// -------------------------------------------------------------------------
@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));
}
