapolation.price(new EuropeanVanillaOption(CutoffDw - eps, expiry, true));
double firstInt =
(-2.0 * optionPriceInt + 1.5 * boundaryValue + 0.5 * optionPriceIntDw) / eps;
assertEquals(firstInt, firstExt, eps);
double secondExt = (optionPriceExtUp + boundaryValue - 2.0 * optionPriceExt) / eps / eps;
double secondInt = (boundaryValue + optionPriceIntDw - 2.0 * optionPriceInt) / eps / eps;
assertEquals(secondInt, secondExt, Math.abs(secondInt) * 1.0e-3);
// Checking volatility function
double volInt = funcBDK.getVolatility(CutoffDw);
double volExt = funcBDK.getVolatility(CutoffUp);
double volBoundary = funcBDK.getVolatility(strikes[nStrikes - 1]);
assertEquals(volBoundary, volInt, eps);
assertEquals(volBoundary, volExt, eps);
double volExtUp = funcBDK.getVolatility(CutoffUp + eps);
double volFirstExt = (2.0 * volExt - 0.5 * volExtUp - 1.5 * volBoundary) / eps;
double volIntDw = funcBDK.getVolatility(CutoffDw - eps);
double volFirstInt = (-2.0 * volInt + 1.5 * volBoundary + 0.5 * volIntDw) / eps;
assertEquals(volFirstInt, volFirstExt, eps);
double volSecondExt = (volBoundary + volExtUp - 2.0 * volExt) / eps / eps;
double volSecondInt = (volIntDw + volBoundary - 2.0 * volInt) / eps / eps;
assertEquals(volSecondInt, volSecondExt, Math.abs(volSecondInt) * 1.0e-3);
}
}
/**
* 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);
}
/**
* 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 SABRInterestRateDataBundle sabrData) {
final SABRInterestRateParameters sabrParameter = sabrData.getSABRParameter();
final SwapFixedCoupon<? extends Payment> underlyingSwap = cmsCapFloor.getUnderlyingSwap();
final double forward = underlyingSwap.accept(PRC, sabrData);
final double discountFactorTp =
sabrData
.getCurve(underlyingSwap.getFixedLeg().getNthPayment(0).getFundingCurveName())
.getDiscountFactor(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;
}