/**
* Copyright (C) 2015 - present by OpenGamma Inc. and the OpenGamma group of companies
*
* Please see distribution for license.
*/
package com.opengamma.strata.pricer.impl.volatility.smile;
import java.io.Serializable;
import java.util.Set;
import org.joda.beans.BeanDefinition;
import org.joda.beans.ImmutableBean;
import org.joda.beans.JodaBeanUtils;
import org.joda.beans.MetaBean;
import org.joda.beans.Property;
import org.joda.beans.impl.light.LightMetaBean;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import com.google.common.math.DoubleMath;
import com.google.common.primitives.Doubles;
import com.opengamma.strata.basics.value.ValueDerivatives;
import com.opengamma.strata.collect.ArgChecker;
import com.opengamma.strata.collect.array.DoubleArray;
import com.opengamma.strata.pricer.model.SabrVolatilityFormula;
/**
* The Hagan SABR volatility function provider.
* <p>
* This class provides the functions of volatility and its sensitivity to the SABR model parameters based on the original
* Hagan SABR formula.
* <p>
* Reference: Hagan, P.; Kumar, D.; Lesniewski, A. & Woodward, D. "Managing smile risk", Wilmott Magazine, 2002, September, 84-108
* <p>
* OpenGamma documentation: SABR Implementation, OpenGamma documentation n. 33, April 2016.
*/
@BeanDefinition(style = "light")
public final class SabrHaganVolatilityFunctionProvider
extends VolatilityFunctionProvider<SabrFormulaData>
implements SabrVolatilityFormula, ImmutableBean, Serializable {
/**
* Default implementation.
*/
public static final SabrHaganVolatilityFunctionProvider DEFAULT = new SabrHaganVolatilityFunctionProvider();
private static final Logger log = LoggerFactory.getLogger(SabrHaganVolatilityFunctionProvider.class);
/* internal parameters */
private static final double CUTOFF_MONEYNESS = 1e-12;
private static final double SMALL_Z = 1e-6;
private static final double LARGE_NEG_Z = -1e6;
private static final double LARGE_POS_Z = 1e8;
private static final double BETA_EPS = 1e-8;
private static final double RHO_EPS = 1e-5;
private static final double RHO_EPS_NEGATIVE = 1e-8;
private static final double ATM_EPS = 1e-7;
private static final double MIN_VOL = 1e-6; // Minimal volatility, to avoid negative volatility for extreme parameters
//-------------------------------------------------------------------------
@Override
public double volatility(double forward, double strike, double timeToExpiry, SabrFormulaData data) {
ArgChecker.notNull(data, "data");
double alpha = data.getAlpha();
double beta = data.getBeta();
double rho = data.getRho();
double nu = data.getNu();
return volatility(forward, strike, timeToExpiry, alpha, beta, rho, nu);
}
@Override
public double volatility(
double forward,
double strike,
double timeToExpiry,
double alpha,
double beta,
double rho,
double nu) {
ArgChecker.isTrue(forward > 0.0, "forward must be greater than zero");
ArgChecker.isTrue(strike >= 0.0, "strike must be greater than zero");
ArgChecker.isTrue(timeToExpiry >= 0.0, "timeToExpiry must be greater than zero");
if (alpha == 0.0) {
return 0.0;
}
double cutoff = forward * CUTOFF_MONEYNESS;
double k;
if (strike < cutoff) {
log.info("Given strike of {} is less than cutoff at {}, therefore the strike is taken as {}", new Object[] {
strike, cutoff, cutoff});
k = cutoff;
} else {
k = strike;
}
double vol, z, zOverChi;
double beta1 = 1 - beta;
if (DoubleMath.fuzzyEquals(forward, k, ATM_EPS)) {
double f1 = Math.pow(forward, beta1);
vol = alpha *
(1 + timeToExpiry *
(beta1 * beta1 * alpha * alpha / 24 / f1 / f1 + rho * alpha * beta * nu / 4 / f1 + nu * nu *
(2 - 3 * rho * rho) / 24)) /
f1;
} else {
if (DoubleMath.fuzzyEquals(beta, 0, BETA_EPS)) {
double ln = Math.log(forward / k);
z = nu * Math.sqrt(forward * k) * ln / alpha;
zOverChi = getZOverChi(rho, z);
vol = alpha * ln * zOverChi *
(1 + timeToExpiry * (alpha * alpha / forward / k + nu * nu * (2 - 3 * rho * rho)) / 24) / (forward - k);
} else if (DoubleMath.fuzzyEquals(beta, 1, BETA_EPS)) {
double ln = Math.log(forward / k);
z = nu * ln / alpha;
zOverChi = getZOverChi(rho, z);
vol = alpha * zOverChi * (1 + timeToExpiry * (rho * alpha * nu / 4 + nu * nu * (2 - 3 * rho * rho) / 24));
} else {
double ln = Math.log(forward / k);
double f1 = Math.pow(forward * k, beta1);
double f1Sqrt = Math.sqrt(f1);
double lnBetaSq = Math.pow(beta1 * ln, 2);
z = nu * f1Sqrt * ln / alpha;
zOverChi = getZOverChi(rho, z);
double first = alpha / (f1Sqrt * (1 + lnBetaSq / 24 + lnBetaSq * lnBetaSq / 1920));
double second = zOverChi;
double third = 1 + timeToExpiry * (beta1 * beta1 * alpha * alpha / 24 / f1 +
rho * nu * beta * alpha / 4 / f1Sqrt + nu * nu * (2 - 3 * rho * rho) / 24);
vol = first * second * third;
}
}
//There is nothing to prevent the nu * nu * (2 - 3 * rho * rho) / 24 to be large negative, and hence the volatility negative
return Math.max(MIN_VOL, vol);
}
/**
* Computes the implied volatility in the SABR model and its derivatives.
* <p>
* The derivatives are stored in an array with:
* <ul>
* <li>[0] derivative with respect to the forward
* <li>[1] derivative with respect to the strike
* <li>[2] derivative with respect to the alpha
* <li>[3] derivative with respect to the beta
* <li>[4] derivative with respect to the rho
* <li>[5] derivative with respect to the nu
* </ul>
*
* @param forward the forward value of the underlying
* @param strike the strike value of the option
* @param timeToExpiry the time to expiry of the option
* @param data the SABR data
* @return the volatility and associated derivatives
*/
@Override
public ValueDerivatives volatilityAdjoint(double forward, double strike, double timeToExpiry, SabrFormulaData data) {
ArgChecker.notNull(data, "data");
double alpha = data.getAlpha();
double beta = data.getBeta();
double rho = data.getRho();
double nu = data.getNu();
return volatilityAdjoint(forward, strike, timeToExpiry, alpha, beta, rho, nu);
}
/**
* Computes the implied volatility in the SABR model and its derivatives.
* <p>
* The derivatives are stored in an array with:
* <ul>
* <li>[0] derivative with respect to the forward
* <li>[1] derivative with respect to the strike
* <li>[2] derivative with respect to the alpha
* <li>[3] derivative with respect to the beta
* <li>[4] derivative with respect to the rho
* <li>[5] derivative with respect to the nu
* </ul>
*
* @param forward the forward value of the underlying
* @param strike the strike value of the option
* @param timeToExpiry the time to expiry of the option
* @param alpha the SABR alpha value
* @param beta the SABR beta value
* @param rho the SABR rho value
* @param nu the SABR nu value
* @return the volatility and associated derivatives
*/
@Override
public ValueDerivatives volatilityAdjoint(
double forward,
double strike,
double timeToExpiry,
double alpha,
double beta,
double rho,
double nu) {
ArgChecker.isTrue(forward > 0.0, "forward must be greater than zero");
ArgChecker.isTrue(strike >= 0.0, "strike must be greater than zero");
ArgChecker.isTrue(timeToExpiry >= 0.0, "timeToExpiry must be greater than zero");
double cutoff = forward * CUTOFF_MONEYNESS;
double k = strike;
if (k < cutoff) {
log.info(
"Given strike of {} is less than cutoff at {}, therefore the strike is taken as {}",
new Object[] {k, cutoff, cutoff});
k = cutoff;
}
double betaStar = 1 - beta;
double rhoStar = 1.0 - rho;
if (alpha == 0.0) {
double alphaBar;
if (DoubleMath.fuzzyEquals(forward, k, ATM_EPS)) { //TODO should this is relative
alphaBar = (1 + (2 - 3 * rho * rho) * nu * nu / 24 * timeToExpiry) / Math.pow(forward, betaStar);
} else {
//for non-atm options the alpha sensitivity at alpha = 0 is infinite. Returning this will most likely break calibrations,
// so we return an arbitrary large number
alphaBar = 1e7;
}
return ValueDerivatives.of(0d, DoubleArray.of(0, 0, alphaBar, 0, 0, 0));
}
// Implementation note: Forward sweep.
double sfK = Math.pow(forward * k, betaStar / 2);
double lnrfK = Math.log(forward / k);
double z = nu / alpha * sfK * lnrfK;
double rzxz;
double xz = 0;
if (DoubleMath.fuzzyEquals(z, 0.0, SMALL_Z)) {
rzxz = 1.0 - 0.5 * z * rho; //small z expansion to z^2 terms
} else {
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS)) {
if (z < 1.0) {
xz = -Math.log(1.0d - z);
rzxz = z / xz;
} else {
throw new IllegalArgumentException("can't handle z>=1, rho=1");
}
} else {
double arg;
if (z < LARGE_NEG_Z) {
arg = (rho * rho - 1) / 2 / z; //get rounding errors due to fine balanced cancellation for very large negative z
} else if (z > LARGE_POS_Z) {
arg = 2 * (z - rho);
} else {
arg = (Math.sqrt(1 - 2 * rho * z + z * z) + z - rho);
}
if (arg <= 0.0) { //Mathematically this cannot be less than zero, but you know what computers are like.
rzxz = 0.0;
} else {
xz = Math.log(arg / (1 - rho));
rzxz = z / xz;
}
}
}
double sf1 = sfK * (1 + betaStar * betaStar / 24 * (lnrfK * lnrfK) + Math.pow(betaStar, 4) / 1920 * Math.pow(lnrfK, 4));
double sf2 = (1 + (Math.pow(betaStar * alpha / sfK, 2) / 24 + (rho * beta * nu * alpha) /
(4 * sfK) + (2 - 3 * rho * rho) * nu * nu / 24) * timeToExpiry);
double volatility = Math.max(MIN_VOL, alpha / sf1 * rzxz * sf2);
// Implementation note: Backward sweep.
double vBar = 1;
double sf2Bar = alpha / sf1 * rzxz * vBar;
double sf1Bar = -alpha / (sf1 * sf1) * rzxz * sf2 * vBar;
double rzxzBar = alpha / sf1 * sf2 * vBar;
double zBar;
double xzBar = 0.0;
if (DoubleMath.fuzzyEquals(z, 0.0, SMALL_Z)) {
zBar = -rho / 2 * rzxzBar;
} else {
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS)) {
if (z < 1.0) {
xzBar = -z / (xz * xz) * rzxzBar;
zBar = 1.0d / xz * rzxzBar + 1.0d / (1.0d - z) * xzBar;
} else {
throw new IllegalArgumentException("can't handle z>=1, rho=1");
}
} else {
if (z < LARGE_NEG_Z) {
zBar = 1 / xz * rzxzBar + xzBar / (xz * xz) * rzxzBar;
} else if (z > LARGE_POS_Z) {
zBar = 1 / xz * rzxzBar - xzBar / (xz * xz) * rzxzBar;
} else {
xzBar = -z / (xz * xz) * rzxzBar;
zBar = 1 / xz * rzxzBar + 1 / ((Math.sqrt(1 - 2 * rho * z + z * z) + z - rho)) *
(0.5 * Math.pow(1 - 2 * rho * z + z * z, -0.5) * (-2 * rho + 2 * z) + 1) * xzBar;
}
}
}
double lnrfKBar = sfK * (betaStar * betaStar / 12 * lnrfK + Math.pow(betaStar, 4) / 1920 * 4 * Math.pow(lnrfK, 3)) * sf1Bar +
nu / alpha * sfK * zBar;
double sfKBar = nu / alpha * lnrfK * zBar + sf1 / sfK * sf1Bar -
(Math.pow(betaStar * alpha, 2) / Math.pow(sfK, 3) / 12 + (rho * beta * nu * alpha) / 4 / (sfK * sfK)) * timeToExpiry *
sf2Bar;
double strikeBar = -1 / k * lnrfKBar + betaStar * sfK / (2 * k) * sfKBar;
double forwardBar = 1 / forward * lnrfKBar + betaStar * sfK / (2 * forward) * sfKBar;
double nuBar = 1 / alpha * sfK * lnrfK * zBar +
((rho * beta * alpha) / (4 * sfK) + (2 - 3 * rho * rho) * nu / 12) * timeToExpiry * sf2Bar;
double rhoBar;
if (Math.abs(forward - k) < ATM_EPS) {
rhoBar = -z / 2 * rzxzBar;
} else {
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS)) {
if (z >= 1) {
if (rhoStar == 0.0) {
rhoBar = Double.NEGATIVE_INFINITY; //the derivative at rho = 1 is infinite - this sets it to some arbitrary large number
} else {
rhoBar = xzBar * (1.0 / rhoStar + (0.5 - z) / (z - 1.0) / (z - 1.0));
}
} else {
rhoBar = (0.5 * Math.pow(z / (1 - z), 2) + 0.25 * (z - 4.0) * Math.pow(z / (1.0 - z), 3) / (1.0 - z) *
rhoStar) * xzBar;
}
} else {
rhoBar = (1 / (Math.sqrt(1 - 2 * rho * z + z * z) + z - rho) *
(-Math.pow(1 - 2 * rho * z + z * z, -0.5) * z - 1) + 1 / rhoStar) * xzBar;
}
}
rhoBar += ((beta * nu * alpha) / (4 * sfK) - rho * nu * nu / 4) * timeToExpiry * sf2Bar;
double alphaBar = -nu / (alpha * alpha) * sfK * lnrfK * zBar +
((betaStar * alpha / sfK) * (betaStar / sfK) / 12 + (rho * beta * nu) / (4 * sfK)) * timeToExpiry * sf2Bar +
1 / sf1 * rzxz * sf2 * vBar;
double betaBar = -0.5 * Math.log(forward * k) * sfK * sfKBar - sfK *
(betaStar / 12 * (lnrfK * lnrfK) + Math.pow(betaStar, 3) / 480 * Math.pow(lnrfK, 4)) * sf1Bar +
(-betaStar * alpha * alpha / sfK / sfK / 12 + rho * nu * alpha / 4 / sfK) * timeToExpiry * sf2Bar;
return ValueDerivatives.of(volatility, DoubleArray.of(forwardBar, strikeBar, alphaBar, betaBar, rhoBar, nuBar));
}
/**
* Computes the first and second order derivatives of the Black implied volatility in the SABR model.
* <p>
* The first derivative values will be stored in the input array {@code volatilityD}
* The array contains, [0] Derivative w.r.t the forward, [1] the derivative w.r.t the strike, [2] the derivative w.r.t. to alpha,
* [3] the derivative w.r.t. to beta, [4] the derivative w.r.t. to rho, and [5] the derivative w.r.t. to nu.
* Thus the length of the array should be 6.
* <p>
* The second derivative values will be stored in the input array {@code volatilityD2}.
* Only the second order derivative with respect to the forward and strike are implemented.
* The array contains [0][0] forward-forward; [0][1] forward-strike; [1][1] strike-strike.
* Thus the size should be 2 x 2.
* <p>
* Around ATM, a first order expansion is used to due to some 0/0-type indetermination.
* The second order derivative produced is poor around ATM.
*
* @param forward the forward value of the underlying
* @param strike the strike value of the option
* @param timeToExpiry the time to expiry of the option
* @param data the SABR data.
* @param volatilityD the array used to return the first order derivative
* @param volatilityD2 the array of array used to return the second order derivative
* @return the Black implied volatility
*/
@Override
public double volatilityAdjoint2(
double forward,
double strike,
double timeToExpiry,
SabrFormulaData data,
double[] volatilityD,
double[][] volatilityD2) {
double k = Math.max(strike, 0.000001);
double alpha = data.getAlpha();
double beta = data.getBeta();
double rho = data.getRho();
double nu = data.getNu();
// Forward
double h0 = (1 - beta) / 2;
double h1 = forward * k;
double h1h0 = Math.pow(h1, h0);
double h12 = h1h0 * h1h0;
double h2 = Math.log(forward / k);
double h22 = h2 * h2;
double h23 = h22 * h2;
double h24 = h23 * h2;
double f1 = h1h0 * (1 + h0 * h0 / 6.0 * (h22 + h0 * h0 / 20.0 * h24));
double f2 = nu / alpha * h1h0 * h2;
double f3 = h0 * h0 / 6.0 * alpha * alpha / h12 + rho * beta * nu * alpha / 4.0 / h1h0 + (2 - 3 * rho * rho) /
24.0 * nu * nu;
double sqrtf2 = Math.sqrt(1 - 2 * rho * f2 + f2 * f2);
double f2x = 0.0;
double x = 0.0, xp = 0, xpp = 0;
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) {
f2x = 1.0 - 0.5 * f2 * rho; //small f2 expansion to f2^2 terms
} else {
if (DoubleMath.fuzzyEquals(rho, 1.0, RHO_EPS)) {
x = f2 < 1.0 ? -Math.log(1.0 - f2) - 0.5 * Math.pow(f2 / (f2 - 1.0), 2) * (1.0 - rho) : Math
.log(2.0 * f2 - 2.0) - Math.log(1.0 - rho);
} else {
x = Math.log((sqrtf2 + f2 - rho) / (1 - rho));
}
xp = 1. / sqrtf2;
xpp = (rho - f2) / Math.pow(sqrtf2, 3.0);
f2x = f2 / x;
}
double sigma = Math.max(MIN_VOL, alpha / f1 * f2x * (1 + f3 * timeToExpiry));
// First level
double h0Dbeta = -0.5;
double sigmaDf1 = -sigma / f1;
double sigmaDf2 = 0;
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) {
sigmaDf2 = alpha / f1 * (1 + f3 * timeToExpiry) * -0.5 * rho;
} else {
sigmaDf2 = alpha / f1 * (1 + f3 * timeToExpiry) * (1.0 / x - f2 * xp / (x * x));
}
double sigmaDf3 = alpha / f1 * f2x * timeToExpiry;
double sigmaDf4 = f2x / f1 * (1 + f3 * timeToExpiry);
double sigmaDx = -alpha / f1 * f2 / (x * x) * (1 + f3 * timeToExpiry);
double[][] sigmaD2ff = new double[3][3];
sigmaD2ff[0][0] = -sigmaDf1 / f1 + sigma / (f1 * f1); //OK
sigmaD2ff[0][1] = -sigmaDf2 / f1;
sigmaD2ff[0][2] = -sigmaDf3 / f1;
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) {
sigmaD2ff[1][2] = alpha / f1 * -0.5 * rho * timeToExpiry;
} else {
sigmaD2ff[1][1] = alpha / f1 * (1 + f3 * timeToExpiry) *
(-2 * xp / (x * x) - f2 * xpp / (x * x) + 2 * f2 * xp * xp / (x * x * x));
sigmaD2ff[1][2] = alpha / f1 * timeToExpiry * (1.0 / x - f2 * xp / (x * x));
}
sigmaD2ff[2][2] = 0.0;
// double sigma = alpha / f1 * f2x * (1 + f3 * theta);
// Second level
double[] f1Dh = new double[3];
double[] f2Dh = new double[3];
double[] f3Dh = new double[3];
f1Dh[0] = h1h0 * (h0 * (h22 / 3.0 + h0 * h0 / 40.0 * h24)) + Math.log(h1) * f1;
f1Dh[1] = h0 * f1 / h1;
f1Dh[2] = h1h0 * (h0 * h0 / 6.0 * (2.0 * h2 + h0 * h0 / 5.0 * h23));
f2Dh[0] = Math.log(h1) * f2;
f2Dh[1] = h0 * f2 / h1;
f2Dh[2] = nu / alpha * h1h0;
f3Dh[0] = h0 / 3.0 * alpha * alpha / h12 - 2 * h0 * h0 / 6.0 * alpha * alpha / h12 * Math.log(h1) - rho * beta *
nu * alpha / 4.0 / h1h0 * Math.log(h1);
f3Dh[1] = -2 * h0 * h0 / 6.0 * alpha * alpha / h12 * h0 / h1 - rho * beta * nu * alpha / 4.0 / h1h0 * h0 / h1;
f3Dh[2] = 0.0;
double[] f1Dp = new double[4]; // Derivative to sabr parameters
double[] f2Dp = new double[4];
double[] f3Dp = new double[4];
double[] f4Dp = new double[4];
f1Dp[0] = 0.0;
f1Dp[1] = f1Dh[0] * h0Dbeta;
f1Dp[2] = 0.0;
f1Dp[3] = 0.0;
f2Dp[0] = -f2 / alpha;
f2Dp[1] = f2Dh[0] * h0Dbeta;
f2Dp[2] = 0.0;
f2Dp[3] = h1h0 * h2 / alpha;
f3Dp[0] = h0 * h0 / 3.0 * alpha / h12 + rho * beta * nu / 4.0 / h1h0;
f3Dp[1] = rho * nu * alpha / 4.0 / h1h0 + f3Dh[0] * h0Dbeta;
f3Dp[2] = beta * nu * alpha / 4.0 / h1h0 - rho / 4.0 * nu * nu;
f3Dp[3] = rho * beta * alpha / 4.0 / h1h0 + (2 - 3 * rho * rho) / 12.0 * nu;
f4Dp[0] = 1.0;
f4Dp[1] = 0.0;
f4Dp[2] = 0.0;
f4Dp[3] = 0.0;
double sigmaDh1 = sigmaDf1 * f1Dh[1] + sigmaDf2 * f2Dh[1] + sigmaDf3 * f3Dh[1];
double sigmaDh2 = sigmaDf1 * f1Dh[2] + sigmaDf2 * f2Dh[2] + sigmaDf3 * f3Dh[2];
double[][] f1D2hh = new double[2][2]; // No h0
double[][] f2D2hh = new double[2][2];
double[][] f3D2hh = new double[2][2];
f1D2hh[0][0] = h0 * (h0 - 1) * f1 / (h1 * h1);
f1D2hh[0][1] = h0 * h1h0 / h1 * h0 * h0 / 6.0 * (2.0 * h2 + 4.0 * h0 * h0 / 20.0 * h23);
f1D2hh[1][1] = h1h0 * (h0 * h0 / 6.0 * (2.0 + 12.0 * h0 * h0 / 20.0 * h2));
f2D2hh[0][0] = h0 * (h0 - 1) * f2 / (h1 * h1);
f2D2hh[0][1] = nu / alpha * h0 * h1h0 / h1;
f2D2hh[1][1] = 0.0;
f3D2hh[0][0] = 2 * h0 * (2 * h0 + 1) * h0 * h0 / 6.0 * alpha * alpha / (h12 * h1 * h1) + h0 * (h0 + 1) * rho *
beta * nu * alpha / 4.0 / (h1h0 * h1 * h1);
f3D2hh[0][1] = 0.0;
f3D2hh[1][1] = 0.0;
double[][] sigmaD2hh = new double[2][2]; // No h0
for (int loopx = 0; loopx < 2; loopx++) {
for (int loopy = loopx; loopy < 2; loopy++) {
sigmaD2hh[loopx][loopy] = (sigmaD2ff[0][0] * f1Dh[loopy + 1] + sigmaD2ff[0][1] * f2Dh[loopy + 1] + sigmaD2ff[0][2] *
f3Dh[loopy + 1]) *
f1Dh[loopx + 1] +
sigmaDf1 *
f1D2hh[loopx][loopy] +
(sigmaD2ff[0][1] * f1Dh[loopy + 1] + sigmaD2ff[1][1] * f2Dh[loopy + 1] + sigmaD2ff[1][2] * f3Dh[loopy + 1]) *
f2Dh[loopx + 1] +
sigmaDf2 *
f2D2hh[loopx][loopy] +
(sigmaD2ff[0][2] * f1Dh[loopy + 1] + sigmaD2ff[1][2] * f2Dh[loopy + 1] + sigmaD2ff[2][2] * f3Dh[loopy + 1]) *
f3Dh[loopx + 1] +
sigmaDf3 * f3D2hh[loopx][loopy];
}
}
// Third level
double h1Df = k;
double h1Dk = forward;
double h1D2ff = 0.0;
double h1D2kf = 1.0;
double h1D2kk = 0.0;
double h2Df = 1.0 / forward;
double h2Dk = -1.0 / k;
double h2D2ff = -1 / (forward * forward);
double h2D2fk = 0.0;
double h2D2kk = 1.0 / (k * k);
volatilityD[0] = sigmaDh1 * h1Df + sigmaDh2 * h2Df;
volatilityD[1] = sigmaDh1 * h1Dk + sigmaDh2 * h2Dk;
volatilityD[2] = sigmaDf1 * f1Dp[0] + sigmaDf2 * f2Dp[0] + sigmaDf3 * f3Dp[0] + sigmaDf4 * f4Dp[0];
volatilityD[3] = sigmaDf1 * f1Dp[1] + sigmaDf2 * f2Dp[1] + sigmaDf3 * f3Dp[1] + sigmaDf4 * f4Dp[1];
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) {
volatilityD[4] = -0.5 * f2 + sigmaDf3 * f3Dp[2];
} else {
double xDr;
if (DoubleMath.fuzzyEquals(rho, 1.0, RHO_EPS)) {
xDr = f2 > 1.0 ? 1.0 / (1.0 - rho) + (0.5 - f2) / (f2 - 1.0) / (f2 - 1.0) : 0.5 *
Math.pow(f2 / (1.0 - f2), 2.0) + 0.25 * (f2 - 4.0) * Math.pow(f2 / (f2 - 1.0), 3) / (f2 - 1.0) *
(1.0 - rho);
if (Doubles.isFinite(xDr)) {
volatilityD[4] = sigmaDf1 * f1Dp[2] + sigmaDx * xDr + sigmaDf3 * f3Dp[2] + sigmaDf4 * f4Dp[2];
} else {
volatilityD[4] = Double.NEGATIVE_INFINITY;
}
} else {
xDr = (-f2 / sqrtf2 - 1 + (sqrtf2 + f2 - rho) / (1 - rho)) / (sqrtf2 + f2 - rho);
volatilityD[4] = sigmaDf1 * f1Dp[2] + sigmaDx * xDr + sigmaDf3 * f3Dp[2] + sigmaDf4 * f4Dp[2];
}
}
volatilityD[5] = sigmaDf1 * f1Dp[3] + sigmaDf2 * f2Dp[3] + sigmaDf3 * f3Dp[3] + sigmaDf4 * f4Dp[3];
volatilityD2[0][0] = (sigmaD2hh[0][0] * h1Df + sigmaD2hh[0][1] * h2Df) * h1Df + sigmaDh1 * h1D2ff +
(sigmaD2hh[0][1] * h1Df + sigmaD2hh[1][1] * h2Df) * h2Df + sigmaDh2 * h2D2ff;
volatilityD2[0][1] = (sigmaD2hh[0][0] * h1Dk + sigmaD2hh[0][1] * h2Dk) * h1Df + sigmaDh1 * h1D2kf +
(sigmaD2hh[0][1] * h1Dk + sigmaD2hh[1][1] * h2Dk) * h2Df + sigmaDh2 * h2D2fk;
volatilityD2[1][0] = volatilityD2[0][1];
volatilityD2[1][1] = (sigmaD2hh[0][0] * h1Dk + sigmaD2hh[0][1] * h2Dk) * h1Dk + sigmaDh1 * h1D2kk +
(sigmaD2hh[0][1] * h1Dk + sigmaD2hh[1][1] * h2Dk) * h2Dk + sigmaDh2 * h2D2kk;
return sigma;
}
private double getZOverChi(double rho, double z) {
// Implementation comment: To avoid numerical instability (0/0) around ATM the first order approximation is used.
if (DoubleMath.fuzzyEquals(z, 0.0, SMALL_Z)) {
return 1.0 - rho * z / 2.0;
}
double rhoStar = 1 - rho;
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS)) {
if (z < 1.0) {
return -z / Math.log(1.0d - z);
} else {
throw new IllegalArgumentException("can't handle z>=1, rho=1");
}
}
double rhoHat = 1 + rho;
if (DoubleMath.fuzzyEquals(rhoHat, 0.0, RHO_EPS_NEGATIVE)) {
if (z > -1) {
return z / Math.log(1 + z);
} else if (z < -1) {
if (rhoHat == 0) {
return 0.0;
}
double chi = Math.log(rhoHat) - Math.log(-(1 + z) / rhoStar);
return z / chi;
} else {
return 0.0;
}
}
double arg;
if (z < LARGE_NEG_Z) {
arg = (rho * rho - 1) / 2 / z; //get rounding errors due to fine balanced cancellation for very large negative z
} else if (z > LARGE_POS_Z) {
arg = 2 * (z - rho);
} else {
arg = (Math.sqrt(1 - 2 * rho * z + z * z) + z - rho);
//Mathematically this cannot be less than zero, but you know what computers are like.
if (arg <= 0.0) {
return 0.0;
}
}
double chi = Math.log(arg) - Math.log(rhoStar);
return z / chi;
}
@Override
public int hashCode() {
return toString().hashCode();
}
@Override
public boolean equals(Object obj) {
if (obj == null) {
return false;
}
if (this == obj) {
return true;
}
if (getClass() != obj.getClass()) {
return false;
}
return true;
}
@Override
public String toString() {
return "SABR (Hagan)";
}
//------------------------- AUTOGENERATED START -------------------------
///CLOVER:OFF
/**
* The meta-bean for {@code SabrHaganVolatilityFunctionProvider}.
*/
private static MetaBean META_BEAN = LightMetaBean.of(SabrHaganVolatilityFunctionProvider.class);
/**
* The meta-bean for {@code SabrHaganVolatilityFunctionProvider}.
* @return the meta-bean, not null
*/
public static MetaBean meta() {
return META_BEAN;
}
static {
JodaBeanUtils.registerMetaBean(META_BEAN);
}
/**
* The serialization version id.
*/
private static final long serialVersionUID = 1L;
private SabrHaganVolatilityFunctionProvider() {
}
@Override
public MetaBean metaBean() {
return META_BEAN;
}
@Override
public <R> Property<R> property(String propertyName) {
return metaBean().<R>metaProperty(propertyName).createProperty(this);
}
@Override
public Set<String> propertyNames() {
return metaBean().metaPropertyMap().keySet();
}
//-----------------------------------------------------------------------
///CLOVER:ON
//-------------------------- AUTOGENERATED END --------------------------
}