// -------------------------------------------------------------------------
@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);
}
@SuppressWarnings("null")
private double fdSensitivity(
EuropeanVanillaOption optionData,
double forward,
SabrFormulaData sabrData,
SabrParameter param,
double delta) {
Function<SabrFormulaData, Double> funcC = null;
Function<SabrFormulaData, Double> funcB = null;
Function<SabrFormulaData, Double> funcA = null;
SabrFormulaData dataC = null;
SabrFormulaData dataB = sabrData;
SabrFormulaData dataA = null;
Function<SabrFormulaData, Double> func = getVolatilityFunction(optionData, forward);
FiniteDifferenceType fdType = null;
switch (param) {
case Strike:
double strike = optionData.getStrike();
if (strike >= delta) {
fdType = FiniteDifferenceType.CENTRAL;
funcA = getVolatilityFunction(withStrike(optionData, strike - delta), forward);
funcC = getVolatilityFunction(withStrike(optionData, strike + delta), forward);
} else {
fdType = FiniteDifferenceType.FORWARD;
funcA = func;
funcB = getVolatilityFunction(withStrike(optionData, strike + delta), forward);
funcC = getVolatilityFunction(withStrike(optionData, strike + 2 * delta), forward);
}
dataC = sabrData;
dataB = sabrData;
dataA = sabrData;
break;
case Forward:
if (forward > delta) {
fdType = FiniteDifferenceType.CENTRAL;
funcA = getVolatilityFunction(optionData, forward - delta);
funcC = getVolatilityFunction(optionData, forward + delta);
} else {
fdType = FiniteDifferenceType.FORWARD;
funcA = func;
funcB = getVolatilityFunction(optionData, forward + delta);
funcC = getVolatilityFunction(optionData, forward + 2 * delta);
}
dataC = sabrData;
dataB = sabrData;
dataA = sabrData;
break;
case Alpha:
double a = sabrData.getAlpha();
if (a >= delta) {
fdType = FiniteDifferenceType.CENTRAL;
dataA = sabrData.withAlpha(a - delta);
dataC = sabrData.withAlpha(a + delta);
} else {
fdType = FiniteDifferenceType.FORWARD;
dataA = sabrData;
dataB = sabrData.withAlpha(a + delta);
dataC = sabrData.withAlpha(a + 2 * delta);
}
funcC = func;
funcB = func;
funcA = func;
break;
case Beta:
double b = sabrData.getBeta();
if (b >= delta) {
fdType = FiniteDifferenceType.CENTRAL;
dataA = sabrData.withBeta(b - delta);
dataC = sabrData.withBeta(b + delta);
} else {
fdType = FiniteDifferenceType.FORWARD;
dataA = sabrData;
dataB = sabrData.withBeta(b + delta);
dataC = sabrData.withBeta(b + 2 * delta);
}
funcC = func;
funcB = func;
funcA = func;
break;
case Nu:
double n = sabrData.getNu();
if (n >= delta) {
fdType = FiniteDifferenceType.CENTRAL;
dataA = sabrData.withNu(n - delta);
dataC = sabrData.withNu(n + delta);
} else {
fdType = FiniteDifferenceType.FORWARD;
dataA = sabrData;
dataB = sabrData.withNu(n + delta);
dataC = sabrData.withNu(n + 2 * delta);
}
funcC = func;
funcB = func;
funcA = func;
break;
case Rho:
double r = sabrData.getRho();
if ((r + 1) < delta) {
fdType = FiniteDifferenceType.FORWARD;
dataA = sabrData;
dataB = sabrData.withRho(r + delta);
dataC = sabrData.withRho(r + 2 * delta);
} else if ((1 - r) < delta) {
fdType = FiniteDifferenceType.BACKWARD;
dataA = sabrData.withRho(r - 2 * delta);
dataB = sabrData.withRho(r - delta);
dataC = sabrData;
} else {
fdType = FiniteDifferenceType.CENTRAL;
dataC = sabrData.withRho(r + delta);
dataA = sabrData.withRho(r - delta);
}
funcC = func;
funcB = func;
funcA = func;
break;
default:
throw new MathException("enum not found");
}
if (fdType != null) {
switch (fdType) {
case FORWARD:
return (-1.5 * funcA.apply(dataA) + 2.0 * funcB.apply(dataB) - 0.5 * funcC.apply(dataC))
/ delta;
case BACKWARD:
return (0.5 * funcA.apply(dataA) - 2.0 * funcB.apply(dataB) + 1.5 * funcC.apply(dataC))
/ delta;
case CENTRAL:
return (funcC.apply(dataC) - funcA.apply(dataA)) / 2.0 / delta;
default:
throw new MathException("enum not found");
}
}
throw new MathException("enum not found");
}
/**
* 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;
}
