/**
* Surface Stochastic Volatility Inspired (SSVI) formula.
*
* <p>Reference: Gatheral, Jim and Jacquier, Antoine. Arbitrage-free SVI volatility surfaces.
* arXiv:1204.0646v4, 2013. Section 4.
*/
@BeanDefinition(style = "light")
public final class SsviVolatilityFunction extends VolatilityFunctionProvider<SsviFormulaData>
implements ImmutableBean, Serializable {
/** Default implementation. */
public static final SsviVolatilityFunction DEFAULT = new SsviVolatilityFunction();
/**
* SSVI volatility description diverge for theta -> 0. Lower bound for which time to expiry is
* accepted.
*/
public static final double MIN_TIME_TO_EXPIRY = 1.0E-3;
// -------------------------------------------------------------------------
@Override
public double volatility(
double forward, double strike, double timeToExpiry, SsviFormulaData data) {
ArgChecker.isTrue(
timeToExpiry > MIN_TIME_TO_EXPIRY,
"time to expiry must not be zero to be able to compute volatility");
double volatilityAtm = data.getSigma();
double rho = data.getRho();
double eta = data.getEta();
double theta = volatilityAtm * volatilityAtm * timeToExpiry;
double phi = eta / Math.sqrt(theta);
double k = Math.log(strike / forward);
double w =
0.5
* theta
* (1.0d + rho * phi * k + Math.sqrt(1.0d + 2 * rho * phi * k + phi * k * phi * k));
return Math.sqrt(w / timeToExpiry);
}
/**
* Computes the implied volatility in the SSVI formula 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 time to expiry
* <li>[3] derivative with respect to the sigma (ATM volatility)
* <li>[4] derivative with respect to the rho
* <li>[5] derivative with respect to the eta
* </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 SSVI data
* @return the volatility and associated derivatives
*/
@Override
public ValueDerivatives volatilityAdjoint(
double forward, double strike, double timeToExpiry, SsviFormulaData data) {
ArgChecker.isTrue(
timeToExpiry > MIN_TIME_TO_EXPIRY,
"time to expiry must not be zero to be able to compute volatility");
double volatilityAtm = data.getSigma();
double rho = data.getRho();
double eta = data.getEta();
double theta = volatilityAtm * volatilityAtm * timeToExpiry;
double stheta = Math.sqrt(theta);
double phi = eta / stheta;
double k = Math.log(strike / forward);
double s = Math.sqrt(1.0d + 2 * rho * phi * k + phi * k * phi * k);
double w = 0.5 * theta * (1.0d + rho * phi * k + s);
double volatility = Math.sqrt(w / timeToExpiry);
// Backward sweep.
double[] derivatives = new double[6]; // 6 inputs
double volatilityBar = 1.0;
double wBar = 0.5 * volatility / w * volatilityBar;
derivatives[2] += -0.5 * volatility / timeToExpiry * volatilityBar;
double thetaBar = w / theta * wBar;
derivatives[4] += 0.5 * theta * phi * k * wBar;
double phiBar = 0.5 * theta * rho * k * wBar;
double kBar = 0.5 * theta * rho * phi * wBar;
double sBar = 0.5 * theta * wBar;
derivatives[4] += phi * k / s * sBar;
phiBar += (rho * k + phi * k * k) / s * sBar;
kBar += (rho * phi + phi * phi * k) / s * sBar;
derivatives[1] += 1.0d / strike * kBar;
derivatives[0] += -1.0d / forward * kBar;
derivatives[5] += phiBar / stheta;
double sthetaBar = -eta / (stheta * stheta) * phiBar;
thetaBar += 0.5 / stheta * sthetaBar;
derivatives[3] += 2 * volatilityAtm * timeToExpiry * thetaBar;
derivatives[2] += volatilityAtm * volatilityAtm * thetaBar;
return ValueDerivatives.of(volatility, DoubleArray.ofUnsafe(derivatives));
}
@Override
public double volatilityAdjoint2(
double forward,
double strike,
double timeToExpiry,
SsviFormulaData data,
double[] volatilityD,
double[][] volatilityD2) {
throw new UnsupportedOperationException("Not implemented");
}
// ------------------------- AUTOGENERATED START -------------------------
/// CLOVER:OFF
/** The meta-bean for {@code SsviVolatilityFunction}. */
private static MetaBean META_BEAN = LightMetaBean.of(SsviVolatilityFunction.class);
/**
* The meta-bean for {@code SsviVolatilityFunction}.
*
* @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 SsviVolatilityFunction() {}
@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();
}
// -----------------------------------------------------------------------
@Override
public boolean equals(Object obj) {
if (obj == this) {
return true;
}
if (obj != null && obj.getClass() == this.getClass()) {
return true;
}
return false;
}
@Override
public int hashCode() {
int hash = getClass().hashCode();
return hash;
}
@Override
public String toString() {
StringBuilder buf = new StringBuilder(32);
buf.append("SsviVolatilityFunction{");
buf.append('}');
return buf.toString();
}
/// CLOVER:ON
// -------------------------- AUTOGENERATED END --------------------------
}
/**
* Mock address JavaBean, used for testing.
*
* @author Stephen Colebourne
*/
@BeanDefinition(style = "light")
public final class Light implements ImmutableBean, Serializable {
/** The number. */
@PropertyDefinition private final int number;
/** The number. */
@PropertyDefinition private final boolean flag;
/** The street. */
@PropertyDefinition(validate = "notNull", get = "field")
private final String street;
/** The town. */
@PropertyDefinition(get = "optionalGuava")
private final String town;
/** The city. */
@PropertyDefinition(validate = "notNull")
private final String city;
/** The owner. */
@PropertyDefinition(validate = "notNull")
private final ImmPerson owner;
/** The list. */
@PropertyDefinition(validate = "notNull")
private final ImmutableList<String> list;
/** The currency. */
@PropertyDefinition(get = "optionalGuava")
private final Currency currency;
// -----------------------------------------------------------------------
// manual getter with a different name
public String getStreetName() {
return street;
}
// ------------------------- AUTOGENERATED START -------------------------
/// CLOVER:OFF
/** The meta-bean for {@code Light}. */
private static MetaBean META_BEAN = LightMetaBean.of(Light.class);
/**
* The meta-bean for {@code Light}.
*
* @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 Light(
int number,
boolean flag,
String street,
String town,
String city,
ImmPerson owner,
List<String> list,
Currency currency) {
JodaBeanUtils.notNull(street, "street");
JodaBeanUtils.notNull(city, "city");
JodaBeanUtils.notNull(owner, "owner");
JodaBeanUtils.notNull(list, "list");
this.number = number;
this.flag = flag;
this.street = street;
this.town = town;
this.city = city;
this.owner = owner;
this.list = ImmutableList.copyOf(list);
this.currency = currency;
}
@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();
}
// -----------------------------------------------------------------------
/**
* Gets the number.
*
* @return the value of the property
*/
public int getNumber() {
return number;
}
// -----------------------------------------------------------------------
/**
* Gets the number.
*
* @return the value of the property
*/
public boolean isFlag() {
return flag;
}
// -----------------------------------------------------------------------
/**
* Gets the town.
*
* @return the optional value of the property, not null
*/
public Optional<String> getTown() {
return Optional.fromNullable(town);
}
// -----------------------------------------------------------------------
/**
* Gets the city.
*
* @return the value of the property, not null
*/
public String getCity() {
return city;
}
// -----------------------------------------------------------------------
/**
* Gets the owner.
*
* @return the value of the property, not null
*/
public ImmPerson getOwner() {
return owner;
}
// -----------------------------------------------------------------------
/**
* Gets the list.
*
* @return the value of the property, not null
*/
public ImmutableList<String> getList() {
return list;
}
// -----------------------------------------------------------------------
/**
* Gets the currency.
*
* @return the optional value of the property, not null
*/
public Optional<Currency> getCurrency() {
return Optional.fromNullable(currency);
}
// -----------------------------------------------------------------------
@Override
public boolean equals(Object obj) {
if (obj == this) {
return true;
}
if (obj != null && obj.getClass() == this.getClass()) {
Light other = (Light) obj;
return (number == other.number)
&& (flag == other.flag)
&& JodaBeanUtils.equal(street, other.street)
&& JodaBeanUtils.equal(town, other.town)
&& JodaBeanUtils.equal(city, other.city)
&& JodaBeanUtils.equal(owner, other.owner)
&& JodaBeanUtils.equal(list, other.list)
&& JodaBeanUtils.equal(currency, other.currency);
}
return false;
}
@Override
public int hashCode() {
int hash = getClass().hashCode();
hash = hash * 31 + JodaBeanUtils.hashCode(number);
hash = hash * 31 + JodaBeanUtils.hashCode(flag);
hash = hash * 31 + JodaBeanUtils.hashCode(street);
hash = hash * 31 + JodaBeanUtils.hashCode(town);
hash = hash * 31 + JodaBeanUtils.hashCode(city);
hash = hash * 31 + JodaBeanUtils.hashCode(owner);
hash = hash * 31 + JodaBeanUtils.hashCode(list);
hash = hash * 31 + JodaBeanUtils.hashCode(currency);
return hash;
}
@Override
public String toString() {
StringBuilder buf = new StringBuilder(288);
buf.append("Light{");
buf.append("number").append('=').append(number).append(',').append(' ');
buf.append("flag").append('=').append(flag).append(',').append(' ');
buf.append("street").append('=').append(street).append(',').append(' ');
buf.append("town").append('=').append(town).append(',').append(' ');
buf.append("city").append('=').append(city).append(',').append(' ');
buf.append("owner").append('=').append(owner).append(',').append(' ');
buf.append("list").append('=').append(list).append(',').append(' ');
buf.append("currency").append('=').append(JodaBeanUtils.toString(currency));
buf.append('}');
return buf.toString();
}
/// CLOVER:ON
// -------------------------- AUTOGENERATED END --------------------------
}
/**
* A one-dimensional interpolator. The interpolation is linear on x y^2. The interpolator is used
* for interpolation on integrated variance for options. All values of y must be positive.
*/
@BeanDefinition(style = "light", constructorScope = "public")
public final class TimeSquareInterpolator1D extends Interpolator1D
implements CurveInterpolator, ImmutableBean, Serializable {
/* Level below which the value is consider to be 0. */
private static final double EPS = 1.0E-10;
/** Interpolator name. */
private static final String NAME = "TimeSquare";
// -------------------------------------------------------------------------
@Override
public Double interpolate(Interpolator1DDataBundle data, Double value) {
JodaBeanUtils.notNull(value, "value");
JodaBeanUtils.notNull(data, "data");
ArgChecker.isTrue(value > 0, "Value should be stricly positive");
InterpolationBoundedValues boundedValues = data.getBoundedValues(value);
double x1 = boundedValues.getLowerBoundKey();
double y1 = boundedValues.getLowerBoundValue();
if (boundedValues.getLowerBoundIndex() == data.size() - 1) {
return y1;
}
double x2 = boundedValues.getHigherBoundKey();
double y2 = boundedValues.getHigherBoundValue();
double w = (x2 - value) / (x2 - x1);
double xy21 = x1 * y1 * y1;
double xy22 = x2 * y2 * y2;
double xy2 = w * xy21 + (1 - w) * xy22;
return Math.sqrt(xy2 / value);
}
@Override
public double firstDerivative(Interpolator1DDataBundle data, Double value) {
JodaBeanUtils.notNull(value, "value");
JodaBeanUtils.notNull(data, "data");
ArgChecker.isTrue(value > 0, "Value should be stricly positive");
int lowerIndex = data.getLowerBoundIndex(value);
int index;
if (lowerIndex == data.size() - 1) {
index = data.size() - 2;
} else {
index = lowerIndex;
}
double x1 = data.getKeys()[index];
double y1 = data.getValues()[index];
double x2 = data.getKeys()[index + 1];
double y2 = data.getValues()[index + 1];
if ((y1 < EPS) || (y2 < EPS)) {
throw new UnsupportedOperationException(
"node sensitivity not implemented when one node is 0 value");
}
double w = (x2 - value) / (x2 - x1);
double xy21 = x1 * y1 * y1;
double xy22 = x2 * y2 * y2;
double xy2 = w * xy21 + (1 - w) * xy22;
return 0.5
* (-Math.sqrt(xy2 / value) + (-xy21 + xy22) / (x2 - x1) / Math.sqrt(xy2 / value))
/ value;
}
@Override
public double[] getNodeSensitivitiesForValue(Interpolator1DDataBundle data, Double value) {
ArgChecker.notNull(value, "Value to be interpolated must not be null");
ArgChecker.notNull(data, "Data bundle must not be null");
int n = data.size();
double[] resultSensitivity = new double[n];
InterpolationBoundedValues boundedValues = data.getBoundedValues(value);
double x1 = boundedValues.getLowerBoundKey();
double y1 = boundedValues.getLowerBoundValue();
int index = boundedValues.getLowerBoundIndex();
if (index == n - 1) {
resultSensitivity[n - 1] = 1.0;
return resultSensitivity;
}
double x2 = boundedValues.getHigherBoundKey();
double y2 = boundedValues.getHigherBoundValue();
if ((y1 < EPS) || (y2 < EPS)) {
throw new UnsupportedOperationException(
"node sensitivity not implemented when one node is 0 value");
}
double w = (x2 - value) / (x2 - x1);
double xy21 = x1 * y1 * y1;
double xy22 = x2 * y2 * y2;
double xy2 = w * xy21 + (1 - w) * xy22;
double resultValue = Math.sqrt(xy2 / value);
double resultValueBar = 1.0;
double xy2Bar = 0.5 / resultValue / value * resultValueBar;
double xy21Bar = w * xy2Bar;
double xy22Bar = (1 - w) * xy2Bar;
double y2Bar = 2 * x2 * y2 * xy22Bar;
double y1Bar = 2 * x1 * y1 * xy21Bar;
resultSensitivity[index] = y1Bar;
resultSensitivity[index + 1] = y2Bar;
return resultSensitivity;
}
@Override
public Interpolator1DDataBundle getDataBundle(double[] x, double[] y) {
ArgChecker.notNull(y, "y");
int nY = y.length;
for (int i = 0; i < nY; ++i) {
ArgChecker.isTrue(y[i] >= 0.0, "All values in y must be positive");
}
return new ArrayInterpolator1DDataBundle(x, y);
}
@Override
public Interpolator1DDataBundle getDataBundleFromSortedArrays(double[] x, double[] y) {
ArgChecker.notNull(y, "y");
int nY = y.length;
for (int i = 0; i < nY; ++i) {
ArgChecker.isTrue(y[i] >= 0.0, "All values in y must be positive");
}
return new ArrayInterpolator1DDataBundle(x, y, true);
}
@Override
public String getName() {
return NAME;
}
// ------------------------- AUTOGENERATED START -------------------------
/// CLOVER:OFF
/** The meta-bean for {@code TimeSquareInterpolator1D}. */
private static MetaBean META_BEAN = LightMetaBean.of(TimeSquareInterpolator1D.class);
/**
* The meta-bean for {@code TimeSquareInterpolator1D}.
*
* @return the meta-bean, not null
*/
public static MetaBean meta() {
return META_BEAN;
}
/** The serialization version id. */
private static final long serialVersionUID = 1L;
/** Creates an instance. */
public TimeSquareInterpolator1D() {}
@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();
}
// -----------------------------------------------------------------------
@Override
public boolean equals(Object obj) {
if (obj == this) {
return true;
}
if (obj != null && obj.getClass() == this.getClass()) {
return true;
}
return false;
}
@Override
public int hashCode() {
int hash = getClass().hashCode();
return hash;
}
@Override
public String toString() {
StringBuilder buf = new StringBuilder(32);
buf.append("TimeSquareInterpolator1D{");
buf.append('}');
return buf.toString();
}
/// CLOVER:ON
// -------------------------- AUTOGENERATED END --------------------------
}
/** Dummy trade. Based on a FRA. */
@BeanDefinition(style = "light")
public final class DummyFraTrade implements Trade, ImmutableBean, Serializable {
@PropertyDefinition(validate = "notNull")
private final LocalDate date;
@PropertyDefinition private final double fixedRate;
public static DummyFraTrade of(LocalDate date, double fixedRate) {
return new DummyFraTrade(date, fixedRate);
}
// ------------------------- AUTOGENERATED START -------------------------
/// CLOVER:OFF
/** The meta-bean for {@code DummyFraTrade}. */
private static MetaBean META_BEAN = LightMetaBean.of(DummyFraTrade.class);
/**
* The meta-bean for {@code DummyFraTrade}.
*
* @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 DummyFraTrade(LocalDate date, double fixedRate) {
JodaBeanUtils.notNull(date, "date");
this.date = date;
this.fixedRate = fixedRate;
}
@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();
}
// -----------------------------------------------------------------------
/**
* Gets the date.
*
* @return the value of the property, not null
*/
public LocalDate getDate() {
return date;
}
// -----------------------------------------------------------------------
/**
* Gets the fixedRate.
*
* @return the value of the property
*/
public double getFixedRate() {
return fixedRate;
}
// -----------------------------------------------------------------------
@Override
public boolean equals(Object obj) {
if (obj == this) {
return true;
}
if (obj != null && obj.getClass() == this.getClass()) {
DummyFraTrade other = (DummyFraTrade) obj;
return JodaBeanUtils.equal(date, other.date)
&& JodaBeanUtils.equal(fixedRate, other.fixedRate);
}
return false;
}
@Override
public int hashCode() {
int hash = getClass().hashCode();
hash = hash * 31 + JodaBeanUtils.hashCode(date);
hash = hash * 31 + JodaBeanUtils.hashCode(fixedRate);
return hash;
}
@Override
public String toString() {
StringBuilder buf = new StringBuilder(96);
buf.append("DummyFraTrade{");
buf.append("date").append('=').append(date).append(',').append(' ');
buf.append("fixedRate").append('=').append(JodaBeanUtils.toString(fixedRate));
buf.append('}');
return buf.toString();
}
/// CLOVER:ON
// -------------------------- AUTOGENERATED END --------------------------
}
/**
* Extrapolator that does no extrapolation itself and delegates to the interpolator for all
* operations.
*
* <p>This reproduces the old behaviour in {@link CombinedInterpolatorExtrapolator} when the
* extrapolators were null. This extrapolator is used in place of a null extrapolator which allows
* the extrapolators to be non-null and makes for simpler and cleaner code where the extrapolators
* are used.
*/
@BeanDefinition(style = "light", constructorScope = "public")
public final class InterpolatorExtrapolator
implements CurveExtrapolator, Extrapolator1D, ImmutableBean, Serializable {
/** The interpolator name. */
public static final String NAME = "Interpolator";
// -------------------------------------------------------------------------
@Override
public String getName() {
return NAME;
}
@Override
public Double extrapolate(
Interpolator1DDataBundle data, Double value, Interpolator1D interpolator) {
JodaBeanUtils.notNull(data, "data");
return interpolator.interpolate(data, value);
}
@Override
public double firstDerivative(
Interpolator1DDataBundle data, Double value, Interpolator1D interpolator) {
JodaBeanUtils.notNull(data, "data");
return interpolator.firstDerivative(data, value);
}
@Override
public double[] getNodeSensitivitiesForValue(
Interpolator1DDataBundle data, Double value, Interpolator1D interpolator) {
JodaBeanUtils.notNull(data, "data");
return interpolator.getNodeSensitivitiesForValue(data, value);
}
// ------------------------- AUTOGENERATED START -------------------------
/// CLOVER:OFF
/** The meta-bean for {@code InterpolatorExtrapolator}. */
private static MetaBean META_BEAN = LightMetaBean.of(InterpolatorExtrapolator.class);
/**
* The meta-bean for {@code InterpolatorExtrapolator}.
*
* @return the meta-bean, not null
*/
public static MetaBean meta() {
return META_BEAN;
}
/** The serialization version id. */
private static final long serialVersionUID = 1L;
/** Creates an instance. */
public InterpolatorExtrapolator() {}
@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();
}
// -----------------------------------------------------------------------
@Override
public boolean equals(Object obj) {
if (obj == this) {
return true;
}
if (obj != null && obj.getClass() == this.getClass()) {
return true;
}
return false;
}
@Override
public int hashCode() {
int hash = getClass().hashCode();
return hash;
}
@Override
public String toString() {
StringBuilder buf = new StringBuilder(32);
buf.append("InterpolatorExtrapolator{");
buf.append('}');
return buf.toString();
}
/// CLOVER:ON
// -------------------------- AUTOGENERATED END --------------------------
}
/**
* 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();
/* 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) {
Logger s_logger = LoggerFactory.getLogger(SabrHaganVolatilityFunctionProvider.class);
s_logger.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) {
Logger s_logger = LoggerFactory.getLogger(SabrHaganVolatilityFunctionProvider.class);
s_logger.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 --------------------------
}