/**
* Copyright (C) 2016 - 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 com.opengamma.strata.basics.value.ValueDerivatives;
import com.opengamma.strata.collect.ArgChecker;
import com.opengamma.strata.collect.array.DoubleArray;
/**
* 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 --------------------------
}